arXiv:2104.03268v1 [math.DG] 7 Apr 2021

arX

iv:2

104.

0326

8v1

[m

ath.

DG

] 7

Apr

202

1

GENERALIZED KÄHLER-RICCI FLOW ON TORIC FANO VARIETIES

VESTISLAV APOSTOLOV, JEFFREY STREETS, AND YURY USTINOVSKIY

Abstract. We study the generalized Kähler-Ricci flow with initial data of symplectic type, andshow that this condition is preserved. In the case of a Fano background with toric symmetry, weestablish global existence of the normalized flow. We derive an extension of Perelman’s entropyfunctional to this setting, which yields convergence of nonsingular solutions at infinity. Furthermore,we derive an extension of Mabuchi’s K-energy to this setting, which yields weak convergence of theflow.

1. Introduction

Generalized Kähler structures first appeared through investigations into supersymmetric sigmamodels [40], and were rediscovered in a purely mathematical context in the work of Gualtieri [47]and Hitchin [57]. They have recently attracted interest in both the physics and mathematical com-munities as natural generalizations of Kähler structures. We will focus here entirely on the so calledbiHermitian description of generalized Kähler geometry (cf. [5, 40]). Thus, a generalized Kählermanifold is a smooth manifold M with a triple (g, I, J) consisting of two integrable almost-complexstructures, I and J , together with a Riemannian metric g which is Hermitian with respect to both,such that the Kähler forms ωI and ωJ satisfy

dcIωI =H = −dc

JωJ , dH = 0,(1.1)

where the first equation defines H, and dcI =√−1(∂I − ∂I), dc

J =√−1(∂J − ∂J).

The generalized Kähler-Ricci flow (GKRF) was introduced by the second named author andTian in [76] as a tool for constructing canonical metrics and understanding existence and moduliproblems in generalized Kähler geometry, extending the deeply developed theory in Kähler geometry.Prior global existence and convergence results for GKRF have appeared in for instance [6, 69, 71].More specifically, in this paper we consider a normalized version of the flow, called normalizedgeneralized Kähler Ricci flow. This can be expressed as a flow of ddc

J -closed positive definite (1,1)-forms ωJ(t) on the complex manifold (M,J), together with a flow of I, as (cf. [76]).

∂

∂tωJ = − 2(ρBJ )

1,1 + 2λωJ ,

∂

∂tI = Lθ♯

I−θ♯

JI,

(1.2)

where λ is a fixed real constant and, for any Hermitian metric g on (M,J) defined by its Kähler

form ωJ , the expression (ρBJ )1,1

stands for the (1,1)-part with respect to J of the (closed) Bismut-

Ricci form ρBJ ∈ 2πc1(M,J) of (g, J). Notice that the RHS of (1.2) defines an Aeppli cohomology(1,1)-class on (M,J), and the term normalized refers to the special case when we can chooseλ ≠ 0 such that this class is trivial. It turns out [76] that as far as a solution to (1.2) exists,the triple (gt, It, J) define a generalized Kähler structure in the sense of (1.1). Notice that if theinitial generalized Kähler structure (g, I, J) was Kähler (i.e. d

cJωJ = −H = 0 and I = J in (1.1)),

the Bismut-Ricci form ρB is just the usual Kähler-Ricci form of the Kähler manifold (g, J), and(1.2) reduces to the normalized Kähler Ricci flow. In this setting Cao showed [16] smooth global

Date: June 14, 2021.

1

http://arxiv.org/abs/2104.03268v1

2 VESTISLAV APOSTOLOV, JEFFREY STREETS, AND YURY USTINOVSKIY

existence. The convergence at infinity is a very subtle issue, and many results have appeared inrecent years [20, 29, 66, 80, 81, 82, 83].

In this paper, we will study NGKRF in the case when (M,J) is a smooth toric Fano varietyof complex dimension m, assuming that the initial generalized Kähler structure (g, I, J) is com-patible with a symplectic form F ∈ 2πc1(M,J), and is invariant under the action of a maximalm-dimensional compact torus T in the automorphism group Aut(M,J) of (M,J). More precisely,

suppose that F is a symplectic 2-form on M which tames J , i.e. whose (1,1)-part ωJ ∶= (F)1,1Jis positive definite. We then define a Riemannian metric g, a 2-form b and an almost complexstructure I by

g ∶= −(FJ)sym, b ∶= −(FJ)skew, I ∶= −F−1J∗F.(1.3)

One can show (cf. §2) that when I is also integrable, (g, I, J) is a generalized Kähler structurewith H = db. Furthermore, in this case we say that (g, I, J) is an F -compatible generalized Kählerstructure, or a generalized Kähler structure of symplectic type, referring to the fact [49] that one ofthe corresponding generalized complex structures on TM ⊕ T ∗M is determined completely by thesymplectic 2-form F .

To better understand these structures, first recall that a key feature of generalized Kählergeometry, observed by Hitchin [58] (cf. [5, 67] for the 4-dimensional case) is that there are naturallyassociated holomorphic Poisson structures. More precisely, the tensors

σ = [I, J]g−1, σI = σ −√−1Iσ, σJ = σ −

√−1Jσ(1.4)

define respectively a real Poisson structure and holomorphic Poisson structures with respect to Iand J . Conversely, given a compact Kähler manifold (M,J, g,ω) and a non-trivial holomorphicPoisson structure µ ∈ H0(M,∧2(T 1,0M)), the are deformation results [59, 44, 46, 50] producingnon-Kähler generalized Kähler structures (gt, It, J) with holomorphic Poisson structure σJt = tµ for∣t∣ < ε. This takes a particularly nice and explicit form on a smooth (Kähler) toric variety (M,J,T).Indeed, if ω is a T-invariant Kähler form on (M,J), and A ∈ ∧2

t is a (constant) skew-symmetric2-vector of the Lie algebra t of T, L. Boulanger [15] constructs in an explicit way a T-invariantcomplex structure JA (which is T-equivariantly biholomorphic to J), such that (ω,JA) gives rise toa T-invariant ω-compatible generalized Kähler structure of symplectic type on M . We refer in thispaper to his construction as deformations of type A. More recently, Y. Wang [85, 86] has introducedanother explicit construction of toric generalized Kähler structures, which we call deformations oftype B, such that for a suitable (constant) skew-symmetric 2-vector form B ∈ ∧2

t, one replaces theinitial Kähler 2-form ω with a (T-equivariantly) symplectiomorphic 2-form FB taming JA, and suchthat (FB , JA) gives rise to T-invariant FB-compatible generalized Kähler structure of symplectictype on M . It follows from the results in [86] (cf. Proposition 4.5 below) that, up to pull-back by T-equivariant diffeomorphism, any T-invariant generalized Kähler structure (g, I, J) of symplectic typeon (M,J) with corresponding symplectic 2-form F , is obtained from a T-invariant Kähler structure(g, J) by deformations of type A and B. We furthermore observe in Proposition 4.6 a geometricinterpretation of A and B-type deformations in terms of the holomorphic Poisson structures.

Building upon this, we show in §2 that NGKRF interacts with the associated Poisson struc-tures in a predictable way, and thus is a natural tool for investigating questions on the global moduliof generalized Kähler structures on, as well as the underlying Poisson geometry of, Fano varieties.We formulate certain precise conjectures coming from the formal picture of NGKRF in §2. As afundamental first step in this direction, we establish the sharp global existence of the flow in thecase of toric symmetry.

Theorem 1.1. Let (M,J,T) be a smooth toric Fano variety and F ∈ 2πc1(M,J) a T-invariantsymplectic 2-form which defines a T-invariant generalized Kähler structure (g, I, J) of symplectictype on M . Then, the solution of normalized generalized Kähler Ricci flow (1.2) with this initialdata exists on [0,∞).

GENERALIZED KÄHLER-RICCI FLOW ON TORIC FANO VARIETIES 3

The key geometric observation behind Theorem 1.1 is that in the case of a Fano backgroundwith toric symmetry, the normalized generalized Kähler Ricci flow (1.2) reduces to a flow of T-invariant Kähler metrics ωt ∈ 2πc1(M,J). Using the Abreu–Guillemin reduction [2, 53], one canfurther describe the family of Kähler metrics ωt in terms of a family of strictly convex smoothfunctions φt on R

m, satisfying the scalar PDE

∂

∂tφ = log det((Hess φ)−1 +√−1e−2tB)−1 + 2φ,(1.5)

where B is a skew-symmetric matrix corresponding to the B-type deformation described above.Interestingly, this reduced scalar equation is independent of A-type deformations. Also, whenB = 0, it coincides with the PDE describing the reduction of the Kähler Ricci flow on a smoothtoric Fano variety, studied in [90]. Through a series of estimates based on the maximum principle,we derive a priori L∞ estimates for the associated metric in terms of a C0 bound for φ. Usingthe generalization of Evans-Krylov C2,α/Calabi-Yau C3 estimates for pluriclosed flow ([61, 70]), weobtain C∞ estimates for φ in terms of a C0 estimate for φ. Since φ grows at worst exponentiallyby the maximum principle, we thus conclude the long time existence.

One expects the convergence at infinity to be a delicate issue, and we prove two partial resultsin this direction. First we prove convergence of nonsingular solutions, without the toric symmetryhypothesis. To show this, we first observe that there are several points of view which show that theonly possible smooth limits of NGKRF on a Fano manifold are in fact Kähler-Ricci solitons, andit follows here using a Perelman-type entropy monotonicity. This entropy is closely related to theentropy formula for generalized Ricci flow (cf. [39] Ch. 6), with a further modification arising froma sharp differential inequality satisfied by the torsion potential b defined above. The monotonicityof this entropy shows that the only smooth self-similar limits of NGKRF are in fact Kähler-Riccisolitons, and also shows a uniform κ-noncollapsing result for NGKRF on Fano manifolds.

Theorem 1.2. Let (M,g, I, J) be a generalized Kähler structure of symplectic type with respect toF . Let (gt, It, J) denote the solution to normalized generalized Kähler-Ricci flow with this initialdata. Then

(1) The metric gt is uniformly κ-noncollapsed for all times the flow exists.(2) Suppose (M,J) is Fano, F ∈ 2πc1(M,J). Suppose the solution exists on [0,∞) with uni-

formly bounded curvature. Then any sequence of times tj→∞ admits a subsequence suchthat (gtj , Itj , J) converges in the C∞-Cheeger-Gromov topology to (gKRS , J∞, J∞), wheregKRS is a Kähler-Ricci soliton.

In the convergence statement above, the complex structure J∞ may not be biholomorphic to J ,and as is the case in Kähler-Ricci flow one expects this subtle issue to be related to K-stability of(M,J).

Returning to the setting of toric symmetry, we are able to prove a certain kind of weak, butunconditional, convergence. First, adapting ideas from general GIT theory, we are able to definean extension of the Mabuchi K-energy to toric generalized Kähler structures. Furthermore, weshow that this functional is monotone along NGKRF. Using this monotonicity and exploiting thevariational approach of [9], we obtain a weak convergence.

Theorem 1.3. Let (M,J,T) be a smooth toric Fano variety which admits a Kähler-Einstein metric,and F ∈ 2πc1(M,J) a T-invariant symplectic 2-form which defines a T-invariant generalized Kählerstructure (g, I, J) of symplectic type on M . Let (gt, It, J), t ∈ [0,∞) be the global solution of thenormalized generalized Kähler Ricci flow (1.2) with this initial data given by Theorem 1.1, and ωt

the corresponding family of toric Kähler metrics defined by (1.5). Then, there exists a sequence oftimes j → ∞ and automorphisms τj ∈ TC, such that τ∗j (ωj) converge, with respect to the distance

d1, to a positive (1,1)-current ω∞ of maximal Monge-Ampère mass and finite energy on (M,J). If


we assume furthermore that (M,J,T) is a symmetric toric Fano variety [7], the convergence holdswith τj = id.The precise definitions for the convergence are given in Section 6.4 below, but in the toric case weconsider it can be equivalently characterized [21, 51] in terms of the L1 convergence over the Delzantpolytope of (M,F,T) of the Legendre transforms uj of the convex functions φj satisfying (1.5), upto the addition to uj of affine-linear functions ℓj.

Here is an outline of the rest of the paper. In section 2 we recall fundamental propertiesof generalized Kähler structures and GKRF, focusing on the symplectic-type case. Next in §3we recall results on toric Kähler structures. In §4 we discuss toric generalized Kähler structures,unify the discussions of [15] and [85], [86], and explicitly describe the relevant associated Poissontensors. This leads to a derivation of the underlying Kähler metric and the associated scalarpotentials, and we derive the relationship of these scalar potentials to the general Hamiltonian flowconstructions of [5, 46, 11]. Also we are able to derive a potential function in this setting for theBismut-Ricci curvature. With these geometric results in place, we give the proof of Theorem 1.1in §5. We first derive the scalar reduction of the flow described above, then use this together withmaximum principle arguments to establish the required a priori estimates. In the final § 6, weobtain the monotonicity of the Perelman-type and Mabuchi-type functionals, and give the proofsof Theorems 1.2 and 1.3.

2. The generalized Kähler-Ricci flow in the symplectic-type case

In this section we record fundamental properties of the generalized Kähler-Ricci flow for gen-eralized Kähler structures of symplectic type. We first recall basic curvature identities in Hermitiangeometry relating different Ricci curvature tensors. Next we introduce symplectic-type GK struc-tures from a few different perspectives, and then derive identities for associated Ricci curvaturetensors, a crucial point being the derivation of a local scalar Ricci potential function in Proposition2.10. Using this we discuss the GKRF with symplectic-type initial data, showing that this conditionis preserved by the flow. We end the section by stating some natural conjectures for the GKRF onFano manifolds, as well as their implications.

2.1. Curvature identities for Hermitian metrics. In this subsection we recall the definitionsof the Bismut-Ricci and the Chern-Ricci forms on a Hermitian manifold (M,g,J), extending thenotion of the Kähler-Ricci form of a Kähler manifold.

Definition 2.1. Let (M,g,J) be a Hermitian manifold with fundamental 2-form ωJ = gJ . TheBismut connection is defined by

g(∇BXY,Z) = g(∇XY,Z) − 1

2dcωJ(X,Y,Z),(2.1)

where ∇ stands for the Riemannian connection of g.

Definition 2.2. Let (M,g,J) be a Hermitian manifold with fundamental 2-form ωJ . The Chernconnection is defined by

g(∇CXY,Z) = g(∇XY,Z) + 1

2dcωJ(X,JY,JZ).(2.2)

Both the Chern connection and Bismt connection are Hermitian connections, which necessarilyhave torsion in the non-Kähler case. We are interested in the associated 2-forms, ρB and ρC ,representing up to a scale with 2π the first Chern class of (M,J).


Definition 2.3. Let (M,g,J) be a complex manifold with a Hermitian metric g. The Bismut-Riccicurvature and Chern-Ricci curvature are defined by

ρB(X,Y ) = 12

2m

∑i=1RB(X,Y, ei, Jei),

ρC(X,Y ) = 12

2m

∑i=1RC(X,Y, ei, Jei),

where RB and RC are the curvature tensors associated to the Bismut [13] and Chern connectionsrespectively, and ei is an orthonormal basis for (TM,g). Furthermore we define the Bismut scalarcurvature as

sB =2m

∑i=1ρB(ei, Jei).

Using the formulas (2.1) and (2.2), one can check (see e.g. [41, Rem. 5]) that the inducedunitary connections ∇B and ∇C on the anti-canonical line bundle K−1(M,J) = ∧m(T 1,0M), withrespect to the induced Hermitian metric by g, satisfy

(2.3) ∇BX = ∇C

X + iθJ(JX),where

θJ ∶= ⟨ωJ ,dωJ⟩ = −ω−1J dωJ

is the Lee form of (g, J). It follows from (2.3) that the corresponding Ricci curvatures are relatedby

(2.4) ρB = ρC + d(J ♯θJ),where for a 1-form we have set (J ♯α)(X) ∶= −(J∗α)(X) = −α(JX) for the induced action of J onT ∗M , compatible with the g-duality between TM and T ∗M . Furthermore, as the ∇C induces theCauchy-Riemann operator ∂ on the canonical bundle Ω(m,0)(M,J), the Chern-Ricci form can be

computed from any (local) holomorphic section Θ of Ω(m,0)(M,J), via the formula

(2.5) ρC = −12dd

cJ log

⎛⎝ω[m]J

vΘ

⎞⎠ ,

where vΘ ∶= (√−1)m(Θ ∧Θ). Lastly we record an identity relating the Bismut-Ricci form and theRiemannian Ricci curvature. This follows from ([60], Proposition 3.1), taking in mind the signdifference for the torsion H.

ρB(X,JY ) = Ricg(X,Y ) − 12⟨(ıXH), (ıYH)⟩g + 1

2(Lθ♯g)(X,Y )

− 12(δgH)(X,Y ) + 1

2(dθ)(X,Y ) − 1

2(ıθ♯H)X,Y ,

(2.6)

2.2. Generalized Kähler structures of symplectic type.

Definition 2.4. Given a smooth manifold M , a generalized Kähler structure is a triple (g, I, J)consisting of a Riemannian metric g, and two integrable complex structures I and J compatiblewith g such that

dcIωI =H = −dc

JωJ , dH = 0.Definition 2.5. Given a smooth manifoldM , a generalized Kähler structure (g, I, J) is of symplectictype if

det(I + J) ≠ 0(2.7)


everywhere on M . In this setting we define

F = −2g(J + I)−1, b = −g(J + I)−1(I − J), F ♯ = Fg−1 = −2(J + I)−1.It follows [46] that F is non-degenerate and tames both J and I, and b ∈ ∧2,0+0,2

I∩∧2,0+0,2

J. Fur-

thermore, direct computations yield the basic relationship

−FJ = g + b, −FI = g − b.(2.8)

Lastly, we define

βI = π∧2,0I

b, βJ = π∧2,0J

b.

Several differential operators are relevant for these structures. In particular, we set

dcI = IdI, dcJ = JdJ, dF = F ♯dF ♯Using the integrability conditions for generalized Kähler structures it furthermore follows that (cf.[6, Lemma 2.7])

dF = 0, dcJωJ = −H = −db.

Remark 2.6. It turns out (cf. [36] Proposition 1) that the data above is equivalently described byfixing first a symplectic form F , then asking for an integrable complex structure I which is tamedby F , and furthermore the almost complex structure

J = −F−1IFis integrable. Given these conditions, the formulas in Definition 2.5 can be inverted to derive g andb. This in turn allows for an explicit derivation of the associated generalized complex structures (cf.[47, 48]), one of which is of ‘symplectic type,’ determined explicitly by the global closed pure spinorF .

Kähler metrics are naturally interpreted as symplectic-type generalized Kähler, setting I = J .Furthermore, given a hyperKähler structure (g, I, J,K), the triple (g, I, J) is GK of symplectic type,in fact is of nondegenerate type (cf. [69], [6]). Hamiltonian deformations of these structures areexamples of non-Kähler symplectic-type GK structures ([5]), as are Hitchin’s Poisson deformations ofKähler metrics on Fano surfaces [59]. We further record here some fundamental identities establishedin [6]:

Lemma 2.7. Let (g, I, J) be a generalized Kähler structure of symplectic type and denote by θJ =JδgωJ and θI ∶= IδgωI the Lee forms. Then we have

1

2d log det(I + J) = −(b db) = θJ + ıθ♯

Jb + δgb = θI − ıθ♯

Ib − δgb.

Proof. This follows directly from [6] Lemmas 2.10 and 2.13.

Using this identity, we further identify a key formula relating certian naturally associatedLaplace operators.

Lemma 2.8. Let (g, I, J) be a generalized Kähler structure of symplectic type. Then, for any smoothfunction f ,

(ddF f) ∧ F [m−1]F [m]

= 1

2[∆gf − 1

2⟨df,d log det(I + J)⟩

g] .


Proof. In the computation below, the upper-script ♯ denotes g−1, and we use that F = −2g(I +J)−1,b = −g(J + I)−1(I − J), θ♯

I= IδgI, θ♯

J= JδgJ to obtain

2(ddF f) ∧ F [m−1]

F [m]=

2m

∑i=1[⟨(I + J) (∇ei(I + J)−1df ♯) , ei⟩

g]

= [∆gf −2m

∑i=1⟨ (∇ei(I + J)) ((I + J)−1(df ♯)) , ei⟩

g]

= [∆gf − ⟨(I + J)−1(df ♯), δg(I + J)⟩g]= [∆gf − ⟨df ♯, (I + J)−1(Iθ♯I + Jθ♯J)⟩g]= [∆gf + 1

2⟨df, b(θ♯I − θ♯J) − θI − θJ⟩g] .

The identity then follows from Lemma 2.7.

2.3. The Bismut-Ricci forms for symplectic-type GK structures. Observe that in the con-text of generalized Kähler geometry we have two Bismut connections, defined via

g(∇B,IX Y,Z) = g(∇XY,Z) − 1

2dcIωI(X,Y,Z) = g(∇XY,Z) − 1

2H(X,Y,Z),

g(∇B,JX Y,Z) = g(∇XY,Z) − 1

2dcJωJ(X,Y,Z) = g(∇XY,Z) + 1

2H(X,Y,Z).

For a generalized Kähler structure (g, I, J), we denote by ρCJ , ρCI and ρBJ , ρ

BI the Chern and Bismut-

Ricci forms of the Hermitian structure (g, J) and (g, I), respectively. We next derive a relationshipbetween the two associated Bismut-Ricci forms for GK structures of symplectic type.

Proposition 2.9. Let (g, I, J) be a generalized Kähler structure of symplectic type. Then

ρBJ (X,Y ) − ρBI (X,Y ) = 12(Lθ♯

J−θ♯

IF )(X,Y ).

Proof. An important identity [39, Lemma 9.27] reads as

(2.9) g((Lθ♯I−θ♯

JI)X,Y ) = ρBJ (X, [J, I]Y ),

where the upper script ♯ stands for the bundle map g−1 ∶ T ∗M → TM . Using this together with theanalogous identity involving Lθ♯

I−θ♯

JJ we have

g((Lθ♯I−θ♯

JI)X,Y ) = ρBJ (X, [J, I]Y ),

g((Lθ♯I−θ♯

JJ)X,Y ) = ρBI (X, [J, I]Y ).

Next, using (2.6), we find

(Lθ♯I−θ♯

Jg) (X,Y ) = ρBI (X,IY ) + ρBI (Y, IX) − ρBJ (X,JY ) − ρBJ (Y,JX).

Combining the above for F = −2g(I + J)−1, we find:12(Lθ♯

I−θ♯

JF )(X,Y )

= − Lθ♯I−θ♯

Jg((I + J)−1X,Y ) + g((I + J)−1Lθ♯

I−θ♯

J(I + J)(I + J)−1X,Y )

= ρBJ ((I + J)−1X,JY ) + ρBJ (Y,J(I + J)−1X) − ρBI ((I + J)−1X,IY ) − ρBJ (Y, I(I + J)−1X)− ρBJ ((I + J)−1X, [J, I](I + J)−1Y ) − ρBI ((I + J)−1X, [J, I](I + J)−1Y )

= ρBJ ((I + J)−1X,IY ) + ρBJ (Y,J(I + J)−1X) − ρBI ((I + J)−1X,JY ) − ρBJ (Y, I(I + J)−1X)= − ρBJ (I(I + J)−1X,Y ) − ρBJ (J(I + J)−1X,Y ) + ρBI (J(I + J)−1X,Y ) + ρBJ (I(I + J)−1X,Y )= − ρBJ (X,Y ) + ρBI (X,Y ),


where we used that [J, I](I +J)−1 = (J − I) and that ρBJ (resp. ρBI ) is I invariant (resp.J-invariant),see [39, Lemma 9.26].

Proposition 2.10. Let (g, I, J) be a generalized Kähler structure of symplectic type and vΘ =(√−1)mΘ∧Θ a local holomorphic volume form of (M,J), associated to a local non-vanishing sectionΘ of the canonical line bundle Ωm,0(M,J). Then the smooth function

ΦJ,Θ ∶= −12

⎡⎢⎢⎢⎢⎣log (F [m]

vΘ) + log⎛⎝

F [m]

ω[m]J

⎞⎠⎤⎥⎥⎥⎥⎦

is a potential for ρBI , i.e.

ρBI = ddcJΦJ,Θ.

Proof. Using (2.5) and F [m]/ω[m]J = (2m)det(I + J)− 1

2 , we get

ddcJΦJ,Θ = ρCJ + 1

2dd

cJ log det(I + J)

= ρCJ + 12d (J(θJ + θI) + Jb(θ♯J − θ♯I))

= ρCJ + d(JθJ) + 12d(F (θ♯I − θ♯J))

= ρBJ + 12Lθ♯

I−θ♯

JF

= ρBI ,where for passing from the first to the second line we have used Lemma 2.7, for passing from thesecond to the third line we have used (2.8), and for passing from the third to the fourth line wehave used Proposition 2.9 and (2.4).

2.4. The generalized Kähler-Ricci flow. The generalized Kähler-Ricci flow in the J-fixed gaugeis a solution (gt, It, J) of the geometric equation

∂

∂tωJ = − 2(ρBJ )1,1

J,

∂

∂tI = Lθ♯

I−θ♯

JI.

(2.10)

We refer the reader to [73, 74, 76, 39] for the basic properties of the above equation. Our first goal isto show that the condition of being symplectic type is preserved. As generalized Kähler structuresof symplectic type are quite conveniently described in terms of F and b, we first recast the GKRFin terms of F and b, then use this to show that the symplectic-type condition (2.7) holds for theentire smooth existence time of the flow.

Lemma 2.11. Let (gt, It, J) denote a solution to generalized Kähler-Ricci flow (2.10) which issymplectic-type for all times. Then the associated one-parameter families of two-forms satisfy

(2.11)∂

∂tFt = −2ρB(ωt), ∂

∂tbt =∆gtbt − Lθ♯tbt.

Proof. By (2.6), using that H = db and Lemma 2.7, we obtain

(−2JρB)skew = − δgH + dθ − ıθ♯H =∆gb − Lθ♯b.(2.12)

As π∧

1,1J

F = ωJ and (−FJ)skew = b, equation (2.12) shows that the two equalities in the lemma

are equivalent. Using the identity (2.9) and equations (2.10) we compute below the evolution of


F = −2g(I + J)−1:∂

∂tF (X,Y )= 2( ∂

∂tg) (X, (I + J)−1Y ) + 2g (X, ∂

∂t(I + J)−1Y )

= 2(ρB(JX, (I + J)−1Y ) − ρB(X,J(I + J)−1Y )) − 2g (X, (I + J)−1(Lθ♯I−θ♯

JI)(I + J)−1Y )

= 2(ρB(JX, (I + J)−1Y ) − ρB(X,J(I + J)−1Y )) + 2g ((I + J)−1X, (Lθ♯I−θ♯

JI)(I + J)−1Y )

= 2(ρB(JX, (I + J)−1Y ) − ρB(X,J(I + J)−1Y )) − 2ρB([J, I](I + J)−1´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶J−I

X, (I + J)−1Y )= 2(ρB(IX, (I + J)−1Y ) − ρB(X,J(I + J)−1Y ))= − 2(ρB(X,J(I + J)−1Y ) + ρB(X,I(I + J)−1Y ))= − 2ρB(X,Y ),

where for passing to the last line we have used that ρBJ is I-invariant, see [39, Lemma 9.26].

Proposition 2.12. Let (gt, It, J) denote a solution to generalized Kähler-Ricci flow such that(g0, I0, J) is symplectic-type. Then (gt, It, J) is symplectic-type for all t for which the flow is defined.

Proof. If the maximal interval on which the data is symplectic type is [0, T ), T < ∞, but thesolution exists smoothly on [0, T ]. By integrating the evolution equations over [0, T ] it follows thatthe family of endomorphisms J + It extends smoothly across time T . Using the evolution equationfor F from Lemma 2.11 and integrating in time we also conclude that

(J + It)−1 = −12g−1t Ft

extends smoothly across time T . It follows directly that J + IT is well-defined and invertible,contradicting maximality of T .

2.5. The normalized generalized Kähler Ricci flow in the Fano case. As explained in theintroduction, in the special case when the symplectic form F of a symplectic type generalized Kählerstructure (g, I, J) belongs to 2πc1(M,J), we can consider the normalized GKRF

∂

∂tωJ = − 2(ρBJ )1,1

J+ 2ωJ ,

∂

∂tI = Lθ♯

I−θ♯

JI.

(2.13)

To get a sense of the possible global existence behavior of this in the Fano case, we exhibit thebehavior of some basic quantities.

Proposition 2.13. Let (M2n, J) be a Fano manifold, and suppose (gt, It, J) is a solution of (2.13)of symplectic type such that [F0] = 2πc1(M,J). Then for all t, the cohomology class [Ft] and thePoisson tensor σt = [It, J]g−1t (see 1.4) satisfy

(1) [Ft] = 2πc1(M,J),(2) σt = e−2tσ0.

Proof. The solutions (gt, It) and (gt, It) of (2.10) and (2.13) are related by a reparametrisation

gt ∶= 12e2tg(1−e−2t), It = I(1−e2t).

It follows directly from Lemma 2.11 that

∂

∂tF = −2ρBJ (g) + 2F ,


proving claim (1). To see claim (2), we first note that along a solution of (2.10) one has that σ isfixed in time ([43, Corollary 1.5] cf. also [39, Proposition 9.29]). Claim (2) then follows easily fromthe reparameterization relationships above.

Thus the symplectic form F remains in the canonical class, while the tensor σ, which measuresthe deviation of a symplectic-type GK structure from being Kähler, decays exponentially. Based onthis, as well as the monotonicity formulae described below in §6, one might expect the normalizedGKRF (2.13) to behave similarly to the much studied normalized Kähler-Ricci flow on Fano mani-folds. In particular, by a result of Cao [16] the solution of the normalized Kähler-Ricci flow existsfor all time, and, by results of Tian-Zhu [82, 83], the global solution of the normalized Kähler-Ricciflow composed by automorphisms of (M,J) converges to a Kähler-Ricci soliton, provided that thelatter exists. Recall that a Kähler-Ricci soliton is a Kähler metric ωs ∈ 2πc1(M,J) satisfying

(2.14) ρ(ωs) − ωs = −12LJKωs

for a (uniquely determined) Killing vector field K which is zero precisely when (M,J) admits aKähler-Einstein metric. It is natural to ask whether or not similar results hold true for solutions of(2.13). More precisely we have:

Conjecture 2.14. Suppose (g, I, J) is a symplectic type generalized Kähler structure on a Fanomanifold (M,J), with F ∈ 2πc1(M,J). Then

The solution of (2.13) with this initial data exists for all time t ∈ [0,∞). If (M,J) admits a Kähler-Einstein metric then the global solution of (2.13) converges, in

the C∞(M) topology, to a Kähler-Einstein metric. If (M,J) admits a Kähler-Ricci soliton, with a soliton vector field K, and (g, I, J) is in-

variant by the torus action generated by K, then there exists a smooth family of complexautomorphisms τt of (M,J) such that the global solution gt of (2.13) pulled-back by τt con-verges, in the C∞(M) topology, to a Kähler-Ricci soliton metric in c1(M,J) with solitonvector field K.

Remark 2.15. One can further specify in the last part of Conjecture 2.14 that the family τt =exp(−tJK) is given by the flow of the vector field −JK in (2.14). In this case, as one has that theholomorphic Poisson structure σJ(t) satisfies σJ(t) = e−2tσJ(0) by Proposition 2.13, the convergencestatement in Conjecture 2.14 places the necessary condition

(2.15) λ > −2,for the (real) weight λ of the action of JK on any equivariant component s of σJ(0), i.e. assumingthat LJKs = λs.

One can show that (2.15) is, in fact, a necessary condition for the existence of a Kähler-Riccisoliton on (M,J), reminiscent to Gauntlett-Martelli-Sparks-Yau’s “Lichnerowicz obstruction” [42]

to the existence of Sasaki-Einstein structures. Indeed, if s ∈ H0(M,∧2T1,0J M) is any non-zero

holomorphic section with LJKs = λs,λ ∈ R, Bochner’s formula and (2.14) yield

(∆ωs + LJK) ∣∣s∣∣2ωs≤ 2(2 + λ)∣∣s∣∣2ωs

,

where the Laplace operator and the norm are taken with respect to thew Kähler Ricci solitonmetric ωs. This, combined with the generalized Lichnerowicz inequality λ1 ≥ 2 for the first non-zeroeigenvalue λ1 of the elliptic operator (∆ωs +LJK) (see [38, Cor. 2.4.4]) yields (2.15). It would beinteresting to relate the above “Lichnerowicz obstruction” on the soliton vector field K with theweighted K-stability condition on (M,J,K) [8, 25] corresponding to the existence of a Kähler-Riccisoliton.


3. Toric Kähler structures

In this section we recall fundamental material on toric Kähler structures. This is standardmaterial going back to [2, 53]; we refer to [3, 34] for comprehensive surveys.

3.1. Symplectic potentials and Legendre transformation. Let (M,ω0) be a smooth compactsymplectic manifold of real dimension 2m, endowed with an effective Hamiltonian action of a com-pact m-dimensional torus T. We denote by t the (abelian) Lie algebra of T, and by t

∗ the dualvector space. Let µ ∶M → t

∗ be a fixed momentum map for the action. Delzant’s theorem [26] tellsus that the image of µ(M) is a Delzant polytope P ⊂ t∗, which comes equipped with a minimal setof defining hyperplanes

P = x ∈ t∗ ∶ Lj(x) = ⟨vj , x⟩ + λj ≥ 0, j = 1, . . . , d,where vj are primitive elements of the lattice Λ ⊂ t of circle subgroups of T, i.e. T = t/2πΛ. Wedenote by L = L1(x), . . . ,Ld(x) the set of the defining affine-linear functions of P as above, whichsometimes is referred to as a labelling of P, see [64].

According to [26], the data (P,L) in turn identify (M,ω0), up to a T-equivariant symplec-tomorphism, with the Kähler reduction of C

d (endowed with its flat Kähler structure) by a realtorus of dimension d−m. This gives rise to a canonical ω0-compatible T-invariant Kähler structure(gc, Jc, ω0) on M .

We next describe the set KT(M,ω0) of T-invariant, ω0-compatible complex structures on(M,ω0), or equivalently, the set of T-invariant Kähler metrics (g, J,ω0) with fixed Kähler form ω0.

Notice that Jc ∈ KT(M,ω0) by construction. On the union M0 ∶= µ−1(P) of the generic orbits of

the T-action (where P denotes the interior of P), any Kähler metric g determined by an element Jin KT(M,ω0) has a general expression due to V. Guillemin [53]. In this description, the momentummap µ ∶ M0 → t

∗ is supplemented by angular coordinates (depending on J) θ ∶ M0 → t/2πΛ, suchthat the kernel of dθ is orthogonal to the tangent space of the torus orbits. These momentum-angular coordinates (µ, θ) identify each tangent space to M0 with t ⊕ t

∗, and the symplectic formis ω0 = ⟨dµ∧ dθ⟩, where ⟨⋅, ⋅⟩ denotes contraction of t and t

∗. Hence invariant ω0-compatible Kählermetrics on M0 have the form

(3.1) g = ⟨dµ,G(µ),dµ⟩ + ⟨dθ,H(µ),dθ⟩,where G(µ) is (the pull back by µ) of a positive definite S2

t-valued function on P (with S2t denoting

the symmetric tensor product of t), H is its point-wise inverse in S2t∗ (at each point, G and H

define mutually inverse linear maps t∗ → t and t→ t

∗) and ⟨⋅, ⋅, ⋅⟩ denotes the point-wise contractiont∗ × S2

t × t∗ → R or the dual contraction. The corresponding complex structure J is defined by

Jdθ = −⟨G(µ),dµ⟩, Jdµ = ⟨H(µ),dθ⟩,(3.2)

and J is integrable if and only if G = Hess(u) is the Hessian of a smooth strictly convex function

u(x) on P, called symplectic potential of (g, J,ω0) (cf. [2, 53]). This applies in particular to thecanonical complex structure Jc ∈ KT(M,ω0) coming from the Delzant construction, giving rise tocanonical angular coordinates θc and symplectic potential [53]

(3.3) uc(x) ∶= 1

2

d

∑j=1

Lj(x) logLj(x).Necessary and sufficient conditions for the symplectic potential u to come from a globally definedKähler metric (g, J) on M are obtained in [2, 4, 32]. Here we mention the conditions establishedin [32]:

Proposition 3.1. A smooth strictly convex function u(x) on P is a symplectic potential of a globallydefined ω0-compatible Kähler metric in KT(M,ω0) if and only if u satisfies the following boundaryconditions:


u(x) is smooth and strictly convex in the interior of P, and extends as a continuous functionon P which is smooth and strictly convex on the interior of each face of P; u(x) − uc(x) extends to a smooth function over P, where uc(x) is given by (3.3).

Definition 3.2. We denote by S(P,L) the space of functions u satisfying the conditions of Propo-sition 3.1.

A subtle point in the above description is the fact that the angular coordinates θ dependupon J , but one can show [4, Lemma 3] that by pulling back J by a T-equivariant symplectomor-phism of (M,ω0), we can assume that θ = θc, a normalization we shall implicitly use below withoutfurther notice. It follows that the space KT(M,ω0) modulo pull-backs by T-equivariant symplecto-morphisms can be identified with the space S(P,L), modulo additions with affine-linear functions.Because of this correspondence, for any u ∈ S(P,L), with a slight abuse of notation, we denote by(gu, Ju) the corresponding T-invariant Kähler metric on M (where we implicitly use the canonicalangular coordinates θc to define (gu, Ju)).

M. Abreu [1] obtained the expression of the Ricci form and scalar curvature of a Kählermetric (g, J) ∈ KT(M,ω0) in terms of the momentum/angular coordinates and the correspondingu ∈ S(P,L) as follows. Let us fix a basis of the Lie algebra t = Rm, which gives rise to a dual basisof t∗ = (Rm)∗. We will always take Z-bases of the lattice Λ. We denote by (x1, . . . , xm) the inducedlinear coordinates on t

∗, by (µ1, . . . , µm, θ1, . . . , θm) the induced momentum/angular coordinatesaround each point of M0, and by K1, . . . ,Km the fundamental vector fields corresponding to thebasis. We thus let

G(x) = Hess(u(x)) = (u,ij(x)), H(x) = ((G−1)ij(x)),(G−1)ij(µ) = gu(Ki,Kj) = u,ij(µ)(3.4)

where the lower indices following a comma stand for partial derivatives and the upper indicesfollowing a comma stand for the inverse of the Hessian. In this notation, we have on M0:

Kj = ∂

∂θj, ω0 =

m

∑j=1

dµj ∧ dθj, J = −m

∑i=1(Gij(µ)dµi ⊗ ∂

∂θj− (G−1)ij(µ)dθi ⊗ ∂

∂µj),(3.5)

whereas the Ricci form ρJ of (g, J) is given by

(3.6) ρJ = 1

2dd

cJ( log det (Hess(u))(µ)) = −1

2

m

∑i,j,k=1

(G−1)ij,ik(µ)dµk ∧ dθj.It is a basic fact of the theory (see e.g. [64]) that any two elements of KT(M,ω0) are biholo-

morphic under a T-equivariant diffeomorphism, which acts trivially on the cohomology class of [ω0],i.e. for any J ∈ KT(M,ω0), there exists a T-equivariant diffeomorphism Φ, such that Φ ⋅ J = Jc andΦ∗ω0 is the Kähler form of a T-invariant Kähler metric in the Kähler class [ω0] on the fixed complexmanifold (M,Jc). The converse is also true by the equivariant Moser lemma. These two equivalentdescriptions are often referred to as symplectic versus complex point of view, respectively, and willbe used in our study below. We thus give next an explicit description of this correspondence, basedon an observation from [53].

For any u ∈ S(P,L), we let

y = ∇u, φ(y) + u(x) = ⟨y,x⟩(3.7)

be the Legendre transform of the strictly convex smooth function u on P. Using the compactnessof P and the strict convexity of u, one can see that

y = ∇u ∶ P→ t ≅ Rm


is a diffeomorphism whereas φ(y) is a strictly convex smooth function on Rm. We denote by C(Rm)

the space of such functions on Rm and let

T ∶ S(P,L)↦ C(Rm), T (u(x)) = φ(y)be the induced map on the corresponding Frechét spaces. Notice that we can recover u(x) from itsimage φ(y) by letting x = (∇φ)(y) and using (3.7). A simple computation using the definition (3.7)shows that the differential of T is given by

(δuT )(u) = −φ.(3.8)

It turns out that the Legendre transform T underlines the geometric correspondence betweenthe symplectic and complex points of view mentioned above. To see this, for any J = Ju ∈ KT(M,ω0),we shall implicitly identify the abstract variable y = ∇u ∈ t with the function y ∶M → t given by thecomposition y = ∇u(µ), and use y(µ), θc as a system of coordinates on M0. It is easily seen from(3.5) and (3.7) that the 1-forms dy1, . . . ,dym,dθc1, . . . ,dθcm form a dual basis (at each point of M0)of the basis of commuting real holomorphic vector fields −JK1, . . . ,−JKm,K1, . . . ,Km. We can

thus consider two systems of holomorphic coordinates on (M0, J): y1 +√−1θc1, . . . , ym +√−1θcmand z1, . . . , zm with zj ∶= eyj+

√−1θcj . Geometrically, the action of T gives rise to a holomorphic

action of the complex torus TC ≅ (C∗)m on (M,J), which is the linear action in the coordinatesz1, . . . , zm on M0. In other words, letting xu ∈ P0 be the pre-image of 0 ∈ t under ∇u (or,equivalently, the point of minima of u) and pu the point in M0 such that µ(pu) = xu and θc(pu) = 0,we have the identification (M0, J) = (C∗)m ⋅pu with the orbit of TC at pu, with zj = eyj+

√−1θcj being

the natural complex coordinate on the j-th factor C∗ (reflecting the fact that zj(pu) = 1). The key

observation in [53] is that the Kähler form ω0 is written on (M0, J) as

(3.9) ω0 = ddcJφ(y(µ)) =

m

∑i,j=1

φ,pq(y)dyp ∧ dθcq,where φ = T (u) and, by the Legendre duality, (φ,pq(y)) = (uij(x))−1 =G−1(x) =H(x).

The above description holds on M0 in complex coordinates with respect to J = Ju. Torelate it to the canonical complex structure Jc of M , let (zc1, . . . , zcm) be the respective holomorphiccoordinates on (M0, Jc), obtained from the canonical potential uc(x) defined by (3.3). We letp0 ∶= pu0

be the corresponding point in M0 so that (M0, Jc) = (C∗)m ⋅ p0 with (zc1, . . . , zcm) beingthe standard coordinates on (C∗)m. Then, for any Ju corresponding to a u ∈ S0(P,L), we considerthe T-equivariant diffeomorphism Φu on M0 which maps the complex coordinates (z1, . . . , zm) withrespect to Ju to the complex coordinates (zc1, . . . , zcm) with respect to Jc = Juc . This gives rise toa (C∗)m-equivariant biholomorphism Φu ∶ (M0, Ju) → (M0, Jc). The boundary conditions for uformulated in Proposition 3.1 yield (see the proof of Lemma 4.3 below) that Φu extends to M asa (C∗)m-equivariant biholomorphism between (M,Ju) and (M,Jc). By (3.9), Φu sends the initialsymplectic form ω0 to the (1,1)-form on (M0, Jc)

ωφ ∶= ddcJcφ(yc),

where, we recall, (yc1, . . . , ycm) = (12 log ∣zc1∣2, . . . , 12 log ∣zcm∣2) and φ = T (u). By construction, ωφ

extends as a positive definite (1,1)-form on (M,Jc). Using the boundary conditions for u, one canalso check (see e.g. [33, pp. 395-396]) that the difference φc(yc)−φ(yc) is a smooth function on M ,where φc(y) = T (uc) is the Legendre transform of the canonical potential uc. It follows that thecorresponding Kähler form ωφ = ω0 + dd

c(φc(yc) −φ(yc)) on (M,Jc) belongs to the Kähler class ofω0 = ωφc . Conversely, each T-invariant Kähler form ω on (M,Jc) in this Kähler class can be written

on (M0, Jc) = (C∗)m ⋅ p0 as ω = ddcφ for a smooth T-invariant function φ (see [53]), determined upto an additive constant by the conditions that φ−φc is smooth on M . Writing φ = φ(yc) we obtaina convex function φ on R

m, and the inverse Legendre transform to φ(y) gives rise to momentum


coordinates (µφ)j = −JKj ⋅ φ of ωφ, sending M to P (by [26], as µφ equals to µ at the fixed pointsof the action), and thus a function u(x) ∈ S(P,L), determined up to an additive constant. Wesummarize the discussion in the following:

Proposition 3.3. The Legendre transform T defines a bijective map from the space of symplecticpotentials S(P,L) to the space K(Rm, ω0) of strictly convex smooth functions φ(y) on R

m, satisfyingthe following conditions:

the function defined on (C∗)m ⋅ p0 = (M0, Jc) by

φ(z) ∶= φ(12log ∣z1∣2, . . . , 1

2log ∣zm∣2)

gives rise to a Kähler metric ωφ = ddcφ on (M,Jc). φ(z) − φc(z) extends to a smooth function on (M,Jc).

In particular, T gives rise to a bijection between the space of T-invariant ω0-compatible Kählermetrics on M , modulo the action of T-equivariant symplectomorphisms of (M,ω0), and the space ofT-invariant Kähler metrics in the Kähler class [ω0] on (M,Jc), modulo the action of TC = (C∗)m.

3.2. Kähler-Ricci solitons. We now restrict to the special case when the compact symplecticmanifold (M,ω0,T) satisfies the topological condition 2πc1(M,Ju) = 1

λ[ω0] for some positive con-

stant λ > 0, and any Ju ∈ KT(M,ω0). From the symplectic point of view developed in Section 3,the Fano condition is equivalent (up to a modification of the moment map by a translation) to theassumption that (P,L) satisfies

Lj(0) = λ, j = 1, . . . d,(3.10)

for a positive constant λ. We refer to a Delzant polytope (P,L) satisfying (3.10) as barycentred. Inthis setting we obtain a ω-relative smooth Ricci potential as follows. By (3.6) and (3.9), we havefor any (gu, Ju)(3.11) ρJu −

1

λω = ddc(1

2log det (Hess(u(µ))) − 1

λ(−u(µ) + m

∑i=1µiu,i(µ))).

Using the boundary conditions for u in Proposition 3.1 (which imply that detHess(u) = δ(x)∏d

j=1 Lj(x)for a positive smooth function δ on P, see [2]) and the Fano condition (3.10), we have that thefunction appearing at the RHS of (3.11) satisfies

hu(µ) ∶= 1

2log det (Hess(u(µ))) − 1

λ(−u(µ) + m

∑i=1µiu,i(µ))

= 1

2

d

∑j=1( − logLj −

1

λ(−Lj logLj +

m

∑i=1µi(Lj logLj),i)) + smooth

= 1

2

d

∑j=1( − logLj −

1

λ(−Lj logLj + (Lj − λ) logLj)) + smooth

= smooth.

(3.12)

It thus follows that ρJu −1λω0 = dd

cJuhu globally on M . By rescaling of ω0, we shall furthermore

assume that λ = 1, i.e. ω0 ∈ 2πc1(M,Jc).In this setting, the normalized Kähler-Ricci flow can be reduced to a scalar parabolic problem

of symplectic potentials ut(x) ∈ S(P,L)(3.13)

∂

∂tut = 2hut = log det (Hess(ut)) − 2(−ut +

m

∑i=1xi(ut),i) ,


or, via the Legendre transform (see Proposition 3.3), to a parabolic PDE for convex functions φt(y)on R

m [90]:

(3.14)∂

∂tφt(y) = log det (Hess(φt(y))) + 2φt(y).

In this setup, a Kähler metric (gu, Ju) ∈ KT(M,ω0) is a Kähler-Ricci soliton (see (2.14)) ifthe corresponding symplectic potential u ∈ S(P,L) satisfies (see [34, Sec. 3.1] and [84])

hu(x) = 1

2⟨ξ, x⟩,(3.15)

for some ξ ∈ t, where hu(x) is defined in (3.12) with λ = 1. Indeed, it is easy to check that thecorresponding Kähler metric (ω,Ju) then satisfies (2.14) with K =Kξ, the induced vector field by ξ.Furthermore, it turns out that [34, 89] the element ξ ∈ t in the RHS of (3.15) is uniquely determined

from the barycentered polytope (P,L) by the condition ∫P e⟨ξ,x⟩xidx = 0. We have then followingfundamental results, which follow from the works [82, 83, 84, 90].

Theorem 3.4. Let (M,ω0,T) be a Fano toric manifold with [ω0] ∈ 2πc1(M,Jc), and (P,L) thecorresponding barycentred Delzant polytope. Then (3.15) admits a solution us ∈ S(P,L), which isunique up to the addition of an affine-linear function. Furthermore, the solution of (3.14) existsfor all t ∈ [0,∞) and for a suitable choice of real constants ct and points yt ∈ Rm, φt(y + yt) + ctconverges in C∞ to the Legendre transform φs(y) of a symplectic potential us(x) corresponding toa Kähler-Ricci soliton. When (M,ω0,T) admits a Kähler-Einstein metric, i.e. the element ξ in(3.15) vanishes, the convergence of φt(y) in C∞ holds with yt = 0.

4. Toric generalized Kähler structures

Inspired by the theory of toric Kähler structures, L. Boulanger [15] and Y. Wang [85, 86]gave a similar description in momentum-angular coordinates of the T-invariant generalized Kählerstructures compatible with a symplectic 2-form F on a smooth compact toric symplectic manifold(M,F ). We start with the setting and notation of Section 3: (M,ω0,T) is a given toric symplecticmanifold with momentum map µ ∶M → t

∗ and θ ∶M0 → t/2πΛ are the standard angular coordinateson M0 (i.e. θ = θc); choosing a lattice basis of t, we have

(4.1) ω0 =m

∑j=1

dµj ∧ dθj.

4.1. Toric generalized Kähler deformations of type A. Following [15], we can construct aT-invariant almost complex structure J on M0 from a smooth non-degenerate field Ψ(x) of bilinear

forms on P, with inverse Ψ−1(x), by letting

(4.2) J ∶=m

∑i,j=1(Ψij(µ)dµi ⊗ ∂

∂θj− (Ψ−1)ij(µ)dθi ⊗ ∂

∂µj),

where (Ψij(x)) is the Gram matrix (in the fixed basis of t and t∗) of Ψ(x). This is consistent with

(3.5), in which case we have Ψ(x) =G(x) = Hess(u(x)).The integrability of the almost complex structure J given by (4.2) reads as (see the proof of

[15, Theorem 6])

Ψij,k =Ψik,j.

The almost-complex structure I = −ω−10 J∗ω0 is expressed in momentum-angular coordinates by

I =m

∑i,j=1((ΨT)ijdµi ⊗ ∂

∂θj− (ΨT)−1

ijdθi ⊗

∂

∂µj).


Furthermore, ω0 tames J if and only if the the symmetric tensor

(4.3) g ∶= −(ω0J)s = m

∑i,j=1((Ψs)ijdµidµj + (Ψ−1)sijdθidθj)

is positive definite on M0. In the above formula (and in what follows) we use upper-indices s

and a to denote respectively the symmetric and skew-symmetric parts of bilinear forms and thecorresponding Gram matrices. We shall make an abundant use of the following identities which holdfor a non-degenerate matrix Ψ = (Ψij) (they can be checked easily using the polar decompositionof Ψ).

Ψ(Ψ−1)sΨT =Ψs, Ψ(Ψ−1)aΨT = −Ψa;

Ψs(Ψ−1)a +Ψa(Ψ−1)s = 0, Ψs(Ψ−1)s +Ψa(Ψ−1)a = Id,(4.4)

where ΨT stands for the transposed matrix. It follows from (4.3) that g is positive definite over M0

if and only if Ψs(x) is positive definite over P.A special case when the almost complex structures J and I are both integrable and tamed

by ω0 on M0 is obtained by letting

(4.5) Ψ(x) = Hess(u(x)) +Awhere u(x) is a smooth strictly convex function on P and A ∈ ∧2

t is a (constant) skew-symmetricbilinear form on t

∗, thus extending the Kähler setting (3.5), which is obtained by letting A = 0.Thus, (4.5) gives rise to a T-invariant generalized structure on M0 which is compatible with

the symplectic form ω0 in the sense of (1.3), as explained in the introduction. The following resultis established in [15, Theorem 11] for m = 2, and in [85, Theorems 4.9 & 4.11] in general.

Proposition 4.1. The complex structure JA defined on M0 by (4.2) and Ψ(x) of the form (4.5)

for some smooth strictly convex function u(x) on P and A ∈ ∧2t extends to a globally defined

ω0-compatible generalized Kähler structure (gA, IA, JA) on M if and only if u ∈ S(P,L).Definition 4.2. The generalized Kähler structures (gA, JA, IA) obtained from a toric ω0-compatibleKähler structure (gu, Ju) via Proposition 4.1 (for some and hence any skew-symmetric bilinear formA on t

∗) will be referred to as generalized Kähler deformations of type A of (gu, Ju) .

For a toric generalized Kähler structure (gA, JA, IA) as above, with corresponding u ∈ S(P,L)and A ∈ ∧2

t, we introduce pluriharmonic functions on (M0, J) by

(4.6) yj(µ) ∶= u,j(µ) + m

∑k=1

Akjµk, j = 1, . . . ,m.Indeed, it is easily checked from (4.2) that, at each point, dy1, . . . ,dym,dθ1, . . . ,dθm is the dualbasis of the basis of commuting real holomorphic vector fields −JK1, . . . ,−JKm,K1, . . . ,Km. Asin the case of compatible toric Kähler structures, we can think of y as abstract variables on t, definedby the transformation

(4.7) y ∶= (∇u)(x) +A(x),where A is thought of as a linear map from t

∗ → t. Using that u is strictly convex on P (and thatA is a skew-symmetric bilinear form on t

∗ ≅ (Rm)∗) one can show as in the case of the Legendre

transform (3.7) that y(x) is a diffeomorphism from P to Rm. Similarly to the construction in the

Kähler case, we let xu be the pre-image of 0 ∈ t under (4.7) and pu the point in M0 with µ(pu) = xuand θ(pu) = 0 so that zj = eyj+

√−1θj are identified with standard holomorphic coordinates on the

orbit (Cm) ⋅ pu = (M0, J). With this understood, we have the canonical biholomorphism betweentwo complex structures given by deformations of type A.


Lemma 4.3 (Canonical biholomorphism between (M,JA) and (M,JA0)). Let JA and JA0

be T-invariant complex structures on M , obtained by deformations of type A of the Kähler structures(gu, Ju) and (gu0

, Ju0) in KT(M,ω0) with corresponding symplectic potentials u,u0 ∈ S(P,L), re-

spectively. Then, the T-equivariant diffeomorphism Φ on M0, which sends the complex coordinates(z1, . . . , zm) of JA to the respective complex coordinates (z01 , . . . , z0m) of JA0, extends to a (C∗)m-

equivariant biholomorphism from (M,JA) to (M,JA0). In particular, independent of u and A,(M,JA,T) is isomorphic, as a complex toric variety, to (M,Jc,T), where Jc is the canonical com-

plex structure in KT(M,ω0).Proof. Given some J = JA as in the statement, we shall build a holomorphic (C∗)m-equivariantatlas on (M,J) consisting of complex charts ψv ∶ Mv → C

m associated to each vertex v of P. Tothis end, we suppose that the chosen basis e of t, defining a dual basis of t∗ and the holomorphiccoordinates (y1, . . . , ym) via (4.6) on (M0, J), is a Z-basis of Λ. By Delzant theory [26], each vertexv ∈ P corresponds to a point pv ∈ M fixed by the action of T. We change the momentum mapµv ∶= µ − v by a translation, so that µ(pv) = 0 ∈ t∗. Furthermore, we change the initial Z-basis oft with a Z-basis ev consisting of the primitive normals of the labels Lj ∈ L vanishing at v, withcorresponding transition matrix of bases Nv = (nij) ∈ GL(m,Z). Using the new basis ev and themodified momentum map µv, we define via (4.6), new J-holomorphic coordinates (zv1 , . . . , zvm) on

(M0, J) by zvk = e(yvk+√−1θvk). It follows that on (M0, J) ≅ (C∗)m ⋅ pu, the coordinates z and zv are

related by

zv = e−λv(A)zNv ,(4.8)

which is an abbreviation to the map

(zv)k = e−(NvA(v))j (z1)nk1(z2)nk2⋯(zm)nkm , k = 1, . . . m.We now show that (zv1 , . . . , zvm) can be extended from (M0, J) to define a (C∗)m-equivariant holo-morphic chart on (M,J) centered at pv. To this end, notice that with respect to the basis ev andwith the normalization for the momentum map µv, the labels Lk ∈ L vanishing at v = µ(pv) = 0 areLk(µv) = µvk, k = 1, . . . m. We will show that (zv1 , . . . , zvm) extend at any point in the pre-image M0

v byµv of the union of v and the interiors of all faces of P containing the vertex v. Let us take a point p∗in the pre-image of the interior Σk of the facet defined by µvk = 0. (The argument for a face of higherco-dimension is similar.) Letting x∗ = µv(p∗), by the second boundary condition for u in Proposi-tion 3.1, we have that u,k(µv) = 1

2log(µvk)+ϕ(µv) for a smooth function ϕ(x) defined in a closed ball

B∗ around x∗; it thus follows that on (µv)−1(B∗)∩M0, we have ∣zvk ∣2 = (µvk)eϕ(µv), showing that zvktends to zero for any sequence of points in M0 which converges to p∗. The same boundary conditionalso ensures that the functions yvr (x) with r ≠ k are extendable as smooth functions on Σk. Usingthat first boundary condition of Proposition 3.1 and the definition [26] of the canonical angular

coordinates θr = θCR with r ≠ k on µ−1v (Σk), it follows that (z1, . . . , zk−1,0, zk+1, . . . , zm) gives rise to

a (C∗)m-equivariant coordinate system on (µv)−1(Σk). Similar arguments on any face containingv yield that (zv1 , . . . , zvm) defines a global (C∗)m-equivariant chart ψv ∶Mv → C

m. Considering suchcharts for all vertices of P gives rise to a holomorphic atlas on (M,J).

We now let ψ0v ∶ (Mv, J0)→ Cm be the corresponding chart at pv for the complex structure J0 =

JA0(obtained by a deformation of type A from u0 ∈ S(P,L) and A0 ∈ ∧2

t). Using (4.8), it followsthat in the charts ψv, ψ

0v , the map Φ ∶ (M0, J) → (M0, J0) sending (z1, . . . , zm) to (z01 , . . . , z0m)

becomes

zv ↦ eλv(A0−A)(z0)v,which is well-defined on the whole C

m.


4.2. Toric generalized Kähler deformations of type B. Following [85], let (gA, IA, JA) be ageneralized Kähler deformation of type A on (M,ω0,T), defined via (4.2) and (4.5) by u ∈ S(P,L)and A ∈ ∧2

t, and take another element B ∈ ∧2t. Then the 2-form

(4.9) FB ∶= ω0 + ⟨dµ,B,dµ⟩ = m

∑j=1

dµi ∧ dθi +m

∑i,j=1

Bijdµi ∧ dµj

is T-invariant and symplectic on M , with moment map µ and associated Delzant polytope (∆,L).In particular, (M,ω0,T) and (M,FB ,T) are isomorphic as symplectic toric manifolds, but themomentum-angular coordinates (µ, θ) of ω0 are no longer momentum-angular coordinates of FB :one needs to take instead (µ, θF ∶= θ −B(µ)).

It is observed in [85] that for any B ∈ ∧2t, the almost complex structure IA,B ∶= −(FB)−1(JA)∗FB

is integrable too. Indeed, with respect to the frame ( ∂∂θi, ∂∂µj) on TM0, JA and IA,B are represented

by matrices

(4.10) JA ∼ ( 0 ΨT

−(ΨT)−1 0) , IA,B ∼ ( 2BΨ

−1Ψ + 4BΨ

−1B

−Ψ−1 −2Ψ−1B) ,

where Ψ is given by (4.5). One can then check from the above representations that

(4.11) y ∶= ∇u −A(µ), θ ∶= θ − 2B(µ) = θF −B(µ)are pluriharmonic coordinates with respect to IA,B and hence IA,B is integrable.

Suppose now that JA is tamed by FB . This gives rise to a T-invariant generalized Kählerstructure (gA,B , IA,B , JA), where the induced Riemannian metric gA,B ∶= −(FBJA)s is represented

in the frame ( ∂∂θi, ∂∂µj) on TM0 by the Gram matrix

gA,B ∼( (Ψ−1)s Ψ−1B

−B(ΨT)−1 Ψs) = ( Ψ

−1 0

0 Id)( Ψ

s B

−B Ψs )( (ΨT)−1 0

0 Id) .(4.12)

The equality of matrices in the above formula is deduced by using (4.4). It follows that FB tamesJA on M0 if and only if the Hermitian matrix

Hess(u) +√−1B > 0(4.13)

is positive definite at any point of P. A similar condition imposed on the interior of each face of Pby virtue of Proposition 3.1 guarantees that FB tames J on M . Notice that this is independent ofthe transformation of type A used to obtain J .

Definition 4.4. The generalized Kähler structure (gA,B , IA,B , JA) on M described above will becalled a generalized Kähler deformation of type B of (gA, IA, JA).

By the results in [86, Sect. 4], we have the following generalization of the description of toricKähler metrics.

Proposition 4.5. [86] Any T-invariant generalized Kähler structure of symplectic type on M , com-patible with a T-invariant symplectic form F on M with associated labelled Delzant polytope (P,L),is T-equivariantly isomorphic to a generalized Kähler structure (gA,B , IA,B , JA), obtained from atoric Kähler metric (gu, Ju, ω0) corresponding to a symplectic potential u ∈ S(P,L), by generalizedKähler transformations of type A and B.

Convention 1. In view of the above result, and to ease the notation, we shall omit the indices A,Band implicitly assume (without loss of generality) that a toric generalized Kähler structure (g, I, J)compatible with a T-invariant symplectic form F onM and corresponding Delzant polytope (P,L) isalways obtained from a T-invariant Kähler structure (gu, Ju), via generalized Kähler deformations


of type A and B for respective elements A,B ∈ ∧2t, where u ∈ S(P,L), ω0 is a symplectic form

associated to (P,L) via the Delzant construction, and (gu, Ju) is a ω0-compatible Kähler metricwith symplectic potential u and canonical momentum-angular coordinates (µ, θc) on (M0, ω0).

The computations (4.7) and (4.11) of the corresponding pluriharmonic coordinates of J andI underline the general symmetry of the construction:

(4.14) (u,A,B) ←→ (u,−A,−B)which, geometrically, corresponds to switching the roles of J and I of the corresponding toricbihermitian structure (g, I, J). Indeed, this becomes apparent if we express the construction in themomentum/angular coordinates (µ, θF ) of F . Then, the corresponding Kähler structures ω

Juand

ωIu

are linked to the common symplectic form F via the relations

ωJu=

m

∑j=1

dµj ∧ dθj = F −m

∑i,j=1

Bijdµi ∧ dµj

ωIu=

m

∑j=1

dµj ∧ dθj = F +m

∑i,j=1

Bijdµi ∧ dµj.

F = 12(ω

Iu+ ω

Ju),

(4.15)

where we recall θ are defined in (4.11).

4.3. The case A = 0. When A ≠ 0, the integrable almost complex structures J = JA and Ju

associated respectively to the generalized Kähler structure (g, I, J) and the Kähler structure (gu, Ju)are different (even though they are T-equivariantly biholomorphic by virtue of Lemma 4.3). In

special case A = 0, we have that Ju = J and similarly for the other complex structure, i.e. Iu = I.We shall later make use of this simplification, so we recast below some facts specific to this case.In order to ease notation, and to avoid confusion, we shall denote in this case by (g,ω) the toricKähler structure on the complex manifold (M,J) (in accordance with the notation in Sect. 2) andby (g, I, J) the toric generalized Kähler structure with corresponding pluriclosed fundamental 2-form ω = gJ = F 1,1 on (M,J). When we want to further emphasize the different roles of I and J ,we shall denote, correspondingly, by (ωI , I) and (ωJ , J) the Kähler structures and by (ωI , I) and(ωJ , J) the pluriclosed Hermitian structures.

Letting

(4.16) Π = ΠB ∶= −m

∑i,j=1

Bij∂

∂θi∧

∂

∂θj

denote the real Poisson structure on M determined by B ∈ ∧2t, the formula (4.9) yields the following

basic relation between ω and ω on (M,J) (which holds in the case A = 0):(4.17) F = (ω−1 +Π)−1, ω = F 1,1.

It will be also convenient in this case to express the geometry in terms of the J-pluriharmoniccoordinates (yj, θj) obtained via the Legendre transform of u (see (3.7) and Proposition 3.3): lettingyj = u,j(µ) and φ(y) + u(µ) =∑m

j=1 µjyj, we have on (M0, J) (see (3.9)):

ω = ddcφ =m

∑i,j=1(Hess(φ(y)))

ijdyi ∧ dθj,


with Hess(φ(y)) = (Hess(u(x)))−1 whereas the symplectic 2-form F = FB and the pluriclosed

fundamental 2-form ω = (FB)1,1 introduced in Section 4.2 are given by (as we have set A = 0):F = FB =

m

∑i,j=1(φ,ijdyi ∧ dθj + [(Hess(φ))B(Hess(φ))]

ijdyi ∧ dyj)

= ω + ωΠω;ω = ωB =

m

∑i,j=1(φ,ijdyi ∧ dθj + 1

2[(Hess(φ))B(Hess(φ))]

ij(dyi ∧ dyj + dθi ∧ dθj))

= ω + ω (Π1,1)ω.

(4.18)

In this case, (yj, θj) are pluriharmonic coordinates of I (see (4.11)), so similar formulae hold with

respect to I (in which one should replace B by −B and θj by θj). We also notice that

(4.19) ddcJφ = ωJ , dd

cIφ = ωI ,

so, by (4.15), we get

(4.20) F = 1

2(ddc

Jφ + ddcIφ).

4.4. The holomorphic Poisson structure of a toric generalized Kähler structure. Thenext result gives a geometric meaning of the parameters A,B ∈ ∧2

t. In particular, they completelydetermine the associated holomorphic Poisson structures.

Proposition 4.6. Let (g, I, J) be a toric generalized Kähler structure obtained from a Kähler metric(gu, Ju) by generalized Kähler deformations of type A and B, with respective elements A,B ∈ ∧2t.

Then, the corresponding holomorphic Poisson structure on (M,J) given by (1.4) is

σJ = 2m

∑i,j=1(Aij +

√−1Bij)(Ki −

√−1JKi) ∧ (Kj −

√−1JKj),

where K1, . . . ,Km are the fundamental vector fields associated to a basis of t, and Aij and Bij arethe corresponding components of A and B in this basis.

Proof. The above formula follows by straightforward but somewhat tedious matrix computations,using that Kj = ∂

∂θj, (4.10), the identification (which in turn follows from (4.12))

g−1 ∼ ( ΨT 0

0 Id)( X Y

−Y X)( Ψ 0

0 Id) ,

where

X ∶= (Ψs+B(Ψs)−1B)−1, Y ∶= −(Ψs)−1BX

and the identities (4.4).

4.5. Relation to Hamiltonian flow deformations. In [46, 11], the so-called flow deformationof a symplectic type generalized Kähler structure is introduced, generalizing the Joyce deformationsin the non-degenerate case [5]. Let (g, I, J) be a generalized Kähler structure of symplectic type onM , with corresponding real Poisson tensor σ = [I, J]g−1, and f ∈ C∞(M) a smooth function on M .We denote by Xf ∶= 1

2σ(df) the induced σ-Hamiltonian vector field, and let Ψt be its flow. Using

the basic identity (which follows from the fact that σI = σ+√−1Iσ is a holomorphic Poisson tensor)

LXfI = −1

2σ(ddc

If),


it is shown in [11] that the closed 2-form

Ft ∶= F0 +∫t

0Ψ∗s(ddc

If)dsdefines a generalized Kähler structure of symplectic type (gt, It, J) in GK(M,J, [F0]), as long as

(Ft)1,1J > 0. Furthermore, It = Ψ∗t (I) and σt = σ. As the open positivity condition of Ft willremain true for ∣t∣ < ε, the above construction induces a notion of a generalized-Kähler class forsuch structures, extending the setting of [6] from the non-degenerate to the symplectic type case.Any such generalized Kähler class has an induced formal Fréchet structure, modelled on C∞(M)/Rthrough the relations

F = ddcI f , I = −1

2σ(ddc

I f).In the toric case (assuming A = 0) we can relate the T-invariant flow deformation to the one-parameter variation of the potential φ. As a consequence, we can identify the space of T-invariantgeneralized Kähler deformations in a given class with an open convex subset of the space of invariantsmooth functions modulo constants C∞(M)T/R.

Proposition 4.7. Let (gφ, Iφ, J) be a toric generalized Kähler structure with symplectic form Fφ,obtained as a deformation of type B of a toric Kähler structure ωφ on (M,J), with a (fixed) B ∈ ∧2

t,

see (4.18). Let φ be a first order variation of φ, and ω, F and I denote the corresponding first ordervariations of ωφ, Fφ and Iφ. Then

ω = ddcJ φ, F = ddc

Iφ, I = −12σ(ddc

I φ),where σ = −4J (Π(2,0)+(0,2)

B) is the real part of the corresponding J-holomorphic Poisson structure

given by Proposition 4.6.

Proof. In the J-pluriharmonic coordinates (yj, θj), we have by (4.18)

ωφ =m

∑i,j=1

Hijdyi ∧ dθj, Fφ =m

∑i,j=1

Hijdyi ∧ dθj + (HBH)ijdyi ∧ dyj,where, we recall H = (Hess(u))−1 = Hess(φ). Thus

ω =m

∑i,j=1

Hijdyi ∧ dθj, F =m

∑i,j=1

Hijdyi ∧ dθj + (HBH +HBH)ijdyi ∧ dyj.Using that (y, θ) and (y, θ = θ − 2Bµ) with y = ∇u (see (4.11)) are respective pluriharmonic coordi-nates for J and I, we compute

ddcJ φ =

m

∑i,j=1

φ,ijdyi ∧ dθj =m

∑i,j=1

Hijdyi ∧ dθj,

ddcI φ =

m

∑i=1

dI (φ,idyi) = d ⎛⎝m

∑i=1φ,idθi − 2

m

∑i,j,k=1

HjkBkiφ,idyj⎞⎠

=m

∑i,j=1(Hijdyi ∧ dθj + 2(HBH)ijdyi ∧ dyj)

=m

∑i,j=1(Hijdyi ∧ dθj + (HBH + HBH)ijdyi ∧ dyj) ,

which gives the first two equalities. The variation of I follows from the second equality and thegeneral relation (cf. [46])

J − Iφ = 1

2σFφ.


4.6. The Bismut-Ricci form of a toric generalized Kähler structure. In this subsection wederive the Bismut-Ricci forms ρBI , ρ

BJ associated to a toric generalized Kähler structure (g, I, J),

using momentum-angular coordinates of ω0, thus extending Abreu’s formula (3.6).

Lemma 4.8. Let (g, I, J) be a toric generalized Kähler structure, associated to the data (u,A,B).Then, the Bismut-Ricci forms ρBI , ρ

BJ of (g, J) are given on M0 by

ρBI = 1

2dd

cJ( log det (Hess(u))(µ) +√−1B)),

ρBJ = 1

2dd

cI( log det (Hess(u))(µ) +√−1B)).

Proof. We are going to use Proposition 2.10 with respect to the holomorphic section (defined onM0)

ΘJ ∶= ((K1 −√−1JK1) ∧⋯∧ (Km −

√−1JKm))−1 .

To this end, recall that Kj = ∂∂θj

and, by (4.10) and (4.12), we have (in momentum/angular coor-

dinates)

(4.21) ωJ ∼ ⎛⎝Ψ−1B(ΨT)−1 −Ψ−1Ψs

Ψs(ΨT)−1 B

⎞⎠ ,

so we compute, by using the above together with (4.12), (4.21), (4.4)

2−m⎛⎝ω[m]J

vΘ

⎞⎠ = det (g(Ki,Kj) −√−1ωJ(Ki,Kj))= det ((ΨT)−1(Ψs

−√−1B)Ψ−1)

= det (Ψs +√−1B)

(detΨ)2 = det (Hess(u) +√−1B)(detΨ)2 ,

As ω[m]0 = F [m] (see (4.1) and (4.9)), using (4.1) and (4.21), we also get

⎛⎝F [m]

ω[m]J

⎞⎠ =⎛⎝ω[m]0

ω[m]J

⎞⎠ =

detΨ

det (Hess(u) +√−1B) ,and therefore

⎡⎢⎢⎢⎢⎣log (F [m]

vΘ) + log⎛⎝

F [m]

ω[m]J

⎞⎠⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣2 log

⎛⎝F [m]

ω[m]J

⎞⎠ + log

⎛⎝ω[m]J

vΘ

⎞⎠⎤⎥⎥⎥⎥⎦

= − log det (Hess(u) +√−1B) + const.(4.22)

This yields the formula for ρBI (via Proposition 2.10). Using the symmetry (4.14) and that det(Hess(u)+√−1B) = det(Hess(u) −√−1B), we get the formula for ρBJ .

Similarly to the Kähler toric Fano case, the generalized Kähler toric structure correspondingto data (u,A,B) also admits a relative Ricci potential.

Lemma 4.9. Suppose (P,L) is a barycentred Delzant polytope corresponding to a smooth toricFano variety, i.e. (3.10) is satisfied. Then, for any toric generalized Kähler structure (g, I, J)


corresponding to the data u(x) ∈ S(P,L) and A,B ∈ ∧2t, the function

hu,B(x) ∶= 1

2log det (Hess(u(x)) +√−1B) − 1

λ( − u(x) + m

∑j=1

xju,j(x))= 1

2log det ((Hess(φ(y)))−1 +√−1B) − 1

λφ(y)

pulls-back by the momentum map µ of F to a smooth function on M . Furthermore, if A = 0 wehave

ρBJ −1

λωI = ddc

I(hu,B(µ)), ρBI −1

λωJ = ddc

J(hu,B(µ)),1

2(ρBJ + ρBI ) − 1

λF = 1

2(ddc

I(hu,B(µ)) + ddcJ(hu,B(µ))).

Proof. The smoothness of hu,B(µ) follows from similar considerations as the ones in (3.12). TheBismut-Ricci curvature formulas follow from Lemma 4.8, and lines (4.19) and (4.20).

5. The normalized generalized Kähler-Ricci flow on a toric Fano variety

5.1. Scalar Reduction. We now describe the normalized generalized Kähler Ricci flow (1.2), start-ing from a toric generalized Kähler structure (g, I, J) on a smooth toric Fano manifold (M,ω0).We suppose that the initial symplectic form F belongs to 2πc1(M,J0), and that (g, I, J) satisfiesConvention 1 (so we have ω = ω0 ∈ 2πc1(M,J) as well). We thus are interested in the flow (1.2)with λ = 1. The solution of (1.2) is defined in terms of Hermitian metrics (gt, J) on a fixed complexmanifold (M,J), whereas our Convention 1 normalizes (via the action of a T-equivariant diffeomor-phism) the toric generalized Kähler structures on M to be obtained from a Kähler structure (gu, Ju)compatible with the fixed symplectic form ω0 and canonical angular coordinates via deformationsof type A and B. This is similar to the symplectic versus complex viewpoints in the Kähler case(see Proposition 3.3), and the T-equivariant diffeomorphism in Lemma 4.3 connects the two pointsof view in the generalized Kähler case. The next proposition makes this explicit.

Proposition 5.1. Let (g0, I0, J0) be a toric generalized Kähler structure on a smooth Fano toricvariety, corresponding to the data (u0,A0,B0) with respect to a symplectic form ω0 ∈ 2πc1(M,J0).Denote by (gt, It, Jt) the generalized Kähler structure corresponding to the solution (ut,At,Bt) of

∂

∂tu = log (det(Hess(u) +√−1e−2tB0)) − 2( − u + m

∑j=1

xju,j),At = e−2tA0, Bt = e−2tB0.

(5.1)

Then Φ∗t (gt, It, Jt) solves (1.2), where Φ∗t is the canonical biholomorphism between (M,J0) and(M,Jt) from Lemma 4.3.

Proof. We shall do computations on M0, using by Lemma 4.9 that for ut ∈ S(P,L) the RHS of(5.1) is a smooth function on P, so (5.1) is a flow of elements ut ∈ S(P,L) defined for t ∈ [0, T ), thusgiving rise to a smooth path of generalized Kähler structures (gt, It, Jt), t ∈ [0, T ) on M , obtained asgeneralized Kähler deformations of type A and B of the Kähler structures (gut , Jut) in KT(M,ω0)with the 2-vectors At,Bt ∈ ∧2

t.We denote by Jt the complex structure defined by the generalized Kähler deformation of

type A of the complex structures Jut ∈ KT(M,ω0) with the 2-vector At = e−2tA0 ∈ ∧2t, and by

Ft = ω0 + e−2t⟨dµ,B0, dµ⟩ the symplectic form of (gt, Jt, It), with It being the Ft-conjugate of the

almost complex structure Jt. Denote by yt the Jt-pluriharmonic variables (4.6), and consider theT-equivariant diffeomorphism Φt defined in Lemma 4.3, which satisfies Φt(yt) = y0 and Φ∗t (Jt) =J0 =∶ J . We let Ft ∶= Φ∗t (Ft) be the pull-back of Ft, so that Ft is a symplectic form on (M,J)which gives rise to a generalized Kähler structure (gt, It, J) compatible with Ft. Notice that by the


definition of such generalized Kähler structures, the (1,1)-part with respect to J of Ft is the Kählerform ωJ(t) of (gt, J).

We are going to check that with (ut,At,Bt) satisfying (5.1), the symplectic forms Ft satisfy

(5.2)∂

∂tFt = −2ρBJ (gt) + 2Ft,

where ρBJ (gt) is the Bismut-Ricci form with respect to J of (gt, It, J). Taking the (1,1) part withrespect to J of (5.2) yields (1.2) with λ = 1.

For a toric GK structure (g, I, J), corresponding to the data (u,A,B), we consider the pluri-

harmonic variables y with respect to J , defined in (4.6), and a function φ(y) = φ(y), introduced onM0 by

(5.3) φ(y) ∶= −u(µ) + ⟨µ, y⟩ = −u(µ) + ⟨µ, y⟩ = φ(y),where yj = u,j(µ), and φ(y) is the Legendre transform (3.7) of u. We then have:

Lemma 5.2. For a toric generalized Kähler structure (g, I, J) corresponding to the data (u,A,B),the function φ satisfies on M0

(5.4) ddcJ φ = F + dJ(

m

∑i,j=1

Aijµidµj) − d( m

∑i,j=1

Bijµi ∧ dµj),where F is the symplectic structure compatible with (g, I, J).Proof. We compute using (4.10)

ddcJ φ = ddc

Jφ = dJd(m

∑i=1µiu,i − u) = dJ( m

∑i,j=1

µiu,ijdµj) = d( m

∑i,j,k=1

µiu,ij(ΨT)−1jkdθk)= d( m

∑i,j,k=1

µi((ΨTij +Aij))(ΨT)−1jkdθk)

= F + d( m

∑i,j,k=1

µiAij(ΨT)−1jkdθk) −m

∑i,j=1

Bijdµi ∧ dµj

= F + dJ( m

∑i,j=1

Aijµidµj) − d ⎛⎝m

∑i,j=1

Bijµi ∧ dµj⎞⎠ ,

as required.

By the above lemma, for the function φt(yt) we have that

Ft = Φ∗t (ddcJtφt(yt) − dJt(

m

∑i,j=1(At)ijµidµj) + m

∑i,j=1(Bt)ijdµi ∧ dµj)

= ddcJ φt(y0) − dJ(

m

∑i,j=1(At)ijµtidµtj) +

m

∑i,j=1(Bt)ijdµti ∧ dµtj),

where µt ∶= µ Φt. Using (5.1), we have

∂

∂tFt = ∂

∂t(ddc

J φt(y0) − dJ(m

∑i,j=1(At)ijµtidµtj))

= dJd(∂φt∂t(y0)) + 2dJ( m

∑i,j=1(At)ijµtidµtj)

− dJ( m

∑i,j=1((At)ij(∂µti

∂t)dµtj) + ((At)ijµtid(∂µ

tj

∂t)).

(5.5)


We thus need to compute the t derivatives of φt and µt, We start with computing the t derivativeof φt. By (5.3) we have

∂φ

∂yr(y) = µr + m

∑j=1

yj∂µj

∂yr−

m

∑j=1

u,j∂µj

∂yr

= µr −m

∑j,k=1

Ajk

∂µj

∂yrµk = µr +

m

∑k=1(A(ΨT)−1)

krµk,

so that, by (4.6) and (5.1), we obtain

(5.6)∂ytr∂t= ( log det (Hess(ut) +√−1B)),r − 2

m

∑ℓ=1(ΨT

t )rℓµℓ.

Therefore, we have

∂φt

∂t(yt) = − m

∑r=1(∂φt∂yr)(∂yr

∂t) − ∂ut

∂t(µ) + m

∑r=1

µr(∂yr∂t)

= − ∂ut∂t(µ) − m

∑r,k=1(At(ΨT

t )−1)krµk(∂yr∂t )= − ∂ut

∂t(µ) − m

∑k,r=1(At(ΨT

t )−1)kr( log det (Hess(ut) +√−1B))

,rµk

= − log det (Hess(ut) +√−1B) + 2φt(yt)−

m

∑k,r=1(At(ΨT

t)−1)

kr( log det (Hess(ut) +√−1B)),r,

(5.7)

where for passing from the second line to the third we have used (5.6) and the fact that Aij isskew-symmetric, and for passing from the third line to the fourth we have used (5.1) and the fact

that φt(yt) = φt(yt) (see (5.3)).We now compute the derivative of µt = µ Φt. Using that µtj(yt) = µj and (5.6), we get

∂µtj

∂t(yt) = − m

∑r=1(∂µj∂yr)(∂yr

∂t) = − m

∑r=1(ΨT

t )−1jr (∂yr∂t )= 2µj −

m

∑r=1(ΨT

t )−1jr ( log ∣det(Hess(ut) +√−1B)∣)

,r.

(5.8)


Thus, substituting back in (5.5) and using Lemmas 5.2 and 4.8, we obtain

(Φ∗t )−1( ∂∂tFt)= ddc

Jt[2φt − log det (Hess(ut) +√−1B) −m

∑k,r=1(At(ΨT

t )−1)krµk( log (detHess(ut) +√−1B))

,r]

− 2dJt[ m

∑k,r=1(At)krµkdµr] − dJt[ m

∑k,r=1(At(ΨT

t )−1)kr( log det (Hess(ut) +√−1B))

,rdµk]

+ dJt[ m

∑k,r=1

µkd((At(ΨTt )−1)kr( log detHess(ut)),r]

= 2Ft − dJtd[ log det (Hess(ut) +√−1B)]− 2d[ m

∑ℓ,r=1(Ψ−1t A(ΨT

t )−1)ℓr( log det (Hess(ut) +√−1B))

,rdθℓ)]

= 2Ft − dJtd[ log det (Hess(ut) +√−1B) − 2d[ m

∑ℓ,r=1(Ψ−1t )arℓ( log detHess(ut)),rdθℓ)]

= 2Ft − 2ρBJt(gt),

as required.

We apply the Legendre transform (3.7) to ut(x) and use the formula (3.8) to rewrite theevolution equation (5.1) of symplectic potentials ut(x) as an evolution equation of convex functionsφt(y) on R

m (similar to the reduction (3.14) of the normalized Kähler-Ricci flow):

Proposition 5.3. Let (ut,At,Bt) denote a solution to (5.1) on a toric Fano manifold. The asso-ciated one-parameter family of potentials φt(y) on R

m satisfies

(5.9)∂

∂tφt = log det((Hess(φt))−1 +√−1e−2tB0)−1 + 2φt.

Proof. This follows from (5.1) and the basic properties of the Legendre transform: Hess(φt(y)) =(Hess(ut(x)))−1 and (3.8).

Remark 5.4. Note that it follows directly from Propositions 4.6 and 5.1 that the holomorphicPoisson tensor on (M,J) evolves by (σJ)t = e−2t(σJ)0 along the normalized generalized KählerRicci flow, as pointed out in Proposition 2.13.

The results of Proposition 5.1 and 5.3 yield that the flow of A completely decouples from theevolution equations for u and B. Because of this it suffices to study the flow in the case A = 0, anobservation which greatly simplifies the technicalities to follow.

Corollary 5.5. Let (g0, I0, J0) be a toric generalized Kähler structure on a smooth Fano toricvariety, corresponding to the data (u0,A0,B0) with respect to a symplectic form ω0 ∈ 2πc1(M,J0).Let (ut,0, e−2tB0) be the unique maximal solution of (5.1) with the initial data (u0,0,B0). Denote by(gt, It, Jt) the generalized Kähler structure corresponding to the one-parameter family (ut, e−2tA0, e

−2tB0),and by Φ∗t the canonical biholomorphism between (M,J0) and (M,Jt) from Lemma 4.3. ThenΦ∗t (gt, It, Jt) is the solution to (1.2) with initial data (g0, I0, J0).

By now restricting to the case A = 0 using Corollary 5.5, we are able to give an explicitdescription of the pluriclosed metric ωt along the generalized Kähler Ricci flow in the toric setting,


as a certain transformation of a flow of associated Kähler metrics ωt = ωφt = ddcJφt on a fixed

complex manifold (M,J). The precise relation is given by (4.18), i.e.

(5.10) ωt = ωt + e−2t [ωt (Π1,1

B0)ωt] ,

where ωt ∶= ωφt is the flow of Kähler metrics corresponding to a solution φt of (5.9) and Π1,1B0

denotes the (1,1)-part of the real Poisson tensor (4.16). As ωt are Kähler metrics with [ωt] ∈ [Ft] =2πc1(M,J), and satisfy ω

[m]t = F [m]t (this follows from ω

[m]0 = F [m]B , see (4.1) and (4.9)), ωt is

precisely the Kähler reduction of (1.2) introduced in Sect. 2 whereas (5.10) provides the preciserelation between ωt and ωt.

We next recapture the Kähler flow ωt in terms of ω0-relative Kähler potentials (where ω0 canbe the initial Kähler metric, or any other reference T-invariant Kähler metric in 2πc1(M,J)).Proposition 5.6. Let (g0, I0, J) denote a toric generalized Kähler structure on (M,J), with corre-sponding data (u0,0,B0), and let ω0 ∈ 2πc1(M,J) denote an initial toric Kähler metric with Riccipotential h0. Suppose ϕt satisfies

∂

∂tϕt = log⎛⎝

ω[m]ϕt

ω[m]0

⎞⎠ + log

⎛⎜⎝ω[m]ϕt

ω[m]φt

⎞⎟⎠ + 2ϕt − 2h0 + ct, ϕ(0) = 0,ωϕt = ω0 + dd

cJϕt, ωϕt = ωϕt + e

−2t (ωϕtΠ1,1B0ωϕt) .

(5.11)

Then, ωt ∶= ωϕt is a flow of Kähler metrics in 2πc1(M,J), and ωt = ωϕt an associated flow ofpluriclosed Hermitian metrics gt = −ωtJ on (M,J), defining a generalized Kähler structure (gt, It, J)with corresponding symplectic form

Ft ∶= ωt + e−2tωtΠB0

ωt,

and satisfying the evolution equations

(5.12)∂

∂tωt = −2(ρBIt(gt) − ωt), ∂

∂tωt = −2(ρBJ (gt)1,1 − ωt) .

Proof. We let ϕt(y) ∶= φt(y) − φ0(y) which, by Proposition 3.3, pulls back to a smooth function onM (still denoted by ϕt) such that ωt = ω0 + dd

cJϕt. Using that in the (yi, θj) coordinates we have

(see (4.18))

(5.13)ω[m]

ω[m]= (det (G +

√−1B))(detH)2

detH0

,ω[m]

ω[m]0

= detH

detH0

,

where ω0 is any background toric Kähler metric (and H0 = ω0(Ki, JKj) = Hess(φ0) is the corre-sponding smooth matrix), we can rewrite the flow (5.9) in terms of ϕt as (5.11). The remainingclaims follow from Corollary 5.5 and (5.10).

Remark 5.7. In Proposition 5.6, ct can be any smooth function of t, which allows for differentnormalizations of the ω0-relative Kähler potentials ϕt, without changing the corresponding metricsωϕt and ωϕt.

5.2. Evolution equations. Here we determine the evolution of various quantities along NGKRF.To begin we recall the Chern Laplacian and derive an explicit form for its action on (p,0)-forms.

Definition 5.8. Let (M,g,J) denote a Hermitian manifold, with associated Chern connection. Let∆C

g ∶ Γ(∧p,q(M))→ Γ(∧p,q(M)), denote the Chern Laplacian, defined by

∆Cg ψ =

m

∑i=1(∇C)2

Ei,Eiψ,


where Ei ∶= ei −√−1Jei, i = 1, . . . m for a J-adapted orthonormal frame ei, Jei. For a smoothfunction f one has

∆Cg f = ⟨ω,ddcf⟩ = −ω−1 ddcf.

Lemma 5.9. Fix a Hermitian manifold (M2n, g, J) with Chern connection ∇c and let H = −dcω.For ψ ∈ ∧p,0, ∂ψ = 0, and Z1, . . . Zp ∈ T 1,0M , one has

(∆Cg ψ)(Z1, . . . ,Zp) = (∆gψ +H ⋆ (dψ) + d (H ⋆ ψ) −Lθ♯ψ)

Z1,...,Zp

,

where for a (q + 1)-form α, we let

(H ⋆ α)X1,...Xq ∶= 1

2

2m

∑i,k=1

q

∑j=1(−1)jH(ei, ek,Xj)α(ei, ek,X1, . . . , Xj , . . . ,Xq).

Proof. Using that ∇C preserves J and the (1,1)-part of the torsion T c of ∇C vanishes, we compute.

(∆Cg ψ)(Z1, . . . ,Zp) = m

∑i=1((∇C)2

EiEiψ)(Z1, . . . ,Zp) = m

∑i=1∇

CEi(dψ)(Ei,Z1, . . . ,Zp)

= −2(δ∇C

dψ)(Z1, . . . ,Zp),where the real operator δ∇

C

∶ Γ(∧q(M)) → Γ(∧q−1(M)) is introduced by the formula

(δ∇C

α)(X1, . . . ,Xq−1) ∶= − 2m

∑i=1(∇C

eiα)(ei,X1, . . . ,Xq−1)

= (δgα)(X1, . . . ,Xq−1) + (θ α)(X1, . . . ,Xq−1)−1

2

p−1

∑j=1(−1)j 2m

∑i,k=1

α(ei, ek,X1, . . . , Xj , . . . ,Xq−1)H(ei, Jek, JXj).(5.14)

Specializing (5.14) to dψ and vector fields of type (1,0), we thus get

(∆Cg ψ)(Z1, . . . ,Zp) = − 2(δgdψ)(Z1, . . . ,Zp) − 2(θ dψ)(Z1, . . . ,Zp)

+√−1

p−1

∑j=1(−1)j 2m

∑i,k=1

dψ(ei, ek,Z1, . . . , Zj , . . . ,Zp)H(ei, Jek,Zj)= − 2(δgdψ)(Z1, . . . ,Zp) − 2(θ dψ)(Z1, . . . ,Zp).

(5.15)

For the last line we have used ∂ψ = 0, so that for fixed vectors Z1, . . . ,Zp, the 2-form (dψ)(⋅, ⋅,Z1 , . . . , Zj , . . . ,Zp)is of type (1,1), and (using H = −dcω) the 2-form H(⋅, ⋅,Zj) is of type (1,1) + (0,2).

Using d(dψ) = 0, ∂ψ = 0, (T c)1,1 = 0, and g(T c(Zj ,Ei),X) = √−1H(Zj ,Ei, JX), we alsocompute

(∆Cg ψ)(Z1, . . . ,Zp) = m

∑i=1(∇C

Eidψ)(Ei,Z1, . . . ,Zp)

= − 2√−1d(ω dψ)Z1,...,Zp

+√−1

2m

∑i,k=1

p

∑j=1(−1)jH(ei, Jek,Zj)(dψ)(ei, ek,Z1, . . . , Zj , . . . ,Zp)

+

2m

∑i,k=1

p

∑j=1(−1)jH(Jei, Jek,Zj)(dψ)(ei, ek,Z1, . . . , Zj , . . . ,Zp)

= − 2√−1d(ω dψ)Z1,...,Zp + 2 (H ⋆ (dψ))Z1,...,...,Zp.


For the last line, we have again used the type decomposition of the two forms (dψ)(⋅, ⋅,Z1 , . . . , Zj , . . . ,Zp)and H(⋅, ⋅,Zj). Finally, by the identity

2√−1(ω (dψ))

Z1,...,Zp−1

= −m

∑i=1(dψ)(Ei,Ei,Z1, . . . ,Zp−1)

= −m

∑i=1(∇C

Eiψ)(Ei,Z1, . . . ,Zp−1) = 2(δ∇C

ψ)(Z1, . . . ,Zp−1).we get

∆Cg ψ = − 2dδ∇C

ψ + 2(H ⋆ (dψ)).(5.16)

The claim follows by summing up (5.15) and (5.16) and using (5.14) for α = ψ and vector fields oftype (1,0) (noting that ψ is a (p,0)-form).

Lemma 5.10. Let ωφ = ddcJφ be a toric Kähler metric on (M,J), described by its toric Kähler

potential φ on Rm, associated to a toric generalized Kähler structure (gφ, Iφ, J) corresponding to

A = 0,B ∈ ∧2t. Then, for a variation φs = φ0 + sφ of φ one has

∂

∂s∣s=0

log det ((Hess(φs))−1 +√−1B)−1 =∆Cgφφ.

Proof. We denote by H ∶= Hess(φ(y)) = (Hess(u(x)))−1 and notice that

(H−1 +√−1B)−1 =X +√−1Ywith X = (H−1 +BHB)−1, Y = −HBX. Thus, we have

δφ( log det (H−1 +√−1B))(φ) = −tr((X +√−1Y)H−1HH−1)

= −tr(H−1XH−1Hess(φ))

= −tr([H[H−1 +BHB]H]−1Hess(φ))= −tr([H +HBHBH]−1Hess(φ)).

Notice that for any T-invariant smooth function f on M , viewed as a function f(y) in the holomor-

phic coordinates (yj +√−1θj), we have

ddcJf(y) =

m

∑i,j=1(Hess(f(y)))

ijdyi ∧ dθj.

whereas the inverse of ωφ is computed from (4.18) to be

(ωφ)−1 = − m

∑i,j=1

Xij( ∂∂yi∧

∂

∂θj) + 1

2Yij( ∂

∂yi∧

∂

∂yj+∂

∂θi∧

∂

∂θj)

with

X = [H +HBHBH]−1, Y = BHX.

We thus get

∆Cgφf = tr([H +HBHBH]−1Hess(f(y)))

and the claim follows.

Lemma 5.11. Let φt be a solution of (5.11). Then one has

∂

∂tϕt =∆C

gtϕt + 2ϕt − 2∣∣bt∣∣2gt .(5.17)


Proof. Notice that by Proposition 3.3, ϕt(y) = φt(y) − φ0(y) extends to a smooth function on M ,

thus so is φt = ϕt. For the second part, we take derivative with respect to t in (5.9) and useLemma 5.10 to compute

∂

∂tφt = ∆C

gt φt + 2tr[BtHtBtXt] + 2φt= ∆C

gt φt + 2m − 2tr[H−1t Xt] + 2φt= ∆C

gt φt + 2m − 2tr[Id +HtBtHtBt]−1 + 2φt,where we have set Bt = e−2tB0, and Ht and Xt are the objects defined in the proof of Lemma 5.10with respect to φt.

As FtJ = −gt − bt, we have

∣∣Ft∣∣2gt = 1

2∣∣FtJ ∣∣2gt =m + ∣∣bt∣∣2gt ,

noting that, by our convention, the norm of a 2-form is half the norm of the corresponding tensor.Using that Ft = −2gt(J + It)−1 we obtain

2m + 2∣∣bt∣∣2gt = 4∣∣(J + It)−1∣∣2gt = −4tr(It + J)−2whereas by (4.10) (recalling that A = 0 and H =Ψ−1 in our case) we compute

(J + It)2 ∼ −4( Id +BtHtBtHt 0

0 Id +BtHtBtHt) ,

so that

m + ∣∣bt∣∣2gt = tr[Id +BtHtBtHt]−1.The claim follows.

Using Lemma 5.9, we can refine the result of Lemma 2.11, giving a useful evolution equationfor the (2,0) piece of b, which yields a key differential inequality.

Proposition 5.12. Let (M2n, gt, It, J) be a solution to GKRF with symplectic-type initial data, andset βJ = π∧2,0

J

b. Then

∂

∂tβJ = ∆C

gtβJ ,

∂

∂t∣βJ ∣2 = ∆C

gt∣βJ ∣2 − ∣∇βJ ∣ − ∣∂βJ ∣2 − ∣βJ H2,1∣2 .

Proof. We apply Lemma 5.9 to βJ . In this case, H = db and dβ =H2,1, so that the term (H⋆dβ)Z1,Z2

is zero. By Lemma 2.7, the term bH is exact, and thus (d(H ⋆β))Z1,Z2= (d(bH))Z1,Z2

= 0. Weconclude that

∆Cg βJ = (∆gβJ − Lθ♯

JβJ)2,0.

As ∆g −Lθ♯J

is a real operator, it follows from Lemma 2.11 that the evolution equation of βt under

the GKRF is∂

∂tβJ =∆C

gtβJ ,

which is the formula obtained in [70], noticing that in this context βJ = ∂α and b = R(∂α) in the

notation of [70]. The evolution of ∣βJ ∣2 then follows from the established evolution for βJ and aBochner formula (cf. [71] Lemma 4.7).


5.3. Long-time existence. In this subsection we give the proof of Theorem 1.1. We first derive anestimate for the tensor b which holds without any symmetry hypotheses. Next we derive two key apriori estimates for the potential φ potential in the toric setting. Using these and further maximumprinciple arguments we prove the long time existence, giving a sharper statement yielding the natureof the estimates.

Proposition 5.13. Let (M2n, gt, It, J) be a solution to GKRF with symplectic-type initial datadefined on [0, T ). Then

supM×[0,T )

∣βJ ∣2 ≤ supM×0

∣βJ ∣2 .Proof. This follows by applying the maximum principle to the second relation in Proposition 5.12.

Proposition 5.14. Let ϕt denote a solution to (5.11) on a toric Fano manifold. There exists aconstant C > 0 so that

supM×t

(∣ϕ∣ + ∣ϕ∣) ≤ Ce2t.Proof. Appying the maximum principle directly to (5.17) yields the upper bound φ ≤ Ce2t. ByProposition 5.13), we have ∣∣bt∣∣2gt ≤ C. Using this and applying the maximum principle to (5.17)

yield the lower bound ϕ ⩾ −Ce2t. Using the estimate of ϕ the bound for ϕ follows by integration intime.

Proof of Theorem 1.1. By Proposition 4.5 the given invariant data is equivalent to a triple (u0,A0,B0).By Corollary 5.5 the solution to NGKRF is described by a triple (ut, e−2tA0, e

−2tB0), where ut isa solution to (5.1). Taking Legendre transform, by Proposition 5.6 this solution is equivalentlydescribed by ϕt which solves (5.11). We will prove the long time existence of the flow ϕt which inturn proves the long time existence of (1.2) by the discussion above.

Equation (5.11) is strictly parabolic for ϕ, thus the short-time existence is guaranteed bygeneral theory. To establish global existence, we first aim to establish uniform parabolicity of theequation, which amounts to uniform equivalence of the time-varying metrics along the flow. Wewill first show an upper bound for the metric, then estimate the determinant to obtain the lowerbound. To obtain the upper bound we first recall that by [6, Lemma 6.9], we have

( ∂∂t−∆C

gt) log trω0

ωt ≤ ∣∣T ct ∣∣2gt +C trωt(ω0),(5.18)

where T c is the torsion of the Chern connection. To control the two terms on the right hand side ofthis inequality, we will augment our test function with two auxiliary terms. We first observe thatby Proposition 5.12, using that db =H and dropping some negative terms we have

( ∂∂t−∆C

gt) ∣bt∣2gt ≤ − ∣T c∣2gt .(5.19)

Furthermore, observe that

( ∂∂t−∆C

gt)ϕ = ϕ − trωt dd

cJϕ

= ϕ − trωt ωt + trωt ω0.

(5.20)

Notice that

trωt(ωt) = tr [Id +HtBtHtBt]−1 =m + ∣bt∣2gt ≤ C,


where the last inequality follows from Proposition 5.13. Using this together with Proposition 5.14in (5.20) yields

−C + trωt ω0 ≤ ( ∂∂t−∆C

gt)φ ≤ C + trωt ω0(5.21)

Now set

Ψ1 ∶= log trω0ωt +A(∣bt∣2gt +ϕ).

For A chosen sufficiently large, it follows from (5.18, 5.19, and 5.21) that

( ∂∂t−∆C

gt)Ψ1 ≤ CA.

Applying the maximum principle, and using that ∣φ∣ is already bounded by Proposition 5.14 weconclude that for any T > 0 there is a constant C so that

supM×[0,T ]

trω0ωt ≤ C.

To obtain the lower bound we get a lower bound for the volume form. First, by [6, Lemma 6.7], wehave

( ∂∂t−∆C

gt) log ωmt

ωm0

⩾ −C trωt ω0.(5.22)

Now set

Ψ2 = log ωmt

ωm0

+Aϕ,

By (5.21) and (5.22) we obtain

( ∂∂t−∆C

gt)Ψ2 ⩾ −C.

Applying the maximum principle establishes the lower bound for the determinant. It follows thatC−1g0 ≤ gt ≤ Cg0 on any finite time interval. As ∣bt∣2gt is bounded, we can now apply [61, Theorem 1.2]to obtain the higher order regularity and hence the long time existence.

6. Monotone functionals and behavior at infinity

In this section we analyze the behavior of the normalized GKRF at infinity. First we derivean extension of Perelman’s shrinker entropy to the GKRF (without toric symmetry), and use it toderive the uniform κ-noncollapsing estimate and convergence of nonsingular solutions of Theorem1.2. Next we derive an extension of the Mabuchi K-energy to the GK setting, and show that it ismonotone along the GKRF with toric symmetry. This monotonicity is the key point behind thatconvergence statements of Theorem 1.3.

6.1. Entropy monotonicity. In this subsection we observe that a modification of Perelman’sentropy functional [65] is monotone along generalized Kähler-Ricci flow for structures of symplectictype. Our discussion is a direct adaptation of [39, Chapter 6], (cf. also [75]). There the monotonicityformulae and applications are contingent upon a further monotone quantity dubbed a torsion-bounding subsolution. In the setting of generalized Kähler structures of symplectic type the torsionpotential b plays this role, and we make this explicit below

Definition 6.1. Let (M2m, g, I, J) be a generalized Kähler structure of symplectic type. Fixf ∈ C∞(M,R), τ∗ > 0, and denote τ = τ∗ − t. The shrinker entropy associated to this data is

W(g,H, f, τ) ∶= ∫M[τ (∣∇f ∣2 +R − 1

12∣H ∣2) − ∣b∣2 + f − 2m] (4πτ)−me−fdV.


Furthermore, let

µ (g,H, τ) ∶= inff ∣ ∫M(4πτ)−me−fdV =1

W (g,H, f, τ) ,and

ν(g,H) ∶= supτ>0

µ− (g,H, τ) .Proposition 6.2. Let (M2m, gt, It, J) be a solution to generalized Kähler-Ricci flow on a compactmanifold such that (g0, I0, J) is symplectic type. Then, setting τ = τ∗ − t for some τ∗ > 0 we have

d

dtW(g,H, f, τ)= ∫

M[2τ ∣Rc−1

4H2+∇

2f −1

2τg∣2 + τ

6∣d∗H + i∇fH ∣2 + 5

6∣H ∣2] (4πτ)−me−fdV.

Proof. This follows immediately by combining the results of ([39] Proposition 6.26) noting that ∣b∣2is a ‘torsion-bounding subsolution’ via Proposition 5.12 and the fact that ∣∂βJ ∣2 = ∣T c∣2 = ∣H ∣2 byDefinitions 2.1 and 2.2.

Remark 6.3. Proposition 6.2 shows that the only self-similar solutions of the normalized GKRFwith symplectic-type initial data are actually Kähler-Ricci solitons. In contrast to the shrinkingcase, there exist non-Kähler steady solitons for GKRF [68, 77, 78].

Proof of Theorem 1.2. Using the entropy monotonicity of Proposition 6.2 the uniform κ-noncollapsingestimate follows from Perelman’s original argument ([65], cf. [39] Theorem 6.29). Now assume wehave a solution to the normalized flow on [0,∞) which satisfies a uniform curvature estimate. Bythe uniform κ-noncollapsing it follows that the volumes of unit balls are uniformly bounded belowalong the flow, and there is a uniform lower bound on the injectivity radius. Since the volume formis bounded above pointwise by Fm

t , the total volume is bounded above by ∫M Fmt = (2πc1(M,J))m.

It follows that there is a uniform upper bound for the diameter along the flow. With these estimatesin place, it follows from Cheeger-Gromov compactness theory (cf. [55] Theorem 2.3 for our simpli-fied case) that for any sequence tj →∞ there exists a subsequence, still denoted tj, such that(M,gtj ) converges in Cheeger-Gromov sense to a smooth limiting Riemannian manifold (M,g∞),where the limiting manifold is still M due to the diameter upper bound. Also, it follows from ([72],[39] Ch. 5) that H and all its covariant derivatives are uniformly bounded along the flow, and thusby choosing a further subsequence H can be assumed to converge to H∞. We claim that by choosinga further subsequence the generalized Kähler structure also converges, which requires establishingC∞ estimates for I and J . First, the g-norms of I and J are uniformly bounded since g is compatiblewith both. By a standard argument using the fact that the Bismut ∇B,I preserves I, and that thecoefficients of this connection are already converging by the discussion above, it follows that thereare uniform C∞ estimates for I, and the same argument using ∇B,J yields estimates for J . It followsthat we can choose a further subsequence such that (M,gtj , Itj , J) converges in Cheeger-Gromovsense to a limiting generalized Kähler manifold (M,g∞, I∞, J∞).

Now note that by rescaling the result of Proposition 6.2 holds for the normalized flow, settingτ ≡ 1. This also implies that the corresponding functional µ is monotone increasing. Since thesolution is nonsingular, there are uniform bounds on µ and all of its derivatives. It thus follows thatfor an arbitrary sequences of times tj→∞ one has

limj→∞∣ ddtµ(gtj ,Htj)∣ = 0.

For such a sequence tj, let (M,g∞, I∞, J∞) denote the generalized Kähler limit constructed asabove. It follows from the entropy monotonicity formula that the limiting torsion H∞ must vanish,


and furthermore g∞ is a shrinking soliton. Since H∞ vanishes, the pairs (g∞, I∞) and (g∞, J∞) areboth Kähler; this shows that the symplectic 2-form F∞ taming I∞ and J∞ is parallel with respectto g∞, and hence must be a (1,1)-form by the Fano condition of (M,J∞). It then follows that wemust have I∞ = J∞.

6.2. The extended Mabuchi functional. In this section we give an extension of the Mabuchifunctional to the setting of GK structures of symplectic type on a Kähler background. To beginwe first observe that it is possible to canonically associate a Kähler metric to every symplectic-typeGK structure on a Kähler manifold. First, using Hodge decomposition, the deRham class α = [F ]defines an element α ∈ H1,1(M,R). Using that F tames J and a result of Demailly-Paun [27], itfollows that α is a Kähler class of (M,J). By Yau’s theorem [88], we can find a unique Kählermetric ω ∈ α with the property

(6.1) ω[m] = cF [m], c = α[m]

[F ][m] .Definition 6.4. On a Kähler background (M,J), for a GK structure (g, I, J) we shall refer to theunique Kähler metric ω satisfying (6.1) as the Kähler reduction of (g, I, J). Fixing a backgroundKähler metric ω0, we refer to the Kähler reduction of a given F as ω = ωϕ = ω0 + dd

cJϕ, where we

recall, the smooth function ϕ is defined only up to an additive constant.

Remark 6.5. Kähler reductions exist in smooth families along a solution to GKRF of symplectictype. In general it might not be possible to derive an explicit formula for the evolution of ω, but inthe toric Fano case this is what is achieved in Proposition 5.6.

We now recall the definitions of the Aubin-Mabuchi functionals I and Iρ0 which are defined

on the space of ω0-relative Kähler potentials ϕ by

(δϕI)(ϕ) = ∫Mϕω[m]ϕ , I(0) = 0,

(δϕIρ0)(ϕ) = ∫Mϕρ0 ∧ ω

[m−1]ϕ , I

ρ0(0) = 0(6.2)

where ρ0 is the Ricci form of ω0. As ρ0 ∈ 2πc1(M,J), letting a ∶= 4π c1(M,J)⋅[α][m−1][α][m] the expression

aI(ϕ) − 2Iρ0(ϕ) does not change if we add to ϕ a real constant, i.e. it introduces a functional,denoted by (aI − 2Iρ0) (ωϕ), acting on the Kähler metrics ωϕ ∈ α.

Definition 6.6. For F ∈GK(M,J,α) we define the extended Mabuchi energy by

M(F ) ∶= ( αm

[F ]m)∫M⎡⎢⎢⎢⎢⎣log⎛⎝F [m]

ω[m]0

⎞⎠ + log

⎛⎝F [m]

ω[m]J

⎞⎠⎤⎥⎥⎥⎥⎦F [m] + (aI − 2Iρ0) (ωϕ),

a ∶= 4πc1(M,J) ⋅ α[m−1]α[m]

,

(6.3)

Remark 6.7. In the case when α = α is a Kähler class, one has that ω[m]ϕ = F [m] ≥ ω[m]J . This

implies the basic inequality

(6.4) M(F ) ≥M(ωϕ),where M(ωϕ) is the usual Mabuchi energy of the Kähler metric ωϕ ∈ α, see e.g. [17].

6.3. The extended Mabuchi energy in the toric case. On a compact smooth toric Kählermanifold (M,J,ω0,T) with Delzant polytope (P,L), Donaldson [31] found concise expressions forthe Futaki invariant and the Mabuchi energy in terms of (P,L) and the space S(P,L). We use thispoint of view in order to express the extended Mabuchi energy as a functional acting on the data


(u,A,B) associated to (P,L). Following [31], we introduce a linear functional defined on the spaceof continuous functions on P by

(6.5) F(f) ∶= −a∫Pf(x)dx + 2∫

∂Pf(x)dσL, a ∶= 2Vol (∂P,dσL)

Vol(P,dx) ,

where, in the above expression, dx is a fixed Lebesgue measure on t∗ (associated with a chosen basis

of t) and dσL is the induced measure by dx and the inward normal dLj on each facet Pj ⊂ ∂P.If we take f(x) = ⟨ξ, x⟩ + λ to be affine-linear, (6.5) computes, up to a dimensional multiplicativeconstant, the Futaki invariant of the holomorphic vector JKξ on (M,J, [ω0]).Lemma 6.8. Let (g, I, J) be a toric generalized Kähler structure, corresponding to the data (u,A,B)with respect to ω0. Then

1

(2π)mM(F ) = F(u) − ∫Plog det(Hess(u) +√−1B)dx +∫

Plog det (Hess(u0))dx,

where u0 is the symplectic potential of a background toric Kähler metric.

Proof. On a toric manifold H2(M,C) = H1,1(M,C), so we are in the case α = α with F [m] = ω[m]ϕ .By (6.6) and using (4.22), we compute

M(F ) =M(ωϕ) + (2π)m ∫Plog ( detHess(u)

det(Hess(u) +√−1B))dx.The claim follows by the above, taking in mind that ωϕ = dd

cJAφ(y), where φ is the Legendre

transform of u ∈ S(P,L) (see (3.7)) and y are the pluriharmonic coordinates of JA (see (4.7)), andthe fact that (see [31])

M(ωϕ) = (2π)m (F(u) − ∫Plog det(Hess(u))dx + ∫

Plog det(Hess(u0))dx) .

We next derive fundamental variational properties for the extended Mabuchi energy. Recallour expression for the metric

X =R[Hess(u) +√−1B]−1.It turns out that the quantity

(6.6) κ(u,A,B) ∶= − m

∑i.j=1

Xij,ij

is identified in [15, 87] with the momentum map for the action of the group of T-equivariant Hamil-tonian symplectomorphisms of (M,ω0) on the space of compatible toric generalized Kähler struc-tures, and thus κ extends the notion of scalar curvature to the generalized Kähler context, viathe momentum map picture of [30, 37]. We call κ the generalized scalar curvature. This point ofview is extended to generalized Kähler manifolds of symplectic type (without toric symmetry) in[45]. We thus conclude that the critical points of M correspond to toric GK structures for whichGscal(g) = κ(u,A,B) = a.Proposition 6.9. Given a one-parameter family of symplectic potentials us = u + su and fixedmatrices A,B, we obtain

(δuM)(u) = ∫Pu (κ(u,A,B) − a)dx.


In particular, the critical points of M on the space of toric generalized Kähler structures of symplectictype on (M,J) with fixed cohomology class and holomorphic Poisson tensor have constant generalizedKähler scalar curvature. Furthermore,

(δ2uM)(u, u) = ∫Ptr[((Hess(u) +√−1B)−1Hess(u))2]dx.

Proof. These follow by observing that the linearization of the entropy term is

∫Ptr[(Hess(u) +√−1B)−1 Hess(u)]dx = ∫

Ptr(X Hess(u))dx.

Using that F is linear, and differentiating one more time the above expression yields the formula forthe Hessian of M. To obtain the first formula, we need to integrate by parts the RHS of the above

expression as in [31]. To this end, we observe that X = [Hess(u) +B(Hess(u))−1B]−1 is a smoothsymmetric matrix on P, which satisfies the first order boundary conditions of [4, Prop. 1] (see alsoRemark 4 in that reference). The second variation follows easily by passing another s derivativethrough the variation formula above and using that u = 0.

Remark 6.10. By the convexity property of M established in Lemma 6.9 (and using that S(P,L)is linearly convex) for given A,B, there exists at most one, up to an addition with an affine-linearfunction, u ∈ S(P,L) which satisfies κ(u,A,B) = a. One can also show that for any u ∈ S(P,L),the corresponding X is a smooth, matrix-valued function on P which satisfies the positivity andboundary conditions of [4, Prop. 1], and thus (6.6) gives rise to an almost Kähler structure withconstant scalar curvature in the sense of [63]. The necessary condition for the existence of such u

is the one of uniform relative K-stability of (P,L) (see [31, 18, 91]) which implies that there existconstants λ, δ > 0 such that

(6.7) F(u) ≥ λ∫P∣u∣dx − δ, ∀u ∈ S(P,L),

where u(x) = u(x) + ℓ(x) for an affine-linear function ℓ(x) is the unique normalization of u, i.e.which satisfies u(x) ≥ u(x0) = 0 for a fixed point x0 in the interior of P.

Conversely, it is now established (as a consequence of deep recent work by Chen-Cheng [19]and previous work by Zhou-Zhu [91]) that any smooth toric variety with uniform relative K-stableDelzant polytope (P,L) in the sense of (6.7) admits a compatible toric Kähler metric of constantscalar curvatue, thus showing that the existence of toric generalized Kähler structures of constantgeneralized Kähler scalar curvature is equivalent to the existence of Kähler ones.

Lastly in this subsection we establish the monotonicity of the extended Mabuchi energy alongNGKRF in the toric Fano setting. We build this up through a series of lemmas.

Lemma 6.11. On a toric Fano manifold (M,J,T), along the flow (5.11)

d

dtM(Fϕt) = [−∫

M∣dϕt∣2ωϕt

F [m]ϕt+ ∫

M(sBJ (gϕt) − 2m)F [m]ϕt

] ,where sBJ (gϕt) denotes the Bismut scalar curvature of the pluriclosed Hermitian structure (gϕt , J).Proof. To simplify the notation, we let ωt = ωϕt , Ft = Fϕt and ωt = ωϕt. Referring to (6.3) (withα = α and a = 2m according to our normalization), Fm

t = ωmt and using (5.11)-(5.12) (with c(t) ≡ 0),


we compute

d

dtM(Ft) = 2∫

Mtrωt (ρBJ (gt) − ωt)ω[m]t + ∫

M(ϕt − 2ϕt + 2h0)(ddcϕt) ∧ ω[m−1]t

+ 2m∫Mϕtω

[m]t − 2∫

Mϕtρ0 ∧ ω

[m−1]t

= ∫M(sBJ (gt) − 2m)ω[m]t − ∫

M∣dϕt∣2ωt

ω[m]t

− 2∫Mϕt(ωt − ω0) ∧ ω[m−1]t + 2∫

Mϕt(ρ0 − ω0) ∧ ω[m−1]t

+ 2m∫Mϕtω

[m]t − 2∫

Mϕtρ0 ∧ ω

[m−1]t

= ∫M(sBJ (gt) − 2m)ω[m]t − ∫

M∣dϕt∣2ωt

ω[m]t .

The claim follows by using ω[m]t = F [m]t again.

Lemma 6.12. Suppose (M,g, I, J) is a compact generalized Kähler manifold of symplectic type.Then

∫M[trωJ

(ρBJ ) − trF (ρBJ )]F [m] = −18∫M[∣d log det(I + J) − θI − θJ ∣2

g+ ∣θI − θJ ∣2g]F [m].

Proof. We denote by sBJ and sBI the Bismut-Ricci scalar curvatures of (g, J) and (g, I), respectively.By Proposition 2.9 and [60], we know that

ρBJ − ρBI = 1

2L(θ♯

J−θ♯

I)F, sBJ − s

BI = −δg(θJ − θI).

Using that F = −2g(I + J)−1 and the above formula, we compute

2 [trωJ(ρBJ ) − trF (ρBJ )] = sBJ − ⟨ρBJ , ωJ + ωI⟩g = 1

2sBJ − ⟨ρBJ , ωI⟩g

= 1

2(sBJ − sBI ) − 1

2⟨L(θ♯

J−θ♯

I)F,ωI⟩

g

= − 1

2δg(θJ − θI) − 1

2⟨L(θ♯

J−θ♯

I)F,ωI⟩

g.

We next use that, by (2.8),

L(θ♯J−θ♯

I)F = d (F (θ♯J − θ♯I)) = d (I(θJ − θI) + Ib(θ♯J − θ♯I)) ,

so we continue the computation

2 [trωJ(ρBJ ) − trF (ρBJ )] = −1

2δg(θJ − θI) − 1

2⟨d (I ((θJ − θI) + b(θ♯J − θ♯I))) , ωI⟩

g

= 1

2(δg (b(θ♯J − θ♯I)) + ⟨θJ − θI + b(θ♯J − θ♯I), θI⟩g) ,

where we have used the identity

⟨dIα,ωI⟩g = −δgα − ⟨α, θI⟩g.We next use the identity (which follows from Lemma 2.7)

d log det(I + J) = θI + θJ + b(θ♯J − θ♯I)in order to express the terms containing b(θ♯

J− θ♯

I). We thus get

2 [trωJ(ρBJ ) − trF (ρBJ )] = 1

2(−∆g log det(I + J) − δg(θI + θJ) + ⟨ − 2θI + d log det(I + J), θI⟩g)

= 1

2(−∆g log det(I + J) − δg(θI + θJ) − 2∣θI ∣2g + ⟨d log det(I + J), θI⟩g) .


We finally use that (as F = −2g(I + J)−1)(6.8) F [m] = 2m det(I + J)− 1

2 ω[m]J

and integration by parts to get

∫M[trωJ

(ρBJ ) − trF (ρBJ )]F [m]= 2m−2 ∫

M(−∆g log det(I + J) + ⟨d log det(I + J), θI⟩g − δg(θI + θJ) − 2∣θI ∣2g)det(I + J)− 1

2 ω[m]J

= 2m−2 ∫M(−∆g log det(I + J) + ⟨d log det(I + J), θI⟩g − ∣θI ∣2g − ∣θJ ∣2g − 2δgθJ)det(I + J)− 1

2 ω[m]J

= 2m−2 ∫M[−1

2∣d log det(I + J)∣2

g+ ⟨d log det(I + J), θI + θJ⟩g − ∣θI ∣2g − ∣θJ ∣2g]det(I + J)− 1

2 ω[m]J

= −18∫M[∣d log det(I + J) − θI − θJ ∣2

g+ ∣θI − θJ ∣2g]F [m]

where we have used the relation δgθI + ∣θI ∣2g = ∣H ∣2g = δgθJ + ∣θJ ∣2g from [60].

Corollary 6.13. On a toric Fano manifold, the Mabuchi functional M(Fϕt) is monotone decreasingalong NGKRF.

Proof. This follows directly from Lemma 6.11 and Lemma 6.12, noting that trω(ρBJ (g)) = sBJ2

and

∫M

trF (ρBJ (g))F [m] = ∫MρBJ (g) ∧ F [m−1] =m

by the Fano condition and the normalization [F ] ∈ 2πc1(M,J).

6.4. Weak convergence. We suppose through this section that (M,J,T) is a smooth toric Fanomanifold, ω0 ∈ 2πc1(M,J) is an initial Kähler form with corresponding barycentered Fano polytope(P,L), and we further assume that the Futaki invariant of (M,J, [ω0]) vanishes. In this case,by [84], (M,J) admits a Kähler-Einstein metric and, by [24, 79], the Mabuchi energy is coerciverelative to the connected component of the identity Aut0(M,J) of the group of automorphisms of(M,J). We first recall what this precisely means. Denote by

H(M,ω0) = ϕ ∈ C∞(M) ∣ωϕ ∶= ω0 + ddcϕ > 0

the space of ω0-relative Kähler potentials on M , and consider the Aubin-Mabuchi functional I onH(M,ω0), defined in (6.2). Notice that I(ϕ+ c) = I(ϕ)+ cVol(M,ω0), so we can introduce the slice

H(M,ω0) ∶=H(M,ω0) ∩ I−1(0),and consider the induced action of G ∶= Aut0(M,J) on H(M,ω0) via the natural action of G on

the space of T-invariant Kähler metrics in [ω0] by pullbacks, i.e. for any ϕ ∈ H(M,ω0) and any

τ ∈ G, we let τ[ϕ] denote the uniquely determined ω0-relative Kähler potential in H(M,ω0) of theKähler metric τ∗(ωϕ). We thus have

τ[ϕ] = τ[0] +ϕ τ.We next consider the d1-distance on H(M,ω0), introduced via the L1-length of a smooth

curve ϕ(t) ∈H(M,ω0):l1(ϕ(t)) = ∫ 1

0∫M∣ϕ(t)∣ (ωϕ(t))[m].

By the results in [22],

d1(ϕ0, ϕ1) ∶= inf l1(ϕ(t)) ∣ϕ(t, x) ∈ C∞([0,1] ×M), ϕ(t, ⋅) ∈H(M,ω0), ϕ(0) = ϕ0, ϕ(1) = ϕ1defines a distance. We then have


Theorem 6.14. [24] If (M,J, [ω0]) admits a Kähler-Einstein metric, then the Mabuchi energy M

is G-invariant and G-coercive on H(M,ω0), i.e. there are uniform constants λ > 0, δ such that

M(ϕ) ≥ λ infτ∈G

d1(0, τ[ϕ]) − δ, ∀ϕ ∈ H(M,ω0).An important ingredient in proving the above result is the weak compactness, established in

[9, 10]. To state it, we denote by PSH(M,ω0) the space of ω0-relative plurisubharmonic functionson M , i.e. the usc functions ϕ ∈ L1(M,ω0) such that ωϕ = ω0 + dd

cϕ ≥ 0 in the sense of currents.Furthermore, we denote by E(M,ω0) ⊂ PSH(M,ω0) the subspace of elements of full Monge-Ampèremass, i.e.

E(M,ω0) = ϕ ∈ PSH(M,ω0) ∣ ∫Mω[m]ϕ = ∫

Mω[m]0 = Vol(M,ω0).

By the results in [52, 14], each ϕ ∈ E(M,ω0) can be weakly approximated (as a current) by a sequence

of smooth functions (ϕj)j ∈ H(M,ω0) and for any such sequence limj→∞ω[m]ϕj= ω[m]ϕ in the sense

of measures. We can take the latter as a definition of ω[m]ϕ when ϕ ∈ E(M,ω0). Following [52], one

further introduces the sub-space of elements with full Monge-Ampère mass and finite energy

E1(M,ω0) = ϕ ∈ E(M,ω0) ∣ ∫

M∣ϕ∣ω[m]ϕ <∞.

The central fact [22] in this theory is that d1 extends to define a complete distance on E1(M,ω0), suchthat (E1(M,ω0), d1) is a complete geodesic space in which H(M,ω0) is densely embedded, and thed1-convergence on E1(M,ω0), d1) is stronger than both the L1(M,ω0) and the weak convergence of(1,1)-currents. Furthermore, M naturally extends to a continuous and lsc functional on E1(M,ω0),such that

Theorem 6.15. [9, 10] If (ϕj)j ∈ E1(M,ω0) is a sequence with

d1(0, ϕj) < C, M(ϕj) < Cthen there exists a subsequence (ϕjk)k which converges with respect to d1 to an element ϕ0 ∈E1(M,ω0).

In the toric case considered in this paper, we shall rather work with the subspace

HT(M,ω0) ∶=H(M,ω0) ∩C∞T (M)of T-invariant relative Kähler potentials, and consider instead of G = Aut0(M,J) the complex torusTC. The coercivity principle of [24, Theorem 3.4] still applies in the T-relative setting, noting thatT ⊂ Aut0(M,J) is maximal and TC acts transitively on the space of the T-invariant Kähler-Einsteinmetrics in [ω0] (see [12] for the toric case and [62, Thm. 2] for a more general setting), so we havethe following T-relative version of Theorem 6.14 (see [56, 91] or [3, Prop. 3.4] for the toric case, and[54, Thm. 1.6] for a more general statement):

Theorem 6.16. If (M,J, [ω0],T) admits a Kähler-Einstein metric, then the Mabuchi energy M ∶

HT(M,ω0)→ R is TC-invariant and TC-coercive, i.e. there are uniform constants λ > 0, δ such that

M(ϕ) ≥ λ infτ∈TC

d1(0, τ[ϕ]) − δ, ∀ϕ ∈ HT(M,ω0).We next use Theorem 6.16 in order to get a weak convergence of the global solution of NGKRF

in the toric Kähler-Einstein case.

Proof of Theorem 1.3. Let ϕt ∶= φt−φ0 ∈HT(M,ω0) be the relative Kähler potentials along the Käh-

ler reduction (5.11) of the normalized generalized Kähler Ricci flow, and ϕt ∈ HT(M,ω0) = I−1(0)the corresponding normalized ω0-relative Kähler potentials. Using Theorem 6.16, Corollary 6.13


and (6.4), we have along the Kähler reduction (5.11) of the normalized generalized Kähler Ricciflow,

λ infτ∈TC

d1(0, τ[ϕt]) ≤M(ωϕt) + δ ≤M(Fϕt) + δ ≤M(F0) + δ.Furthermore, by [24, Prop. 6.8] (using that TC is reductive), for each t ∈ [0,∞) there exists τt ∈ TC

such that

infτ∈TC

d1(0, τ[ϕt]) = d1(0, τt[ϕt]).We thus get uniform estimates

d1(0, τt[ϕt]) ≤ C, M(τt[ϕt]) =M(ϕt) < Cwhich, by Theorem 6.15 and the fact that E1

T(M,ω0) is d1-closed as a subset of E1(M,ω0) [24],

proves the first part of the theorem.For the second part, according to [7], on a symmetric toric Fano manifolds the Tian α-invariant

[79] satisfies αT(M,J) > mm+1

. However, the above estimate of αT(M,J) also yields (see e.g. [28,Theorem 1.4]) that the Mabuchi energy is strongly coercive, i.e. that there exist constants λ > 0, δsuch that

M(ωϕ) ≥ λd1(0, ϕ) − δ, ∀ϕ ∈HT(M,ω0),where we have used [24, Prop. 5.4] to connect with the definition of coercivity in terms of theJ-functional used in [28]. The same argument as above concludes the proof.

Remark 6.17. One would expect to further improve the unconditional weak convergence in Theo-rem 1.3, by showing that the limiting curent ω∞ is in fact a Kähler–Einstein metric. In the case ofthe normalized KRF on a Fano manifold with strongly coercive Mabuchi energy, such a conclusionis indeed achieved in [9] by further using the interplay between the Mabuchi and Ding functionals.(Similar arguments appear in [23] concerning the normalized Ricci iterations.)

In our Fano generalized Kähler setting, one can use the definition of the extended Mabuchienergy to also define an extension of the Ding functional as follows. For a generalized Kählerstructure (g, I, J) of symplectic type with symplectic form F ∈ 2πc1(M,J) and corresponding

Kähler reduction ω, we denote by hF the normalized ρBI -potential, defined by

ρBI (g) − ω = ddcJ hF , ∫

Me2hFF [m] = 1,

and introduce the corresponding Ding functional (see [35]) by the formula

(6.9) D(F ) ∶=M(F ) + 2 [∫MhFF

[m]− ∫

Mh0ω

[m]0 ] ,

where ω0 ∈ 2πc1(M,J) is a background Kähler metric with normalized Ricci potential h0, i.e.

ρ0 − ω0 = ddcJ h0, ∫M e2h0ω

[m]0 = 1. Using Proposition 2.10 and writing ω = ω0 + dd

cJϕ, one can

equivalently express D(F ) as

(6.10) D(F ) = −2I(ϕ) − V log⎛⎝∫M e−2ϕ

⎛⎝ω[m]J

F [m]⎞⎠µ0⎞⎠ , µ0 ∶= e2h0ω

[m]0 , V = vol(M,ω0).

When F = ωJ = ω is a Kähler structure, the description (6.10) reduces to the familiar expression ofthe Ding functional on a Fano Kähler manifold. Using Jensen’s inequality in (6.9) and the pointwise

inequality F [m] ≥ ω[m]J in (6.10), we have the inequalities

M(F ) ≥D(F ) + 2∫Mh0ω

[m]0 ≥D(ω) + 2∫

Mh0ω

[m]0 .

Beside these nice properties, and unlike the extended Mabuchi functional M, the behavior of D

along the normalized GKRF (5.11) on a toric Fano manifold is as yet unclear.


References

[1] M. Abreu, Kähler geometry of toric varieties and extremal metrics, Internat. J. Math 9 (1998), 641-651. pages12

[2] M. Abreu, Kähler metrics on toric orbifolds, J. Differential Geom. 58 (2001), 151-187. pages 3, 11, 14[3] V. Apostolov, The Kähler geometry of toric manifolds, Lecture Notes of CIRM winter school, 2019.

http://www.cirget.uqam.ca/∼apostolo/notes.html, pages 11, 39[4] V. Apostolov, D. M. J. Calderbank, P. Gauduchon and C. Tønnesen-Friedman, Hamiltonian 2-forms in Kähler

geometry II Global classification, J. Differential Geom. 68 (2004), 277-345. pages 11, 12, 36[5] V. Apostolov, P. Gauduchon, and G. Grantcharov, BiHermitian structures on complex surfaces, Proc. London

Math. Soc. (3) 79 (1999), 414-428. Corrigendum, 92 (2006), 200-202. pages 1, 2, 4, 6, 20[6] V. Apostolov and J. Streets, The nondegenerate generalized Kähler Calabi-Yau problem, arXiv:1703.08650, to

appear Crelle’s Journal. pages 1, 6, 21, 31, 32[7] V. Batyrev and E. Selivanova, Einstein-Kähler metrics on symmetric toric Fano manifolds, J. Reine Angew.

Math. 512 (1999), 225-236. pages 4, 40[8] R. J. Berman and D. Witt Nystrom. Complex optimal transport and the pluripotential theory of Kähler-Ricci

solitons. arXiv:1401.8264. pages 10[9] R. Berman, S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, Kähler–Einstein metrics and the Kähler–Ricci flow

on log Fano varieties, Journal für die reine und angewandte Mathematik, 751 (2019), 27-89. arXiv:1111.7158.pages 3, 39, 40

[10] R. J. Berman, T. Darvas, C. H. Lu, Convexity of the extended K-energy and the large time behaviour of the weak

Calabi flow, Geometry and Topology 21 (2017), 2945-2988. pages 39[11] F. Bischoff, M. Gualitieri, M. Zabzine, M. Gualtieri, Morita equivalence and the generalized Kähler potential,

arXiv:1804.05412, to appear in J. Differential Geom. pages 4, 20, 21[12] D. Guan, On modified Mabuchi functional and Mabuchi moduli space of Kähler metrics on toric bundles, Math.

Res. Let. 6 (1999), 547–555. pages 39[13] J.M. Bismut, A local index theorem for non Kähler manifolds, Math. Ann. 284 (1989), 681-699. pages 5[14] Z. Blocki and S. Kolodziej, On regularization of plurisubharmonic functions on manifolds. Proc. Amer. Math.

Soc., 135 (2007), 2089-2093. pages 39[15] L. Boulanger, Toric generalized Kähler structures, J. Symplectic Geom. 17 (2019), 973-1019. arXiv:1509.06785.

pages 2, 4, 15, 16, 35[16] H.D. Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent.

Math. 81 (1985), 359-372. pages 1, 10[17] X.X. Chen, The space of Kähler metrics, J. Differential Geom. 56 (2000), 189–234. pages 34[18] B. Chen, A.-M. Li and L. Sheng, Uniform K-stability for extremal metrics on toric varieties, Journal of Differ-

ential Equations 257 (2014), 1487–1500. pages 36[19] X.X. Chen and J. Cheng, On the constant scalar curvature Kähler metrics, existence results, arXiv:1801.006

pages 36[20] X.X. Chen, S. Sun, B. Wang, Kähler-Ricci flow, Kähler-Einstein metric, and K-stability, Geom. Topol. 22(6):

3145-3173 (2018). pages 2[21] T. Darvas, E. George and K. Smith, Optimal asymptotic of the J-functional with respect to the d1-metric,

arXiv:2101.02589. pages 4[22] T. Darvas, The Mabuchi geometry of finite energy class, Adv. Math. 285 (2015), 182–219. pages 38, 39[23] T. Darvas and Y. Rubinstein, Convergence of the Kähler-Ricci iteration, Analysis & PDE 12 (2019), 721-735.

pages 40[24] T. Darvas and Y. Rubinstein, Tian’s properness conjectures and Finsler geometry of the space of Khler metrics,

J. Amer. Math. Soc. 30 (2017), 347–387. pages 38, 39, 40[25] V. Datar, G. Székelyhidi, Kähler–Einstein metrics along the smooth continuity method. Geom. Funct. Anal. 26

(2016), 975–1010. pages 10[26] T. Delzant, Hamiltoniens périodiques et image convexe de l’application moment, Bull. Soc. Math. France 116

(1988), 315-339. pages 11, 14, 17[27] J.-P. Demailly and M. Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann.

of Math. 159 (2004), 1247-1274. pages 34[28] R. Dervan, Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds, Annales de la Faculté

des sciences de Toulouse : Mathématiques, Sér. 6, 25 (2016) 919–934. pages 40[29] R. Dervan, G. Székelyhidi, The Kähler-Ricci flow and optimal degenerations, J. Diff. Geom. 116(1): 187-203

(2020). pages 2[30] S. K. Donaldson, Remarks on gauge theory, complex geometry and 4-manifold topology, Fields Medallists lectures,

384–403, World Sci. Ser. 20th Century Math., 5, World Sci. Publishing, River Edge, NJ, 1997. pages 35

http://www.cirget.uqam.ca/~apostolo/notes.html

http://arxiv.org/abs/1703.08650







[31] S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), 289-349. pages34, 35, 36

[32] S. K. Donaldson, Interior estimates for solutions of Abreu’s equation, Collect. Math. 56 (2005) 103-142. pages11

[33] S.K. Donaldson, Extremal metrics on toric surfaces: a continuity method, J. Differential Geom. 79 (2008),389-432. pages 13

[34] S. K. Donaldson, Kähler geometry of toric manifolds, and some other manifolds with large symmetry. in “Hand-book of Geometric Analysis”, No. 1. Adv. Lect. Math. (ALM), vol. 7, pp. 29-75. International Press, Somerville(2008) pages 11, 15

[35] W. Y. Ding and G. Tian, The generalized Moser-Trudinger inequality, 1992, 57–70. pages 40[36] N. Enrietti, A. Fino, G. Grantcharov, Tamed symplectic forms and generalized geometry, J. Geom. Phys. 71

(2013), 103-116. pages 6[37] A. Fujiki, Moduli space of polarized algebraic manifolds and Kähler metrics, [translation of Sugaku 42 (1990),

231–243], Sugaku Expositions 5, no. 2 (1992), 173–191. pages 35[38] A. Futaki, Kähler–Einstein metrics and Integral Invariants, Lecture Notes in Mathematics, Springer-Verlag 1314,

1988. pages 10[39] M. Garcia-Fernandez, J. Streets Generalized Ricci Flow, AMS University Lecture Series 2021. pages 3, 7, 8, 9,

10, 32, 33[40] S. Gates, C. Hull, M. Rocek, Twisted multiplets and new supersymmetric non-linear σ-models, Nuclear Physics

B248 (1984) 157-186. pages 1[41] P. Gauduchon, Hermitian Connections and Dirac Operators, Boll. U. M. I. 11-B (1997), Supp. facs. 2, 257-288.

pages 5[42] J. P. Gauntlett, D. Martelli, J. Sparks, S.-T. Yau, Obstructions to the existence of Sasaki–Einstein metrics,

Comm. Math. Phys. 273 (2007), 803–827. pages 10

[43] Gibson, M., Streets, J. Deformation classes in generalized Khler geometry, Complex Manifolds, Volume 7, Issue1, 2020. pages 10

[44] R. Goto, Unobstructed K-deformations of generalized complex structures and bi-Hermitian structures. Adv. Math.231 (2012), 1041-1067. pages 2

[45] R. Goto, Scalar curvature as moment map in generalized Kähler geometry, arXiv:1612.08190. pages 35[46] M. Gualtieri, Branes on Poisson varieties. In “The Many facets of Geometry: A Tribute to Nigel Hitchin” (eds.

J.-P. Bourguignon, O. Garcia-Prada and S. Salamon), Oxford University Press, 2010. pages 2, 4, 6, 20, 21[47] M. Gualtieri, Generalized Kähler geometry, PhD Thesis, arXiv:1007.3485. pages 1, 6[48] M. Gualtieri, Generalized complex geometry, Ann. of Math. (2) 174 (2011), 75-123. pages 6[49] M. Gualtieri, Generalized Kähler geometry, Comm. Math. Phys. 331 (2014), 297-331. pages 2[50] M. Gualtieri, Generalized Kähler metrics from Hamiltonian deformations, Geometry and Physics: Volume II: A

Festschrift in honour of Nigel Hitchin (2018). pages 2[51] V. Guedj, The metric completion of the Riemannian space of Kähler metrics, arXiv:1401.7857. pages 4[52] V. Guedj, A. Zeriahi, The weighted Monge-Ampère energy of quasiplurisubharmonic functions, Journal of Func-

tional Analysis, 250 (2007), 442-482. pages 39[53] V. Guillemin, Kähler structures on toric varieties, J. Differential Geom. 40 (1994), 285-309. pages 3, 11, 12, 13[54] J. Han and C. Li, On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations,

arXiv:2006.00903. pages 39[55] R. Hamilton, A compactness property for solutions of the Ricci flow, Amer. J. Math., Jun., 1995, Vol. 117, No.

3, pp. 545-572. pages 33[56] T. Hisamoto, Stability and coercivity for toric polarizations, arXiv:1610.07998. pages 39[57] N. Hitchin, Generalized Calabi-Yau manifolds Q.J. Math., 54 (2003), 281-308. pages 1[58] N. Hitchin, Instantons, Poisson structures and generalized Kähler geometry, Comm. Math. Phys. 265 (2006),

131-164. pages 2[59] N. J. Hitchin, Bihermitian metrics on del Pezzo surfaces. J. Symplectic Geom. 5 (2007), 1-8. pages 2, 6[60] S. Ivanov and G. Papadopoulos, Vanishing theorems and string backgrounds, Class. Quantum Grav. 18 (2001),

1089–1110. pages 5, 37, 38[61] J. Jordan and J. Streets, On a Calabi-type estimate for pluriclosed flow, Adv. Math. 366 (2020), 107097 pages

3, 32[62] A. Lahdili, Convexity of the weighted Mabuchi functional and the uniqueness of weighted extremal metrics, arXiv:

2007.01345. pages 39[63] M. Lejmi, Extremal almost-Kähler metrics, Internat. J. Math. 21 (2010), 1639–1662. pages 36[64] E. Lerman and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer.

Math. Soc. 349 (1997), 4201-4230. pages 11, 12






[65] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159. pages32, 33

[66] Phong, D. H., Song, Jian, Sturm, Jacob, Weinkove, Ben On the convergence of the modified Kähler-Ricci flow

and solitons. Comment. Math. Helv. 86 (2011), no. 1, 91-112. pages 2[67] M. Pontecorvo, Complex structures on Riemannian 4-manifolds, Math. Ann. 309 (1997), 159-177. pages 2[68] J. Streets, Classification of solitons for pluriclosed flow on complex surfaces, Math. Ann. 375, 1555-1595 (2019).

pages 33[69] J. Streets, Generalized Kähler Ricci flow and the classification of nondegenerate generalized Kähler surfaces,

Adv. Math. Vol 316, 20 August 2017, 187-215. pages 1, 6[70] J. Streets, Pluriclosed flow, Born-Infeld geometry, and rigidity results for generalized Kähler manifolds, Comm.

PDE., 41 (2016), 318-374. pages 3, 30[71] J. Streets, Pluriclosed flow on generalized Kähler manifolds with split tangent bundle, J. für die reine und Ang.

Math. 739 (2018), 241-276. pages 1, 30[72] J. Streets, Regularity and expanding entropy for connection Ricci flow, J. Geom. Phys. Vol 58, Issue 7, July 2008,

pp. 900-912. pages 33[73] J. Streets, G. Tian, Hermitian curvature flow, J. Eur. Math. Soc. 13 (2011), 601-634. pages 8[74] J. Streets, G. Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Notices 2010 (2010), 3101-3133.

pages 8[75] J. Streets, G. Tian, Regularity results for the pluriclosed flow, Geom. & Top. 17 (2013) 2389–2429. pages 32[76] J. Streets, G. Tian, Generalized Kähler geometry and the pluriclosed flow, Nuc. Phys. B, 858 (2012) 366–376.

pages 1, 8[77] J. Streets, J., Y. Ustinovskiy, Classification of generalized Kähler-Ricci solitons on complex surfaces,

https://arxiv.org/abs/1907.03819, to appear Comm. Pure. Appl. Math. pages 33[78] J. Streets, J., Y. Ustinovskiy, The Gibbons-Hawking ansatz in generalized Kähler geometry,

https://arxiv.org/abs/2009.00778. pages 33[79] G. Tian, On Kähler-Einstein metrics on certain Kähler manifolds with C1(M) > 0, Invent. Math. 89 (1987),

225–246. pages 38, 40[80] G. Tian, S. Zhang, Z. Zhang, X. Zhu, Perelman’s entropy and Kähler-Ricci flow on a Fano manifold, Trans.

Amer. Math. Soc., 365 (12):6669-6695, 2013. pages 2[81] G. Tian, Z. Zhang, Regularity of Kähler-Ricci flows on Fano manifolds, Acta Math. 216(1): 127-176 (2016).

pages 2[82] G. Tian and X. Zhu, Convergence of Kähler Ricci Flow, Journal Amer. Math. Soc. 20 (2007), 675-699. pages 2,

10, 15[83] G. Tian and X. Zhu, Convergence of the Kähler–Ricci flow on Fano manifolds, J. reine angew. Math. 678 (2013),

223–245. pages 2, 10, 15[84] X.-J. Wang and X. Zhu, Kähler Ricci solitons on toric manifolds with positive first Chern class, Adv. Math. 188

(2004), 87-103. pages 15, 38[85] Y. Wang, Toric generalized Kähler structures I, arXiv:1810.08265. pages 2, 4, 15, 16, 18[86] Y. Wang, Toric generalized Kähler structures II, arXiv:1811.06848. pages 2, 4, 15, 18[87] Y. Wang, Toric generalized Kähler structures III, to appear in Journal of Geometry and Physics. pages 35[88] S.T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, Comm.

Pure Appl. Math. 31 (1978), 339-411. pages 34[89] X. Zhu, Kähler Ricci soliton type equation on compact complex manifolds with c1(M) > 0, J. Geom. Anal. 10

(2000), 759-774. pages 15[90] X. Zhu, Kähler Ricci flow on a toric manifold with positive first Chern class. in “Differential geometry”, pp.

323-336, Adv. Lect. Math. (ALM), 22, Int. Press, Somerville, MA, 2012. pages 3, 15[91] B. Zhou, X. Zhu, Relative K-stability and modified K-energy on toric manifolds, Adv. Math. 219 (2008), 1327–

1362. pages 36, 39

Vestislav Apostolov, Départment de matheématiques, Université du Québec à Montréal, Casepostale 8888, succursale centre-ville Mongtréal (Québec) H3C 3P8

Email address: [email protected]

Jeffrey Streets, Rowland Hall, University of California, Irvine, CA 92617Email address: [email protected]

Yury Ustinovskiy, Courant Institute of Mathematical Sciences, New York University, 251Mercer Street, New York, NY, 10012-1185

Email address: [email protected]



mailto:[email protected]



Documents

arXiv:2104.03268v1 [math.DG] 7 Apr 2021