Contents - sebastien.boucksom.perso.math.cnrs.frsebastien.boucksom.perso.math.cnrs.fr/notes/L2.pdf · 7. The Ohsawa-Takegoshi extension theorem 26 8. Blow-ups and valuations 30 9

SINGULARITIES OF PLURISUBHARMONIC FUNCTIONS

AND MULTIPLIER IDEALS

SEBASTIEN BOUCKSOM

Contents

Introduction 11. Subharmonic functions 12. Plurisubharmonic functions 53. Regularization of quasi-psh functions 104. Basic analysis of first order differential operators 155. Fundamental identities of Kahler geometry 206. L2-estimates for the ∂-equation 237. The Ohsawa–Takegoshi extension theorem 278. Blowups and valuations 309. Fundamental properties of multiplier ideals 3510. Valuative description of multiplier ideals 41References 47

Bibliography 47

Introduction

These notes are based on second year Master course I delivered in Orsay in2015–2017.

1. Subharmonic functions

We use [DemAG] as a reference.

1.1. The Green-Riesz representation formula. Denote by

∆ :=

d∑i=1

∂2

∂x2i

Date: January 21, 2020.

1

2 SEBASTIEN BOUCKSOM

the usual (negative definite) Laplacian on Rd, and recall that a fundamentalsolution of ∆ is given by the Newton kernel

(1.1) Nd(x) =

1

2|x| for d = 1;

1

2πlog |x| for d = 2;

1

(2− d)|Sd−1||x|2−d for d ≥ 3,

where|Sd−1| = d|Bd| = dπd/2/(d/2)!

is the area of the unit sphere. Observe that Nr(x) only depends on r = |x|, with

∂

∂rNd =

1

|Sd−1|rd−1.

Given a smoothly bounded domain Ω b Rd and f ∈ C∞(Ω), v ∈ C∞(∂Ω), ,the Dirichlet problem

(1.2)

∆u = f

u|∂Ω = v

admits a unique solution u ∈ C∞(Ω) (uniqueness being a simple consequence ofthe maximum principle). The resulting isomorphism

C∞(Ω)⊕ C∞(∂Ω) ' C∞(Ω)

gives rise to two kernels, the Green kernel

GΩ : Ω× Ω→ [−∞, 0]

and the Poisson kernelPΩ : Ω× ∂Ω→ R

with the property that

(1.3) u(x) =

ˆΩGΩ(x, y)∆u(y) +

ˆ∂ΩPΩ(x, y)u(y)dσ(y)

for all u ∈ C∞(Ω), with dσ(y) the Lebesgue measure ∂Ω. The Green kernel GΩ

is the unique solution of

(1.4)

∆xGΩ(x, y) = δy

GΩ(·, y)|∂Ω ≡ 0.

It is symmetric in (x, y), smooth outside the diagonal of Ω× Ω, and satisfies

GΩ(x, y) = Nd(x− y) mod C∞

on Ω × Ω. The Poisson kernel PΩ is smooth, and PΩ(x, y)dσ(y) is a probabilitymeasure on ∂Ω for all x ∈ ∂Ω.

Example 1.1. The Poisson kernel of the open ball B(0, r) is given by

Pr(x, y) =1

|Sd−1|rr2 − |x|2

|x− y|d.

SINGULARITIES AND MULTIPLIER IDEALS 3

The unique solution of (1.4) on B(0, r) with y = 0 is given by

Gr(x, 0) = Nd(x)−Nd(re1) =1

|Sd−1|

ˆ |x|r

dt

td−1.

Injecting this in (1.3) and translating at a given a ∈ Rd, we get the fundamnetalGauss-Jensen formula.

Corollary 1.2. For each u ∈ C∞(B(a, r)), the difference between u(a) and themean value

fflS(a,r) u on the sphere S(a, r) is expressed as

(1.5)

S(a,r)

u− u(a) =1

|Sd−1|

ˆ r

0

dt

td−1

ˆB(a,t)

∆u.

1.2. Subharmonic functions. In what follows, Ω ⊂ Rd denotes any open sub-set.

Definition 1.3. A function u : Ω→ [−∞,+∞) is subharmonic if:

(a) u is usc, not identically −∞ on any connected component of Ω;(b) for each ball B(a, r) ⊂ Ω, the mean value inequality u(a) ≤

fflS(a,r) u holds.

Note that u is locally bounded above by (a), so that´S(a,r) u dσ makes sense

in [−∞,+∞). Since

(1.6)

ˆB(a,r)

u(x)dx =

ˆ r

0dt

ˆS(a,t)

u(y)dσ(y),

(b) implies u(a) ≤fflB(a,r) u. Using this, it is easy to see that u is locally integrable.

Example 1.4. The Gauss-Jensen formula (1.5) directly shows that a smoothfunction u is subharmonic if and only if ∆u ≥ 0.

Proposition 1.5. Subharmonic functions satisfy the following properties.

(i) For any family (uα) of subharmonic functions on Ω, locally bounded above,the usc regularization (supα uα)? is subharmonic.

(ii) If u1, . . . , ur are subharmonic on Ω, then χ(u1, . . . , ur) is subharmonic forany convex function χ on Rr that is non-decreasing in each variable.

(iii) If u is subharmonic on Ω and ν is a probability measure supported on acompact set K ⊂ Rd, then u ? ν is subharmonic on any open set U suchthat U +K ⊂ Ω.

Pick a nonnegative radial function ρ ∈ C∞c (Rd), compactly supported in the

unit ball and such that´Rd ρ(x)dx = 1, i.e. ρ(x) = ρ(|x|) with |Sd−1|

´ 10 ρ(r)rd−1dr =

1, and consider the associated approximation of unity

(1.7) ρε(x) = ε−dρ(ε−1x)

We then have the following key monotone regularization result.

Theorem 1.6. Let u be subharmonic function on Ω. Then u ? ρε is smooth andsubharmonic on Ωε = x ∈ Ω | d(x, ∂Ω) > ε, and decreases to u as ε→ 0.


Proof. Since ρε is an approximation of unity, u ? ρε is smooth on Ωε, u ? ρε → uin L1

loc, and lim supε→0 u ? ρε ≤ u, u being usc. Noting that

(1.8) (u ? ρε)(a) = |Sd−1|ˆ 1

0ρ(r)rd−1dr

S(a,εr)

u dσ,

the mean value inequality yields u ≤ u ? ρε. By Proposition 1.5, u ? ρε is subhar-monic, and it therefore remains to show that ε 7→ u? ρε is non-decreasing. Whenu is smooth, it follows from (1.8) and the Gauss-Jensen formula (Corollary 1.2),which shows that

fflS(a,r) u is a non-decreasing function of r. In the general case,

u ? ρδ is smooth and subharmonic on Ωδ for any δ > 0. By the smooth case,ε 7→ (u ? ρε) ? ρδ = (u ? ρδ) ? ρε is thus non-decreasing, and hence so is ε 7→ u ? ρεin the limit as δ → 0.

Corollary 1.7. If u is subharmonic, then ∆u ≥ 0 in the sense of distributions.Conversely, if v ∈ L1

loc(Ω) satisfies ∆v ≥ 0, then there exists a unique subhar-monic function u on Ω with u = v a.e.

Using the monotone regularization procedure of Theorem 1.6, one easily ex-tends the Green-Riesz representation formula as follows.

Corollary 1.8. Any subharmonic function u defined in a neighborhood of theclosure of a smooth bounded domain Ω b Rd is integrable on ∂Ω, and

u(x) =

ˆΩGΩ(x, y)(∆u)(y)dy +

ˆ∂ΩPΩ(x, y)u(y)dσ(y).

for all x ∈ Ω.

By the maximum principle, we have GΩ ≤ 0, which yields the generalizedsubmean value inequality

u(x) ≤ˆ∂ΩPΩ(x, y)u(y)dσ(y)

for x ∈ Ω. In particular, a subharmonic function u defined on a neighborhood ofB(0, 1) satisfies

u(x) ≤ 1

|Sd−1|

ˆS(0,1)

1− |x|2

|x− y|du(y)dσ(y),

for all x ∈ B(0, 1). After translation and scaling, this implies the following basicHarnack inequality.

Corollary 1.9. For any ε > 0, there exists a constant C = C(ε, d) such that

supB(a,(1−ε)r)

u ≤ C S(a,r)

u.

for any non-positive subharmonic function u ≤ 0 defined on a neighborhood of aball B(a, r).


1.3. Regularity properties. As we next show, every subharmonic function be-longs to the Sobolev space W 1,1+ε for 0 < ε 1.

Proposition 1.10. Let u be a subharmonic function on an open subset Ω ⊂ Rd.Then u ∈ Lploc for 1 ≤ p < d

d−2 and ∇u ∈ Lploc for 1 ≤ p < dd−1 .

When d = 2, e−cu is even locally integrable near any given a ∈ Ω for all c > 0small enough. More precisely, this holds as soon as c−1 > 1

π∆u(a).

Proof. A direct check shows that the Newton kernel Nd satisfies these properties.Let χ ∈ C∞c (Ω) be a cut-off function with χ ≡ 1 on a neighborhood U of a givena ∈ Ω, and consider the compactly supported positive measure µ := χ∆u. OnU , the function v−Nd ? µ is harmonic, and hence smooth, and it is thus enoughto show the result for Nd ? µ. For the Lp conditions, this follows directly fromLemma 1.11 below, by homogeneity of χ(t) = tp. Assume now that d = 2, andpick c > 0 with c−1 > 1

π∆u(a). We may then choose suppχ such that the total

mass m of µ satisfies (cm)−1 > 1π . Applying Lemma 1.11 with χ(t) = e−cmt and

ν = m−1µ, we get as desired that

exp(−(cm)Nd ? (m−1µ)

)= exp(−cNd ? µ)

is integrable.

Lemma 1.11. Let χ : [0,+∞) → [0,+∞) be a convex, nondecreasing function,f a measurable function on Rd, ν a compactly supported probability measure onRd. Then ˆ

Rdχ(|f ? ν|) dx ≤

ˆRdχ(|f |) dx.

Proof. Since χ is non-decreasing and convex, Jensen’s inequality yields

χ (|f ? ν|(x)) ≤ χ(ˆ|f(x− y)|dν(y)

)≤ˆχ (|f(x− y)|) dν(y),

and henceˆχ(|f ? ν|(x))dx ≤

ˆdν(y)

ˆχ(|f(x− y)|)dx =

ˆχ(|f |)(x)dx.

2. Plurisubharmonic functions

2.1. First properties. Let Ω ⊂ Cn be an open subset.

Definition 2.1. A function u : Ω → [−∞,+∞) is plurisubharmonic (psh forshort) if u is usc, not identically −∞ on any connected component of Ω, and therestriction of u to each complex line is subharmonic, i.e.

u(a) ≤ 1

2π

ˆ 2π

0u(a+ ξeiθ)dθ

for all a ∈ Ω and ξ ∈ Cn such that a+ zξ | z ∈ D ⊂ Ω.

Proposition 2.2. Plurisubharmonic functions satisfy the following properties.

(i) If u is psh, then it is also subharmonic as a function of 2n real variables.


(ii) If u is smooth, then u is psh iff its complex Hessian(∂2u/∂zj∂zk

)1≤j,k≤n

is pointwise semipositive. Equivalently, i∂∂u ≥ 0.(iii) For any family (uα) of psh functions on Ω, locally bounded above, the usc

regularization (supα uα)? is psh.(iv) If u1, . . . , ur are psh on Ω, then χ(u1, . . . , ur) is psh for any convex func-

tion χ on Rr that is non-decreasing in each variable.(v) If u is psh on Ω and ν is a probability measure supported on a compact

set K ⊂ Rd, then u ? ν is subharmonic on any open set U such thatU +K ⊂ Ω.

(vi) If u is psh and ρε is a radially symmetric approximation of unity as in(1.7), then u ? ρε is smooth, psh, and decreases pointwise to u as ε 0.

(vi) if u is psh on Ω and f : Ω′ → Ω is holomorphic on an open subset of Cp,then u f is psh, or u f ≡ −∞ on some component of Ω′.

The above properties follow from those already established for subharmonicfunctions, except for the key new property (vi). The latter is true in the smoothcase by (ii), and the general case follows by regularization. As a result, pshfunctions make sense on any complex manifold.

2.2. Convexity properties. The class of psh functions contains that of convexfunctions, as follows:

Lemma 2.3. Let ω ⊂ Rn be a convex open subset. A function f on ω is thenconvex if and only if f(x+ iy) is a psh function of z = x+ iy ∈ ω + iRn.

Proof. This follows from the Hessian criterion when f is smooth, and the generalcase is obtained by regularizing f .

As a consequence, we get the following generalized Hadamard three-circle the-orem:

Corollary 2.4. Let u be any psh function on an open subset Ω of Cn and a ∈ Ω.Let also ρr(z) = r−2nρ(r−1z) be a radial approximation of unity. The followingquantities are then convex as functions of log r:

(i) supB(a,r) u;

(ii)fflS(a,r) u;

(iii)fflB(a,r) u;

(iv) (u ? ρr)(a).

Proof. After scaling, we assume for simplicity that B(a, 1) is contained in Ω, sothat f(t) = supB(a,et) u and g(t) =

fflS(a,et) u are both defined on R−. By (1.6)

and (1.8),fflB(a,et) u and (u ? ρet)(a) are both of the form

(2.1)

ˆR+

g(t− s)dµ(s)

with µ a probability measure on R+, and it is thus enough to show that f andg are convex. For ξ ∈ B(0, 1), z 7→ u(a + ezξ) is a subharmonic function on the


half-plane z ∈ C | Re z < 0, and hence so are

supB(a,ez)

u = supξ∈B(0,1)

u(a+ ezξ)

and S(a,ez)

u =1

|S2n−1|

ˆS2n−1

u(a+ ezξ)dσ(ξ).

Being independent of Im z, they are therefore convex functions of Re z.

When u is continuous, the various means of u considered in Corollary 2.4all converge to u(a) as r → 0. As observed in [BK07, Lemma 5], they remainequivalent as r → 0 when u is merely locally bounded:

Lemma 2.5. Assume further that u is locally bounded in Corollary 2.4. Then thedifference between any two of the quantities (i)—(iv), as well as those obtainedby replacing r with by cr for some c > 0, all tend to 0 as r → 0, locally uniformlywith respect to a ∈ Ω:

This fact will later allow us to easily compare the regularization by convolu-tion in various coordinate charts, which is the key point in the Blocki–Kolodziejextension of the Richberg regularization procedure, cf. §3.2.

Proof. Using the above notation, we first claim that there exists C > 0 such that

(2.2) 0 ≤ f(t)− g(t) ≤ C (f(t)− f(t− 1)) .

for all t ∈ R−. Indeed, u−f(t) is a non-positive subharmonic function on B(a, et),and the Harnack inequality of Corollary 1.9 therefore shows that

g(t)− f(t) =

S(a,et)

(u− f(t))

≥ C supB(a,et−1)

(u− f(t)) = C(f(t− 1)− f(t)),

which proves the claim. By convexity of f , we have for all t ∈ R− and s > 0

f(t)− f(t− s)s

≤ f(0)− f(t− s)−t+ s

.

Since u is locally bounded, we infer

(2.3) 0 ≤ f(t)− f(t− δ) ≤ C s

−t+ s

for some constant C > 0, which can be chosen locally uniformly with respect a.Combined with (2.2), this shows that

0 ≤ f(t)− g(t) ≤ C

−t+ 1,

and hence

f(t) = g(t) + o(1) = f(t− s) + o(1)


as t → −∞. We have seen thatfflB(a,et) u and (u ? ρet)(a) are both of the form´

R+g(t−s)dµ(s) with µ a probability measure on R+. The above estimates yield

0 ≤ f(t)−ˆR+

g(t− s)dµ(s) ≤ CˆR+

αt(s)dµ(s).

with αt(s) := s−t+s + 1

−t+s+1 . Since 0 ≤ αt(s) ≤ 2 and limt→−∞ αt(s) = 0 for all

s ∈ R+, dominated convergence yields f(t)−´R+g(t− s)dµ(s)→ 0 for t→ −∞,

and we are done.

2.3. Lelong numbers. In what follows, u still denotes a psh function on anopen subset Ω ⊂ Cn. Pick a ∈ Ω and r > 0 with B(a, r) ⊂ Ω. By convexity off(t) := supB(a,et) u on (−∞, log r],

f(t)− f(log r)

t− log r≥ 0

is a non-decreasing function of t, and f(t)/t therefore admits a limit in R+ ast→ −∞. In other words,

νa(u) := limr→0

supB(a,r) u

log r

exists is R+, and is called the Lelong number of u at a. By convexity of f , wefurther have

f(t)− f(log r) ≤ (t− log r)

(lim

t′→−∞

f(t′)− f(log r)

t′ − log r

)= (t− log r)νa(u).

for t ≤ log r. In other words,

(2.4) u(z) ≤ νa(u) log|z − a|r

+ supB(a,r)

u

for all z ∈ B(a, r). It follows that

(2.5) νa(u) = maxγ ∈ R+ | u(z) ≤ γ log |z − a|+O(1) near a.which intuitively says that νa(u) is the ’vanishing order’ at a of eu, and hencemeasures the ’size’ of the singularity of u at a. Using the estimate (2.4), it is easyto establish:

Lemma 2.6. The Lelong number νa(u) is usc with respect to a ∈ Ω, and alsowith respect the weak (equivalently, L1

loc) topology of psh functions. In particular,if (uj) is an increasing sequence of psh functions converging a.e. to a psh functionu, then νa(uj) νa(u).

While a locally bounded psh function u has zero Lelong numbers, the converseis far from true.

Example 2.7. For each convex, non-decreasing function χ : R− → R, u(z) =χ(log |z|) is subharmonic on the unit disc D ⊂ C, with

ν0(u) = χ′(−∞) = limt→−∞

χ(t)

t.

If χ(t) = − log(−t) or (−t)α with 0 < α < 1, then u(0) = −∞ but ν0(u) = 0.


Recall from Corollary 2.4 thatfflS(a,r) u is a convex function of log r.

Lemma 2.8. For each psh function u on Ω and a ∈ Ω, we have

νa(u) = limr→0

fflS(a,r) u

log r.

Proof. The Harnack inequality of Corollary 1.9 shows that

(2.6) 0 ≤

fflS(a,r) u− supB(a,r) u

log r≤ C

supB(a,r/2) u− supB(a,r) u

log r

for 0 < r 1. The right-hand side clearly tends to 0, hence the result.

Remark 2.9. Arguing as in the proof of Lemma 2.5, one similarly checks that

νa(u) = limr→0

fflB(a,r) u

log r= lim

r→0

(u ? ρr)(a)

log r.

Corollary 2.10. For any two psh functions u, v we have

νa(maxu, v) = minνa(u), νa(v)and

νa(u+ v) = νa(u) + νa(v).

The next result provides an alternative characterization of νa(u) as a limitdensity. Set ω := i

2∂∂|z|2, which is the Kahler form associated to the Euclidian

metric on Cn. The Lebesgue measure of Cn is then equal to ωn/n!. Since u is

psh, T := iπ∂∂u is a closed positive (1, 1)-current, and σT := T ∧ ωn−1

(n−1)! is thus a

positive measure on Ω, called the trace measure of T . The choice of normalizationfor T comes from the following special case:

Example 2.11. When u = log |z1|, the trace measure σT = σH coincides withthe Lebesgue measure on the hyperplane H = (z1 = 0). For n = 1, this amountsto the well-known formula i

π∂∂ log |z| = δ0. Pulling-back by the projection

(z1, . . . , zn) 7→ z1 shows that iπ∂∂ log |z1| is the integration current δH on H

(a special case of the Lelong-Poincare formula), and σH = δH ∧ ωn−1

(n−1)! is thus the

Lebesgue measure on H.

We are now in a position to decribe νa(u) as a ratio of trace measures:

Proposition 2.12. If u is a psh function and T = iπ∂∂u, then

σT (B(a, r))

σH (B(0, r)) νa(u)

as r 0.

Proof. Set f(r) := σT (B(a, r)) /σH (B(0, r)). Since ωn/n! is the Lebesgue mea-sure on Cn, the Gauss-Jensen formula of Corollary 1.2 yields

S(a,r)u− u(a) =

1

|S2n−1|

ˆ r

0

dt

t2n−1

ˆB(a,t)

∆uωn

n!.


Using

f(r) =(n− 1)!

(πr2)n−1

ˆB(a,r)

T ∧ ωn−1

(n− 1)!=

1

πn−1r2n−2

ˆB(a,r)

T ∧ ωn−1,

i∂∂u ∧ ωn−1

(n− 1)!=

1

2∆u

ωn

n!,

and |S2n−1| = 2n|B2n| = 2πn/(n− 1)!, we get

(2.7)

S(a,r)

u− u(a) =

ˆ r

0f(t)

dt

t.

SincefflS(a,r) u is a convex function of log r, it follows that f is non-decreasing,

with

limr→0

f(r) = limr→0

fflS(a,r) u

log r,

and we conclude by Lemma 2.8.

Remark 2.13. When n = 1, Proposition 2.12 says that νa(u) = 12π∆u(a), and

Proposition 1.10 can thus be rewritten as

νa(u) < 1 =⇒ e−u ∈ L2loc near a.

The converse is also true, and follows immediately from (10.1). By a nontrivialresult originally due to Skoda, the above implication remains valid for all n,cf. Corollary 9.18.

3. Regularization of quasi-psh functions

3.1. Quasi-psh functions and singular metrics. As mentioned above, pshfunctions make sense on any complex manifold X.

Definition 3.1. A usc function ϕ : X → [−∞,+∞) on a complex manifold Xis quasi-psh if it is locally of the form ϕ = u+ f where u is psh and f is smooth.

It will be convenient to introduce the real operator ddc := iπ∂∂ (the role of the

normalization factor π will become apparent below). By the Poincare lemma,any closed, real (1, 1)-form θ on X is locally of the form θ = ddcf for a smoothreal-valued function f , called a local potential of θ. In particular, θ is a Kahlerform if and only if its local potentials are strictly psh.

Definition 3.2. A quasi-psh function ϕ is θ-psh if θ + ddcϕ ≥ 0 in the sense ofcurrents; equivalently, f + ϕ is (locally) psh for any choice of local potentials ffor θ.

Let now L be a Hermitian holomorphic line bundle on X. The curvature formof L is the closed, real (1, 1)-form θ locally defined by θ := −ddc log |τ | for anylocal trivializing (holomorphic) section τ of L. This (1, 1)-form is indeed globallywell-defined, as any other local trivializing section is of the form τ ′ = uτ withu ∈ O?X , and ddc log |τ ′| = ddc log |τ | since log |u| pluriharmonic.

This definition of the curvature form θ is simply a concrete description of thecurvature form of the Chern connection of L, normalized so that θ represents


the first Chern class of L in de Rham cohomology, i.e. the image in H2(X,R) ofc1(L) ∈ H2(X,Z).

Any other Hermitian metric on L is of the form | · |′ = | · |e−ϕ with ϕ ∈ C∞(X),and the curvature forms are related by θ′ = θ + ddcϕ. The data of a θ-pshfunction ϕ can thus be interpreted as that of a singular metric | · |e−ϕ on L withsemipositive curvature current θ + ddcϕ.

3.2. Local regularization of quasi-psh functions. Local regularization ofquasi-psh functions is easily achieved by convolution with a smoothing kernel.As above, choose a radially symmetric, nonnegative function ρ ∈ C∞c (Cn) suchthat

Ćn ρ = 1, and consider the corresponding approximation of unity (ρj),

defined by

ρj(z) := j2nρ(jz).

Theorem 3.3. Let X be a complex manifold equipped with a reference positive(1, 1)-form ω. Let θ be a closed, real (1, 1)-form, and ϕ be a θ-psh function onX. Pick local holomorphic coordinates z centered at a given point a ∈ X, anddenote by ϕ?z ρj the (locally defined) smooth function obtained by convolution inthese coordinates. Then the following holds:

(i) ϕ ?z ρj + cj decreases pointwise to ϕ for some sequence cj → 0+;(ii) θ + ddc (ϕ ?z ρj) ≥ −εjω for some sequence εj → 0+;(iii) if ϕ is further locally bounded and z′ is any other local coordinate system

centered at a, then ϕ ?z ρj − ϕ ?z′ ρj tends to 0 locally uniformly.

Proof. Let f be a local potential of θ near a. Since u := f + ϕ is psh, Proposi-tion 2.2 shows that u ?z ρj is psh and decreases pointwise to u. As f is smooth,f ?z ρj → f in C∞ topology. In particular, f ?z ρj → f locally uniformly, and

−εjω ≤ ddc (f ?z ρj − f) ≤ εjω

with εj → 0+, which proves (i) and (ii). Choose another local coordinate systemz′ centered at a, and denote by B(a, r) and B′(a, r) the corresponding Euclidianballs. Since z and z′ are locally Lipschitz equivalent, we have

B(a,C−1ε) ⊂ B′(a, ε) ⊂ B(a,Cε)

for some locally uniform constant C > 0, and hence

supB(a,C−1ε)

u ≤ supB′(a,ε)

u ≤ supB(a,Cε)

u.

By Lemma 2.5, we have

(u ?z′ ρε)(a) = supB′(a,ε)

u+ o(1)

and

(u ?z ρε)(a) = supB(a,C−1ε)

u+ o(1) = supB(a,Cε)

u+ o(1)

locally uniformly with respect to a. As a result, u ?z ρε − u ?z′ ρε → 0 locallyuniformly. By continuity of f , we also have f ?z ρε−f ?z′ ρε → 0 locally uniformly,hence (iii).


3.3. Global regularization of bounded quasi-psh functions. A classicalresult of Richberg [Ric68] (see also [DemAG, §5.E]) states that a continuousθ-psh ϕ on a complex manifold X can be written as the locally uniform limitof a sequence of smooth functions ϕj ∈ C∞(X) such that the negative part ofθ + ddcϕj tends to 0 on compact sets.

The following extension to the case where ϕ is merely locally bounded wasinitially obtained as a consequence of deep regularization results due to Demailly[Dem92], before Blocki and Kolodziej realized in [BK07] that it can in fact beproved along the lines of the Richberg argument. The importance of passing fromthe continuous to the locally bounded will be illustrated by Theorem 3.8 below.

Theorem 3.4. Let X be a complex manifold, equipped with a reference positive(1, 1)-form ω. Given a locally bounded quasi-psh function ϕ on X, there existsa sequence ϕj ∈ C∞(X) with the following properties:

(i) ϕj ϕ pointwise on X;(ii) for each closed, real (1, 1)-form θ such that θ+ddcϕ ≥ 0 and each relatively

compact open U b X, there exists εj → 0+ such that θ + ddcϕj ≥ −εjωon U .

The proof presented below combines [DemAG, §5.E] and [BK07]. Note firstthat the local boundedness assumption cannot be dropped:

Example 3.5. Let E ⊂ X be the exceptional divisor of the blowup of a pointin a compact Kahler surface, and recall that the cohomology class [E] ∈ H2(X)satisfies [E]2 = −1. Pick a Hermitian metric | · | on the line bundle O(E), withcurvature form θ, and let s ∈ H0(X,O(E)) be the canonical section, so thatϕ := log |s| is θ-psh. Assume by contradiction that ϕ can be written as the weaklimit of smooth functions ϕj ∈ C∞(X) such that θ + ddcϕj ≥ −εjω for someKahler form ω on X and εj → 0. Then

0 ≤ˆX

(θ + ddcϕj + εjω)2 = ([E] + εj [ω])2 → [E]2 = −1,

a contradiction.

The Richberg patching technique relies on the use of regularized max functions.

Definition 3.6. Given a non-negative function ρ ∈ C∞(R) with supp ρ ⊂ [0, 1],´R ρ(t)dt = 1 and an r-tuple of positive numbers δ = (δ1, . . . , δr), the regularized

max function maxδx1, . . . xr is defined as the convolution product

maxδx1, . . . , xr =

ˆt∈[0,1]r

maxx1 + δ1t1, . . . , xr + δrtrρ(t1) . . . ρ(tr)dt1 . . . dtr.

The following properties are trivially satisfied:

(i) maxδ is smooth and convex on Rr, non-decreasing in each variable, and

maxδx1 + c, . . . , xr + c = maxδx1, . . . , xr+ c

for all x1, . . . , xr, c ∈ R.(ii) If xi + δi ≤ maxj 6=i(xj − δj), then

maxδx1, . . . , xr = maxδx1, . . . , xi, . . . , xr.


(iii) maxx1, . . . , xr ≤ maxδx1, . . . , xp ≤ maxx1 + δ1, . . . , xr + δr.By (i) and Lemma 3.7 below, maxδ(ϕ1, . . . , ϕr) is thus θ-psh for any choice ofθ-psh functions ϕi.

Lemma 3.7. Suppose that χ : Rr → R is convex, non-decreasing in each variableand satisfies χ(t1 + c, . . . , tr + c) = χ(t1, . . . , tr) + c for all c ∈ R. If ϕ1, . . . , ϕrare θ-psh functions, then so is χ(ϕ1, . . . , ϕr).

Proof. Let f be a local potential of θ. Since f+ϕj is psh, Proposition 2.2 impliesthat

χ(f + ϕ1, . . . , f + ϕr) = f + χ(ϕ1, . . . , ϕr)

is psh, and the result follows.

Proof of Theorem 3.4. Choose a locally finite cover of X by coordinate charts(Vα), and open subsets U ′α b Uα b Vα with (U ′α) still covering K. For each α, thelocal regularization result of Theorem 3.3 yields a sequence of smooth functionsϕα,j defined on a neighborhood of Uα such that

(i) ϕα,j ϕ;(i) for every closed real (1, 1)-form θ with θ+ddcϕ ≥ 0, we have θ+ddcϕα,j ≥−εα,jω with εα,j → 0+;

(ii) supUα∩Uβ |ϕα,j − ϕβ,j | ≤ δα,j with δα,j 0+.

For each α, choose a cut-off function χα ∈ C∞c (Uα) with 0 ≤ χα ≤ 1 and χα ≡ 1

on a neighborhood of U′α, and set for each z ∈ X

ϕj(z) := maxδα,jϕα,j(z) + 2δα,jχα(z)

with α ranging over all indices such that z ∈ Uα. The monotonicity properties ofthe regularized max then imply that

ϕ(z) ≤ ϕj+1(z) ≤ ϕj(z) ≤ maxϕα,j(z) + 2δα,jχα(z) + δα,j | z ∈ Uα,

and hence ϕj(z) ϕ(z). Now pick z0 ∈ X and α0 ∈ A with z0 ∈ U ′α0, and set

V := U ′α0\

⋃z0 /∈Uα

Uα ∪⋃

z0∈∂Uα

suppχα

.

Using the basic properties of regularized max functions, a simple check showsthat for all z ∈ V we have

ϕj(z) = maxδα,jϕα,j(z) + 2δα,jχα(z),

where α now ranges over the fixed set of indices such that z0 ∈ Uα. As a firstconsequence, ϕj is smooth on X. For each compact K ⊂ X, denote by AK ⊂ Athe (finite) set indices such that Vα intersects K, and choose CK > 0 such thatddcχα ≥ −CKω near K for all α ∈ AK . If θ+ ddcϕ ≥ 0 for a given θ, Lemma 3.7yields

θ + ddcϕj ≥ −( minα∈AK

εα,j + 2CK minα∈AK

δα,j)ω

near K, and we are done.


3.4. Global regularization of unbounded quasi-psh functions. We nowshow how to get rid of the local boundedness condition in Theorem 3.4, underadequate extra assumptions. This is only briefly mentioned in [BK07], and weprovide some details.

Recall that a complex manifold X is strongly pseudoconvex if it admits asmooth, strictly psh exhaustion function τ : X → R. If X is Stein, it admitsa closed embedding in some CN , and the restriction to X of |z|2 yields a strictlypsh exhaustion function. The Levi problem asked for the converse, and was pos-itively answered by Grauert in 1958. A proof using L2-estimates for ∂ will beprovided below.

Theorem 3.8. Let ϕ be a quasi-psh function on a complex manifold X, andassume given finitely many closed, real (1, 1)-forms θα such that θα + ddcϕ ≥ 0for all α. Suppose either that X is strongly pseudoconvex, or that θα > 0 for allα. Then we can find a sequence ϕj ∈ C∞(X) with the following properties:

(i) ϕj converges pointwise to ϕ;(ii) for each relatively compact open subset U b X, there exists jU 1 such

that the sequence (ϕj) becomes decreasing with θα+ddcϕj > 0 for j ≥ jU .

Proof. Suppose first that θα > 0 for all α. By Lemma 3.7, the locally boundedfunctions ψj := maxϕ,−j satisfy θα + ddcψj ≥ 0 for all α, j. Note that ψjis generally not continuous, which is why the improvement from continuous tolocally bounded functions in Richberg’s theorem is so crucial.

Choose next an exhaustion of X by compact subsets Kj ⊂ Kj+1. We are goingto construct by induction on j ≥ 1 a sequence ϕj ∈ C∞(X) such that

(i) on Kj+1, ϕj > ψj and θα + ddcϕj > 0 for all α;(ii) on Kj , ϕj ≤ ϕj−1 and ‖ϕj − ψj‖L1(Kj) ≤ 1/j.

Since ψj → ϕ in L1loc, (ϕj) will then satisfy the conditions of the theorem. Start

with any smooth function ψ0 > maxϕ, 0, and assume that ψ0, . . . , ψk−1 havebeen constructed. Let ψj,k ∈ C∞(X) be a sequence obtained by applying Theo-rem 3.4 to the locally bounded quasi-psh function ψj , i.e. ψj,k ψj and

(3.1) θα + ddcψj,k > −εkθα

on Kj+1 for some sequence εk 0 depending on j (but not on α), by positivityof θα. On Kj , we have by induction

limkψj,k = ψj ≤ ψj−1 < ϕj−1,

and hence ψj,k ≤ ϕj−1 − ηj on Kj for some constant ηj > 0 and all k 1, byDini’s lemma. Set δk := 1− (1 + εk)

−1 and

ϕj,k := (1− δk)ψj,k + δk(1 + supKj+1

ψj).

On Kj+1, we have

ϕj,k ≥ (1− δk)ψj + δk(1 + supKj+1

ψj) > ψj ,


and θα+ddcϕj,k > 0 by (3.1). Since ψj,k ≤ ϕj−1−ηj on Kj , we have ϕj,k < ϕj−1

on Kj for k 1, and also ‖ϕj,k−ψj‖L1(Kj) ≤ 1/j since ϕj,k → ψj in L1loc. Setting

ϕj := ϕj,k with k 1 thus completes the inductive step.Assume finally that X is strongly pseudoconvex, with θα not necessarily pos-

itive anymore. Given a smooth, strictly psh exhaustion function τ : X → R, wecan find a positive continuous function f : X → R+ such that f ddcτ + θα > 0for all α. Choosing a smooth, convex, non-decreasing function χ : R → R withfast enough growth to guarantee that χ′′(c) ≥ supτ=c f for all c ∈ R, we get

ddcχ(τ) = χ′′(τ)ddcτ + χ′(τ)dτ ∧ dcτ ≥ f ddcτ.

For each α, θα := θα + ddcχ(τ) is thus positive with θα + ddc[ϕ− χ(τ)] ≥ 0. Thefirst case treated above yields a sequence ϕi ∈ C∞(X) converging pointwise to

ϕ − χ(τ), and which becomes decreasing with θα + ddcϕ′j > 0 for j 1 on any

given relatively compact open subset. Setting ϕj := ϕj + χ(τ) yields the desiredsequence for ϕ.

4. Basic analysis of first order differential operators

4.1. Differential operators. We use [Dem02, §2] as a reference for what follows.Let M be a smooth manifold and E,F be smooth, complex vector bundles onM . A linear differential operator of order (at most) δ is a C-linear map A :C∞(M,E) → C∞(M,F ) such that for any coordinate chart in which E,F aretrivialized, we have

(4.1) A =∑

α∈Nd,|α|≤δ

aαDα

with aα ∈ Hom(E,F ). More invariantly, the space of differential operator oforder δ can be written as

Dδ(M ;E,F ) = C∞(M,Hom(JδE,F ))

with JδE = JδM ⊗ E the vector bundle of δ-jets of sections of E. The jet exactsequence

0→ SδT ?M → JdM → Jδ−1M → 0

gives rise to an isomorphism

Dδ(M ;E,F )/Dδ−1(M ;E,F ) ' C∞(M,SδTM ⊗Hom(E,F )).

The image σA ∈ C∞(M,SδTM ⊗ Hom(E,F )) of A ∈ Dδ(M ;E,F ) is called theprincipal symbol of A, and is more concretely described as the Hom(E,F )-valuedhomogeneous form of degree δ on T ?M

σA(ξ) =∑|α|=δ

aαξα,

when A is locally written in the form (4.1).

Proposition 4.1. Let A : C∞(M,E) → C∞(M,F ) be a differential operator oforder δ.


(i) If A′ : C∞(M,F )→ C∞(M,G) is a differential operator of order δ′, thenA′A is a differential operator of order δ′+δ, and for each ξ ∈ T ?M we have

σA′A(ξ) = σA′(ξ)σA(ξ).

(ii) For δ = 1, we may view σA is an Hom(E,F )-valued vector field, and wehave the commutator relation

(4.2) [A, f ]u := A(fu)− fAu = σA(df)u = (σA · f)u

for each f ∈ C∞(M) and u ∈ C∞(M,E).

In particular, the space of order 1 differential operators E → F with a given(degree 1) symbol is an affine space modeled on the space of bundle homomor-phisms.

Example 4.2. A connection ∇ on E induces a differential operator of order 1

D∇ : C∞(M,Λ•T ?M ⊗ E)→ C∞(M,Λ•T ?M ⊗ E),

with principal symbol ξ ∧ •.

4.2. Adjoint. Now assume given a smooth volume form dV on M and Hermitianscalar products on E and F . If A : C∞(M,E) → C∞(M,F ) is a differentialoperator of order δ, its formal adjoint A? : C∞(M,F ) → C∞(M,E) is uniquelydetermined by the requirement

(4.3) 〈Au, v〉L2 = 〈u,A?v〉L2

for all u ∈ C∞(M,E), v ∈ C∞(M,F ) such that u (or v) has compact support.

Lemma 4.3. Any differential operator A ∈ Dδ(M ;E,F ) admits a formal adjointA? ∈ Dδ(M ;F,E). Further, its principal symbol is given by

σA? = (−1)δσ?A.

Further, given a section v ∈ C∞(M,F ), the value of A?v at a given point onlydepends on the δ-jet of dV and of the metrics of E and F .

Proof. By linearity and a partition of unity, we may assume that E,F are trivialand A = aDα with a ∈ Hom(E,F ). Denote by hE , hF the matrices representingthe scalar product of E, F and write dV = fdx with f > 0. Then

〈Au, v〉L2 =

ˆ〈aDαu, v〉FdV =

ˆ〈Dαu, a?v〉EdV

=

ˆDα(tu)hEa?vfdx = (−1)d

ˆtuDα

(hEa?vf

)dx,

and hence

A?v = (−1)δ th−1E Dα

(thEa

?vf)f−1 = (−1)δa?Dαv + l.o.t.


4.3. Friedrich’s lemma for complete metrics. A differential operator A ∈Dδ(M ;E,F ) acts in a natural way on distributional sections, and (4.3) remainsvalid when u is a distribution and v ∈ C∞c . In particular, A uniquely extendsto a densely defined, closed operator (i.e. with closed graph) A : L2(M,E) 99KL2(M,F ), with u ∈ DomA iff Au ∈ L2. However, in general, (4.3) does not holdfor all u ∈ DomA, v ∈ DomA? (i.e. DomA? does not coincide with the domainof the Hilbert space adjoint of A).

Example 4.4. Let I ⊂ R be a bounded open interval endowed with the Euclidianmetric, and A = d/dx. Then A? = −A, DomA = DomA? is the Sobolev spaceW 1,2(I) ⊂ C0(I), and 〈Au, v〉 = 〈u,A?v〉+ [uv]∂I for u, v ∈W 1,2.

Theorem 4.5. If the Riemannian metric g of M is complete, then any u ∈L2(M,E) can be written as the L2-limit of a sequence of test sections uj ∈C∞c (M,E) with the following property: for each first order linear differentialoperator A ∈ D1(M ;E,F ) with bounded symbol σA, Au ∈ L2(M,F ) impliesAuj → Au in L2.

The regularizing sequence uj will be obtained by truncation and local convo-lution.

Lemma 4.6. Let (M, g) be a Riemannian manifold, possibly with boundary, fixp ∈ [1,∞], and consider the following conditions:

(i) M admits a smooth exhaustion function ψ : M → R with dψ ∈ Lp;(ii) there exists an exhaustion by compact sets Kj ⊂ Kj+1 b M and a se-

quence of cut-off functions χj ∈ C∞c (Kj+1) with 0 ≤ χj ≤ 1, χj ≡ 1 on aneighborhood of Kj, such that ‖dχj‖Lp → 0;

(iii) there exists Kj, χj as in (ii) with dχj bounded in Lp.

Then (i)=⇒(ii)=⇒(iii); when p > 1, we conversely have (iii)=⇒(i). For p =∞,these conditions are also equivalent to the completeness of g.

The case of M = (0, 1] shows that (iii)=⇒(i) fails in general for p = 1. A Rie-mannian manifold (M, g) satisfying the above conditions with p = 2 is classicallycalled parabolic, cf. [Gla83] and the references therein for more information.

Proof. Suppose given ψ as in (i), and pick a smooth function cut-off functionθ ∈ C∞c (R) with 0 ≤ θ ≤ 1, θ ≡ 1 near 0. Then χj(x) = θ(j−1ψ(x)) satisfies (ii).(ii)=⇒(iii) is trivial. Let χj as in (iii) and p > 1. Then

ψ :=∑j≥1

j−1(1− χj)

is a smooth exhaustion function, since we have ψ ≥ Sj outside Kj+1 with Sj =∑jk=1 k

−1 → +∞. Since the supports of the gradients dχj are disjoint, we haveˆM|dψ|pdV ≤

∑j≥1

j−pˆM|dχj |p < +∞,

which proves (i).


Fix a basepoint x0 ∈M . By the Hopf-Rinow Lemma, g is complete iff d(•, x0)is an exhaustion function. If ψ is a smooth exhaustion function with bounded gra-dient, then ψ(x) ≤ ψ(x0)+Cd(x, x0), and d(•, x0) is thus exhaustive. Conversely,if the Lipschitz continuous function d(•, x0) is exhaustive, then a regularizationargument shows the existence of a smooth exhaustion function with boundedgradient.

Proof of Theorem 4.5. By Lemma 4.6, we may find an exhausting sequence ofcut-off functions χj ∈ C∞c (M) with |dχj |g ≤ 1. For each u ∈ L2(M,E), we haveχju→ u in L2. Further, (4.2) yields A(χju) = χjAu+ σA(dχj)u with

|σA(dχj)u| ≤ C|u|

since |dχj |g ≤ 1 and σA is bounded. By dominated convergence, we thus haveσA(dχj)u→ 0 in L2, and hence A(χju)→ Au in L2 whenever the latter is in L2.

We are thus reduced to the case where u has compact support. Using a parti-tion of unity, we can further assume that u is compactly supported in a coordinatechart. We then conclude using Lemma 4.7 below, usually known as Friedrich’slemma.

Lemma 4.7. Pick ρ ∈ C∞c (Rd) with´ρ(x)dx = 1, and set ρε(x) = ε−dρ(ε−1x).

Let also A =∑

α aαDα ∈ D1(Rd) be a first order linear differential operator with

|daα| ≤ C. For each f ∈ L2(Rd), we then have A(f ? ρε) − (Af) ? ρε → 0 inL2(Rd) as ε→ 0.

Proof. The result is clearly true if f is a test function, since f ?ρε → f in C∞. Bydensity of test funtions in L2, it is thus enough to prove the existence of C > 0such that

‖A(f ? ρε)− (Af) ? ρε‖L2 ≤ C‖f‖L2

for all f ∈ L2. To do this, we may assume by linearity that A = a∂j wherea ∈ C∞ has bounded gradient. Write

(a∂jf) ? ρε = (af) ? ∂jρε − ((∂ja)f) ? ρε.

Since L2 ? L1 ⊂ L2, the second term satisfies

‖ ((∂ja)f) ? ρε‖L2 ≤ ‖(∂ja)f‖L2‖ρε‖L1 ≤ C‖f‖L2 .

Next,

((af) ? ∂jρε) (x)− a(x)(∂jf ? ρε)(x) =

ˆ(a(x− y)− a(x))f(x− y)(∂jρε)(y)dy

yields

|(af) ? ∂jρε − a(∂jf ? ρε)| ≤ C |f ? (x∂jρε)| ,

whose L2 norm is bounded above by a uniform multiple of ‖f‖L2 since ‖x∂jρε‖L1 =‖x∂jρ‖L1 is bounded.


4.4. Removable singularities for first order PDEs.

Proposition 4.8. Let A ∈ D1(M ;E,F ) be a first order differential operator,u ∈ L2

loc(M,E), v ∈ L1loc(M,F ), and assume that Au = v holds outside a closed

submanifold N ⊂M of codimension at least 2. Then Au = v on M .

Lemma 4.9. If M is a compact Riemannian manifold with boundary and N ⊂Mis a closed submanifold of codimension at least 2, then M \N is parabolic, i.e. itsatisfies the equivalent condition of Lemma 4.6 with p = 2.

Proof. Using a partition of unity, we are reduced to a local statement. We maythen choose local coordinates x = (x′, x′′) ∈ Rn × Rd such that N = x′ = 0.Pick any smooth function χ : R→ R+ with χ = 0 near 0, χ ≡ 1 on [1,+∞), andset

χj(x) := χ(j‖x′‖

).

Then´|dχj |2dV ≤ Cj2 Vol(‖x′‖ ≤ j−1) ≤ Cj2−n is bounded since n ≥ 2 by

assumption.

Proof of Proposition 4.8. Let χj be a sequence of cut-off functions as in (ii) ofLemma 4.6, and write A(χju) = χjv+σA(dχj)u. We have χju→ u and χjv → vin L1

loc, which implies A(χju) → Au in the sense of distributions. Further,‖σA(dχj)‖L2 → 0 since σA is bounded and dχj → 0 in L2. Since u ∈ L2

loc,it follows that σA(dχj)u→ 0 in L1

loc, and we are done.

More generally, we prove:

Proposition 4.10. Let p ∈ (1,m] with conjugate exponent q ∈ [ md−1 ,∞). Let

A ∈ D1(M ;E,F ) be a first order differential operator, u ∈ Lqloc(M,E), v ∈L1

loc(M,F ), and assume that Au = v as distributions outside a closed subsetF ⊂M of finite (m− p)-Hausdorff measure. Then Au = v on M .

The Heaviside function shows that result fails in general for p = 1. As before,Proposition 4.10 will be a direct consequence of the following result.

Lemma 4.11. Let M be a compact Riemannian manifold with boundary, and letF ⊂ M be a closed subset of finite (m − p)-dimensional Hausdorff measure forsome p ∈ [1,m]. Then M \ F admits an exhausting sequence of cut-off functionswith gradient bounded in Lp, as in (iii) of Lemma 4.6.

Proof. Following [EG92, p.154, Theorem 3], we are going to show that any com-pact set K ⊂M \ F , we can find a cut-off function χ ∈ C∞c (M) with 0 ≤ χ ≤ 1,χ ≡ 1 on a neighborhood of K, and ‖χ‖Lp ≤ C for a uniform constant C > 0.

Since F is compact and has finite Hm−p-measure, there exists a constant C > 0such that F can be covered by finitely many open balls B(xi, ri) of radius ri 1

such that∑

i rm−pi ≤ C.

Let θ : R → R be the piecewise affine function defined by θ(t) = 0 for t ≤ 1,θ(t) = t − 1 for 1 ≤ t ≤ 2, and χ(t) = 1 for t ≥ 2, and consider the Lipschitzcontinuous function

χi(x) = θ(r−1i d(x, xi)

).


Then χ := maxi χi is also Lipschitz continuous, it has compact support in M \F ,and χ ≡ 1 on a neighborhood of K. Further, we have

|dχ| ≤ maxi|dχi| ≤ max

ir−1i 1x|ri≤d(x,xi)≤2ri

a.e. on M , henceˆM|dχ|pdV ≤ C

∑i

r−pi VolB(xi, 2ri) ≤ C ′∑i

rm−pi ≤ C ′′

with C ′′ > 0 independent of K. This proves the claim, after regularizing χ nearthe boundary of χ = 1.

5. Fundamental identities of Kahler geometry

5.1. Some Hermitian algebra. Let V be a finite dimensional complex vectorspace, and denote by

Λ =⊕p,q∈N

Λp,qV ?

the corresponding bigraded exterior algebra. The graded commutator of A,B ∈End(Λ) is defined by

[A,B] = AB − (−1)|A||B|BA

for A,B of pure degree |A|, |B|. It satisfies the graded Jacobi identity

(5.1) [[A,B], C] = [A, [B,C]]− (−1)|A||B|[B, [A,C]],

which expresses that A 7→ ad(A) := [A, •] is a graded Lie algebra homomorphism.For each η ∈ Λ, we also denote by η the associated endomorphism η∧• ∈ End(Λ).Note that [η, η′] = 0 for any two η, η′ ∈ Λ.

Now suppose that V is equipped with a Hermitian scalar product, and denoteby ω ∈ Λ1,1 the corresponding positive (1, 1)-form. Given an orthonormal basis(ξi) of Λ1,0 ' V ?, (ξI ∧ ξJ)|I|=p,|J |=q is an orthonormal basis of Λp,q, and

(5.2) ω = i∑j

ξj ∧ ξj .

For η ∈ Λ, denote by η? ∈ End(Λ) the adjoint of η ∧ •, which has bidegree(−p,−q) when η ∈ Λp,qV ?.

We next discuss in more detail [η, ω?] ∈ End(Λ) for η ∈ Λp,qV ? of bidegree(p, q) = (1, 0) or (1, 1). The next result is the ’linear algebra’ part of the Kahlercommutation identities.

Lemma 5.1. For each ξ ∈ Λ1,0 ' V ?, the following identities hold:

(i) [ξ, ω?] = iξ?;(ii) [ξ?, ω] = iξ;(iii) [ω, ξ?] = iξ;(iv) [ω?, ξ] = iξ?.


Proof. A direct check shows that ξ? is given by contraction against the dual vectorξ# ∈ V . As a result, ξ? is an anti-derivation of Λ, i.e.

(5.3) ξ?(α ∧ β) = ξ?(α) ∧ β + (−1)|α|α ∧ ξ?(β),

or, equivalently, [ξ?, α] = ξ?(α) for all α ∈ Λ. We get (ii) thanks to ξ?(ω) = iξ,which is easily seen using (5.2). Applying adjunction and conjugation yields theremaining identities.

Now fix a real (1, 1)-form θ ∈ Λ1,1. The operator [θ, ω?] will play a crucial rolein the L2 estimates for ∂, and we therefore study it in some detail.

Proposition 5.2. [Dem82, Lemme 3.2] Assume that θ ∈ Λ1,1 is positive, and fixan integer q ≥ 1.

(i) The endomorphism [θ, ω?] = θω? of Λn,q is positive definite. More pre-cisely, if we pick ε > 0 then θ ≥ εω implies [θ, ω?] ≥ qε id on Λn,qV ?;

(ii) for each u ∈ Λn,q, 〈[θ, ω?]−1u, u〉dVω is non-increasing with respect to ωand θ;

(iii) when u ∈ Λn,1, 〈[θ, ω?]−1u, u〉dVω is independent of ω, and is given by

〈[θ, ω?]−1u, u〉dVω = in2u, uθ,

where u, uθ denotes the the image of iu⊗ u under the contraction mor-phism

Λn,1V ? ⊗ Λ1,nV ? '−→ Λ1,1 ⊗ Λn,nV ? trθ−−→ Λn,nV ?.

Proof of Proposition 5.2. Let λ1 ≤ · · · ≤ λn be the eigenvalues of θ with respectto ω, and pick an ω-orthonormal basis (ξj) such that θ = i

∑j λjξj ∧ ξj . Setting

Ω := ξ1 ∧ · · · ∧ ξn ∈ Λn,0V ?, a direct check shows that

(5.4) θω?(Ω ∧ ξJ) = (∑j∈J

λj)Ω ∧ ξJ ,

which implies (i). Pick u ∈ Λn,1V ?, and write u = Ω ∧ v with v =∑

j vj ξj in

Λ0,1. By (5.4), we have

〈[θ, ω?]−1u, u〉 =∑j

λ−1j |vj |

2 = trθ(iv ∧ v),

and (iii) follows since

dVω =∧j

iξj ∧ ξj = in2Ω ∧ Ω.

Monotonicity with respect to θ in (ii) is clear, in view of (5.4). Observe that thequadratic form

qω(u) := 〈[θ, ω?]−1u, u〉ωsatisfies

(5.5) qω(u) = sup0 6=v∈Λn,qV ?

|〈u, v〉ω|2

〈θω?v, v〉ω.


Now let ω′ ≥ ω be another positive (1, 1)-form with eigenvalues µi ≥ 1 withrespect to ω, and let (ζj) be an ω-orthonormal basis of Λ1,0V such that ω′ =∑

j µjζj ∧ ζj (which is however not necessarily diagonalizing θ). Then dV ′ω =µ1 . . . µndVω, and our goal is thus to show that

(5.6) qω′(u)µ1 . . . µn ≤ qω(u).

Setting µ = diag(µj), we have ω′? = µ1/2ω?µ−1/2, and (5.5) yields

qω′(u) = sup06=v∈Λn,qV ?

|〈u, v〉ω′ |2

〈θω′?v, v〉ω′= sup

06=v∈Λn,qV ?

|〈u, v〉ω|2

〈θµ1/2ω?µ1/2v, v〉ω.

Using µj ≥ 1, one easily sees that

〈θµ1/2ω?µ1/2v, v〉ω ≥ µ1 . . . µn〈θω?v, v〉ω,

which yields as desired (5.6).

5.2. Kahler identities. In what follows, (X,ω) is a Kahler manifold, and E isa Hermitian holomorphic vector bundle on X, with Chern connection ∂E + ∂E .We start with the existence of local normal forms for ω and E.

Lemma 5.3. Near each point of X, we may find:

(i) holomorphic coordinates (z0, . . . , zn) such that

〈 ∂∂zj

,∂

∂zk〉ω = δjk +O(|z|2).

(ii) a holomorphic basis (ej) of E such that 〈ej , ek〉E = δjk +O(|z|2).

Conversely, note that a Hermitian metric ω admitting local holomorphic coor-dinates as in (i) is closed, and hence Kahler.

Proof. Locally, we have ω = ddcϕ for a smooth (strictly psh) function ϕ. Chooseholomorphic coordinates w such that ω =

∑j idwj ∧ dwj at 0. The Taylor

expansion of ϕ at 0 then writes

ϕ = |w|2 + Re

∑j

Pj(w)wj

+ ReP (w) +O(|w|4)

where each Pj(w) is a holomorphic quadratic polynomial, and P (w) is a holo-morphic polynomial of degree at most 3. Setting zj = wj − 1

2Pj(w) yields (i).The proof of (ii) is similar and left to the reader.

Theorem 5.4. Assume that (X,ω) is Kahler, let E be a Hermitian holomorphicvector bundle on X, with Chern connection ∂E + ∂E. Then

[∂?E , ω] =i∂E

[∂?E , ω] =− i∂E[ω?, ∂E ] =− i∂?E[ω?, ∂E ] =i∂

?E .


Proof. Since the adjoint of a first order operator only depends on the 1-jets ofthe metrics involves, the normal forms of Lemma 5.3 reduces us to the case ofCn with its Euclidian metric and E the trivial line bundle. In that case, the firstorder operators involved have constant coefficients. It is thus enough to provethat their symbols coincide, which is the content of Lemma 5.1.

The key consequence for us is the following Bochner-type formula, which goesback to the work of Kodaira and Nakano.

Corollary 5.5 (Bochner–Kodaira–Nakano identity). With the previous notation,the Laplacians of ∂E and ∂E satisfy

∆∂E= ∆∂E + [θE , ω

?]

with θE ∈ C∞(X,Λ1,1T ?X ⊗Hom(E,E)) the curvature tensor.

Proof. The above Kahler identities combined with the Jacobi identity (5.1) yield

∆∂E= [∂

?E , ∂E ] = i[[∂E , ω

?], ∂E ]

= i[∂E , [ω?, ∂E ]]− i[ω?, [∂E , ∂E ]]

= [∂E , ∂?E ] + i[[∂E , ∂E ], ω?] = ∆∂E + i[θE , ω

?].

As a first application, we obtain the following classical cohomology vanishingresult.

Theorem 5.6 (Kodaira vanishing). If L is a Hermitian holomorphic line bundlewith positive curvature form θL > 0, then

Hn,q(X,L) = Hq(X,KX + L) = 0

for each q ≥ 1.

Proof. By Hodge theory, it is enough to show that for every L-valued (n, q)-formu with ∆∂L

u = 0 vanishes. The Bochner–Kodaira–Nakano equality yields

0 = ‖∂Lu‖2 + ‖∂?Lu‖2 + 〈[θL, ω?]u, u〉L2 .

By Proposition 5.2, the operator [θ, ω?] is positive definite on (n, q)-forms, hencethe result.

6. L2-estimates for the ∂-equation

The results in this section go back to the fundamental work of Hormander,with successive improvements and generalizations by Andreotti–Vesentini, Skoda,Demailly.


6.1. The smooth, complete case.

Theorem 6.1. Let (X,ω) be a Kahler manifold, and let be L a Hermitian holo-morphic line bundle on X with positive curvature form θ > 0. Assume also thatX admits a complete Kahler metric (not necessarily equal to ω or θ). If

v ∈ L2loc(X,Λ

n,qT ?X ⊗ L)

satisfies ∂v = 0 and ˆX〈[θ, ω?]−1v, v〉dVω < +∞,

then v = ∂u for some u ∈ L2(X,Λn,q−1T ?X ⊗ L) such that

‖u‖2L2 =

ˆX|u|2dVω ≤

ˆX〈[θ, ω?]−1v, v〉dVω.

If v is smooth, then u can be chosen smooth.

Recall that the endomorphism [θ, ω?] = θω? of Λn,qT ?X is positive definite, byProposition 5.2.

Remark 6.2. For q = 1, the result does in fact not involve ω, since

〈[θ, ω?]−1v, v〉dV = in2v, vθ

and

|u|2dV = in2u ∧ u.

As the next result shows, the assumption on X is satisfied by a fairly generalclass of Kahler manifolds.

Lemma 6.3. Assume that X is weakly pseudoconvex, i.e. admits a smooth, pshexhaustion function. Then X admits a complete Kahler metric.

In particular, any Stein manifold admits a complete Kahler metric.

Proof. Let ϕ : X → R be a smooth, psh exhaustion function. Since ϕ is boundedbelow, we may assume that ϕ ≥ 0. Then

i∂∂(ϕ2) = 2ϕ i∂∂ϕ+ 2i∂ϕ ∧ ∂ϕ

shows the gradient of ϕ is bounded with respect to the Kahler metric ω+i∂∂(ϕ2),which is thus complete.

The key tool in the proof of Theorem 6.1 the following consequence of theBochner–Kodaira–Nakano identity.

Lemma 6.4. If ω is complete, then each u ∈ L2(X,Λn,qT ?X⊗L) with ∂u, ∂?u ∈ L2

satisfies

‖∂u‖2L2 + ‖∂?u‖2L2 ≥ˆX〈[θ, ω?]u, u〉dV.


Proof. The symbol of

∂ : C∞(X,Λn,qT ?X ⊗ L)→ C∞(X,Λn,q+1T ?X ⊗ L)

is given by σ∂(ξ) = ξ ∧ •, and is thus bounded. The symbol of its adjointσ∂?(ξ) = −σ∂(ξ)? is thus also bounded, and Theorem 4.5 yields a sequence uj ∈C∞c (X,Λn,qT ?X ⊗ L) with uj → u, ∂uj → ∂u and ∂

?uj → ∂

?u in L2 norm. After

passing to a subsequence, we may assume that uj → u a.e. and henceˆX〈[θ, ω?]u, u〉dV ≤ lim inf

j

ˆX〈[θ, ω?]uj , uj〉dV,

by Fatou’s lemma. For each j, the Bochner–Kodaira–Nakano identity implies

‖∂uj‖2L2 + ‖∂?uj‖2L2 ≥ˆX〈[θ, ω?]uj , uj〉dV,

and we get the desired inequality in the limit.

Proof of Theorem 6.1. Assume first that ω itself is complete. The existence of uclearly implies

(6.1) |〈w, v〉|2 ≤ C‖∂?w‖2L2

for all test forms w ∈ C∞c (X,Λn,qT ?X ⊗L). Assume conversely that this estimate

holds. Then ∂?w 7→ 〈w, v〉 defines a continuous linear form of norm at most C1/2

on

∂?C∞c (X,Λn,qT ?X ⊗ L) ⊂ L2(X,Λn,q−1T ?X ⊗ L).

By the Hahn-Banach theorem, this linear form admits a continuous extension toL2(X,Λn,q−1T ?X⊗L) of norm at most C1/2. By the Riesz representation theorem,

we may thus find u ∈ L2(X,Λn,q−1T ?X ⊗ L) with ‖u‖L2 ≤ C1/2 such that

〈w, v〉 = 〈∂?w, u〉 = 〈w, ∂u〉

for all test forms w, and hence ∂u = v. In order to establish (6.1), consider theorthogonal decomposition w = w′ + w′′ in L2(X,Λn,q−1T ?X ⊗ L) with w′ ∈ ker ∂

and w′′ ∈ (ker ∂)⊥. For each test form η ∈ C∞c (X,Λn,q−2T ?X ⊗ L), ∂η ∈ ker ∂

implies 〈w′′, ∂η〉 = 0, i.e. ∂?w′′ = 0 in the sense of distributions. Since ∂v = 0,

the Cauchy-Schwarz inequality Lemma 6.4 yield

|〈v, w〉|2 = |〈v, w′〉|2 ≤(ˆ

X〈[θ, ω?]−1v, v〉dV

)(ˆX〈[θ, ω?]w′, w′〉dV

)≤ C‖∂?w′‖2 = C‖∂?w‖2,

which proves (6.1).Next, assume that ω is arbitrary, and let ω′ be a complete Kahler metric. For

each ε > 0, ωε := ω + εω′ is then complete as well. The monotonicity propertyof Proposition 5.2 impliesˆ

X〈[θ, ω?ε ]−1v, v〉ωεdVωε ≤ C,


and the first part of the proof yields the existence of uε ∈ L2loc(X,Λ

n,q−1T ?X ⊗L)

with ∂uε = v and´X |uε|

2ωεdVωε ≤ C. In particular, uε remains bounded in L2

loc,and an easy weak compactness argument yields the desired solution u.

To see the final point, note that we can always replace u by its orthogonalprojection in (ker ∂)⊥. As above, this implies ∂

?u = 0 and hence ∆∂u = ∂

?v in

the sense of distributions. If v is smooth, then u is smooth by ellipticity of theLaplacian.

6.2. The singular case.

Theorem 6.5. Let (X,ω) be a Kahler manifold, and assume that X contains aStein Zariski open subset. Let L be a Hermitian holomorphic line bundle on Xwith curvature form θ, and let also ϕ be a quasi-psh function such that θ+i∂∂ϕ ≥η > 0 in the sense of currents for some positive (1, 1)-form η. If

v ∈ L2loc(X,Λ

n,qT ?X ⊗ L)

satisfies ∂v = 0 and ˆX〈[η, ω?]−1v, v〉e−2ϕdVω < +∞,

then v = ∂u for some u ∈ L2(X,Λn,q−1T ?X ⊗ L) such thatˆX|u|2e−2ϕdVω ≤

ˆX〈[η, ω?]−1v, v〉e−2ϕdVω.

Note that θ is not assumed to be positive here. The current θ + ddcϕ shouldbe interpreted as the curvature of the singular metric | · |Le−ϕ. As in Remark 6.2,the result does in fact not involve ω when q = 1.

Lemma 6.6. If u, are L2loc L-valued forms on a complex manifold X such that

∂u = v outside a closed analytic subset A ⊂ X, then ∂u = v on X.

Proof. Arguing by induction on dimA, it is enough to prove that ∂u = v onX \ Asing, which follows from Proposition 4.8 since Areg is a closed submanifoldof X \Asing, of real codimension at least 2.

Proof of Theorem 6.5. By Lemma 6.6, we may assume that X is Stein. By The-orem 3.8, we may then find an exhaustion of X by weakly pseudoconvex opensubsets Ωj such that ϕ|Ωj is the decreasing limit of a sequence ϕj,k ∈ C∞(Ωj)with θ + ddcϕj,k ≥ η. By Lemma 6.3, Ωj admits a complete Kahler metric,and Theorem 6.1 yields the existence of uj,k ∈ L2(Ωj ,Λ

n,q−1T ?Ωj ⊗ L) such that

∂uj,k = v on Ωj andˆΩj

|uj,k|2e−2ϕj,kdVω ≤ˆ

Ωj

〈[η, ω?]−1v, v〉e−2ϕj,kdVω

≤ C :=

ˆX〈[η, ω?]−1v, v〉e−2ϕdVω,

By monotonicity of (ϕj,k)k, we get for k ≥ l´

Ωj|uj,k|2e−2ϕj,ldVω ≤ C, which

shows in particular that (uj,k)k is bounded in L2(Ωj , e−2ϕj,l). After passing to a


subsequence, we thus assume that that uj,k converges weakly in L2(Ωj , e−2ϕj,l)

to uj , which may further be asumed to be the same for all l, by a diagonal

argument. We then have ∂uj = v, and´

Ωj|uj |2e−2ϕj,ldVω ≤ C for all l, hence´

Ωj|uj |2e−2ϕdVω ≤ C by monotone convergence of ϕj,l → ϕ. By a diagonal

argument, we may finally arrange that uj → u weakly in L2(K, e−2ϕ) for eachcompact K ⊂ X, and we obtain the desired solution.

Corollary 6.7. Let (X,ω) be a Kahler manifold, and assume that X contains aStein Zariski open subset. Let L be a Hermitian holomorphic line bundle on Xwith curvature form θ, and let also ϕ be a quasi-psh function such that θ+i∂∂ϕ ≥εω (resp. θ+ i∂∂ϕ+ Ric(ω) ≥ εω > 0) in the sense of currents for some positive(1, 1)-form η. If v ∈ L2(X,Λn,1T ?X ⊗ L) (resp. v ∈ L2(X,Λ0,1T ?X ⊗ L) satisfies

∂v = 0, then v = ∂u for some u ∈ L2(X,Λn,q−1T ?X ⊗L) (resp. L2(X,Λ0,q−1T ?X ⊗L)) such that ˆ

X|u|2e−2ϕdVω ≤

1

qε

ˆX|v|2e−2ϕdVω.

Proof. The case of (n, q)-forms is a direct consequence of Proposition 5.2 andTheorem 6.5. The case of (0, q)-forms follows by viewing a (0, q)-form v withvalues in L as an (n, 1)-form with values in L − KX , which has curvature θ +Ric(ω).

7. The Ohsawa–Takegoshi extension theorem

Extension theorems with L2 estimates have a rather long history, going backto an original result of Ohsawa and Takegoshi. The following version is a slightrefinement of [Ber, Theorem 2.5] (compare [Dem12, Theorem 13.6]).

Recall that for a smooth hypersurface S in a complex manifold X, we havea canonical isomorphism KS ' (KX + S)|S , given by the Poincare residue mapResS .

Theorem 7.1. Let X be a complex manifold containing a Stein Zariski opensubset, and consider the following data:

(i) a smooth complex hypersurface S ⊂ X with canonical section s ∈ H0(X,O(S)),and a smooth metric on O(S) with curvature form θS, such that |s| ≤ e−1;

(ii) a Hermitian holomorphic line bundle L on X with curvature form θL,and a quasi-psh function ϕ such that θL + i∂∂ϕ ≥ 0 and θL + i∂∂ϕ ≥ θS;

Then for each σ ∈ H0(S,KS + L|S) such that´S |σ|

2e−2ϕ < +∞, there exists

σ ∈ H0(X,KX + S + L) with ResS(σ) = σ and

(7.1)

ˆX

|σ|2

|s|2 log2 |s|e−2ϕ ≤ C

ˆS|σ|2 e−2ϕ

with C > 0 a purely numerical constant.

The singular weight 1|s|2 log2 |s| in the left-hand integral is said to have Poincare

growth (by analogy with the Poincare metric |dz|2|z|2 log2 |z| of the punctured disc). In

particular, it is integrable, but more singular that |s|−2c for any c < 1.


Before entering the proof of Theorem 7.1, we state a ’local’ version of theresult.

Corollary 7.2. Let Ω b Cn be a bounded pseudoconvex domain, ϕ a psh functionon Ω, and L ⊂ Cn a (complex) affine subspace. Then every holomorphic function

f ∈ O(L ∩ Ω) such that |f |e−ϕ is L2 admits a holomorphic extension f ∈ O(Ω)with ˆ

Ω|f |2e−2ϕ ≤ C

ˆL∩Ω|f |2e−2ϕ

for a constant C > 0 only depending on n and the diameter of Ω.

Proof. Writing L as a successive intersection of affine hyperplanes, it is enoughto treat the case where L = H is a hyperplane. The canonical bundles of Ω andH ∩ Ω are trivial, and the metrics they inherit from the Euclidian metric on Cnare flat. We thus obtain Corollary 7.2 as a direct consequence of Theorem 7.1,which would even allow to add a denominator |`|2 log |`|2 in the left-hand integral,with ` = 0 an affine equation for H.

The proof of Theorem 7.1 presented below is essentially an adaptation to the’global setting’ of B locki’s exposition [B lo13] of B.Y.Chen’s proof [Chen].

7.1. Reduction to a twisted L2-estimate. By the removable singularity prop-erty of L2 holomorphic functions, it is enough to prove Theorem 7.1 when X itselfis Stein. By cohomology vanishing, σ automatically admits a holomorphic exten-sion σ ∈ H0(X,KX + F ) with F := S + L. Arguing by compactness as in theproof of Theorem 6.5, we may further assume that ϕ = 0, θL > 0 and θL > θS ,

and´X

|σ|2|s|2 log2 |s| < +∞ (but without any uniform estimate!).

A natural approach to prove Theorem 7.1 is to introduce a cut-off function

χε := χ(ε−1|s|2

)with χ ∈ C∞(R) such that 0 ≤ χ ≤ 1, χ(t) = 1 for t ≤ 1/2 and χ(t) = 0 fort ≥ 1, and solve

(7.2) ∂uε = ∂(χεσ) = ε−1χ′(ε−1|s|2

)∂(|s|2)σ

with an adequate L2 estimate with respect to singular weight log |s|. The F -valued (n, 0)-form

σε := χεσ − uεis then also a holomorphic extension of σ, since uε is holomorphic near S, andhence vanishes on S (as a section of KX + F ) thanks to the L2-estimate. Since

limε→0

ˆX

|χεσ|2

|s|2 log2 |s|= 0

by integrability of the Poincare weight, it would thus be enough to solve (7.2)with an estimate of the form

(7.3)

ˆX

|uε|2|s|−2

log2(|s|2 + ε)≤ˆ

supp ∂uε

|σ|2|s|−2.


Indeed,

lim supε→0

ˆsupp ∂uε

|σ|2|s|−2 ≤ CˆS|σ|2

sincesupp ∂uε ⊂ ε/2 ≤ |s|2 ≤ ε,

and we would get the desired extension of σ as a weak L2loc limit of σε.

In view of (7.2), a direct application of Theorem 6.5 with weight functionψ = log |s| yields insteadˆ

X|uε|2|s|−2 ≤ C

ˆε/2≤|s|2≤ε

∣∣ε−1∂(|s|2)∣∣2θ|σ|2|s|−2,

where∣∣ε−1∂

(|s|2)∣∣θ

is a priori unbounded even when |s|2 ∼ ε. The trick will be

to solve (7.2) with a twisted L2-estimate, replacing in particular the right-handside with ˆ

ε/2≤|s|2≤εe2τε

∣∣ε−1∂(|s|2)∣∣2θε|σ|2|s|−2,

where τε is a smooth function such that θε := θL + i∂∂τε > 0 and

eτε∣∣∂ (|s|2)∣∣

θε= O(ε)

when |s|2 ∼ ε.The general principle underlying these twisted estimates is as follows.

Lemma 7.3. Let X be a Stein manifold, F be a Hermitian holomorphic linebundle with curvature form θF , and ψ be a quasi-psh function such that θF +i∂∂ψ ≥ η with η a Kahler form. Pick a ∂-closed v ∈ L2

loc(X,Λn,1T ?X ⊗ F ), and

assume given a strictly η-psh function τ ∈ C∞(X) such that

(i) |∂τ |2ητ ≤ 1/4 on X;

(ii) |∂τ |2ητ ≤ 1/8 on supp v.

Then there exists u ∈ L2(X,Λn,0T ?X ⊗ F ) such that ∂u = v andˆX

(1− 4|∂τ |2ητ

)|u|2e2(τ−ψ) ≤ 10

ˆXin

2v, vητ e2(τ−ψ).

Proof. Arguing by regularization and weak compactness as in the proof of The-orem 6.5, we may assume that ψ = 0. Let u be the minimal solution of ∂u = vin L2. Since u is orthogonal to ker ∂ in L2(ψ), ue2τ is orthogonal to ker ∂ inL2(τ), and hence is the minimal solution of ∂(ue2τ ) = (v + 2u ∂τ)e2τ in L2(τ).By Theorem 6.1, we thus haveˆ

X|u|2e2τ ≤

ˆXin

2v + 2u ∂τ, v + 2u ∂τ

ητe2τ .

Using the Cauchy-Schwarz inequality combined with 2ab ≤ ta2+t−1b2 for a, b ∈ Rand t > 0, we inferˆ

X

(1− 4|∂τ |2ητ

)e2τ |u|2 ≤ t

ˆsupp v

|u|2e2τ + (1 + t−1)

ˆXin

2v, vητ e2τ .

Since t := 1/4 ≤(1− 4|∂τ |2ητ

)/2 on supp v, we get the desired result.


7.2. Proof of Theorem 7.1. Following the above strategy, suppose given τεsuch that

(i) θε := θL + i∂∂τε > 0;(ii) 4|∂τε|2θε ≤ 1 on X;

(iii) 4|∂τε|2θε ≤ 1/2 on ε/2 ≤ |s|2 ≤ ε.Applying Lemma 7.3 to F = L+ S, ψ = log |s| and η = θL, we get a solution uεof (7.2) such thatˆ

Xe2τε

(1− 4|∂τε|2θε

)|uε|2|s|−2 ≤ C

ˆε/2≤|s|2≤ε

e2τε∣∣ε−1∂

(|s|2)∣∣2θε|σ|2|s|−2.

We will thus be done if τε further satisfies

(iv) e2τε(1− 4|∂τε|2θε

)≥ C−1/ log2

(|s|2 + ε

);

(v) eτε∣∣∂ (|s|2)∣∣

θε≤ Cε on ε/2 ≤ |s|2 ≤ ε.

This will achieved by means of the following construction.

Lemma 7.4. Set ρε := − log(|s|2 + ε). Then τε := − log (ρε + log ρε) satisfies

(iv), (v), and∣∣∂τε∣∣2θτε ≤ C/ log ε−1 on ε/2 ≤ |s|2 ≤ ε.

Proof. We have τε = − log f(ρε) with f(t) = t+ log t. Outside S, we compute

−∂ρε =∂|s|2

|s|2 + ε,

(7.4) − ∂τε =f ′(ρε)

f(ρε)∂ρε,

−i∂∂ρε = i∂∂|s|2

|s|2 + ε=

ε

|s|2i∂ρε ∧ ∂ρε −

|s|2

|s|2 + εθS

≥ ε

|s|2

(f(ρε)

f ′(ρε)

)2

i∂τε ∧ ∂τε − θL,

since θL ≥ θS and θL ≥ 0. This yields

i∂∂τε = −f′(ρε)

f(ρε)i∂∂ρε +

(1 +

f(ρε)

f ′(ρε)2ρ2ε

)i∂τε ∧ ∂τε.

Since |s| ≤ e−1, we have ρε ≥ 2, and hence e−τε = f(ρε) ≥ 2, f ′(ρε) ≤ 2. Weinfer

θ + i∂∂τε ≥ e−τε(

ε

2|s|2+ 1 +

1

4 log2(|s|2 + ε)

)i∂τε ∧ ∂τε.

Finally,

∂(|s|2)

=f(ρε)

f ′(ρε)(|s|2 + ε)∂τε,

and the desired estimates easily follow from this.

8. Blowups and valuations

The goal of this section is to recall a number of basic facts on blowups, integralclosures and valuations, in the setting of complex analytic spaces.


8.1. Blowing-up along an ideal. Assume that X is a reduced complex space.If a ⊂ OX is a coherent ideal (sheaf) defining a complex subspace Z ⊂ X, theblowup µ : X ′ → X of X along a (or Z) is defined by setting

X ′ = Bla(X) = ProjX

(⊕m∈N

am

).

Since X is reduced, the graded OX -algebra⊕

m∈N am is reduced, and X ′ is thusreduced as well. The morphism µ is projective, and an isomorphism over X \ Z;it is thus a proper bimeromorphic morphism, i.e. a modification. Note also thatthere is canonical isomorphism Blam(X) ' Bla(X) for any nonzero m ∈ N.

The blowup can be more concretely understood as the graph of a meromorphicmap, as follows. After shrinking X near a given point, we can pick generators(f1, . . . , fr) of a. This induces a surjection

OX [t1, . . . , tr]⊕m∈N

am,

and hence a closed embeddding X ′ → X × Pr−1 over X, which can be shownto coincide with the graph of the bimeromorphic map X 99K Pr−1 defined by[f1 : · · · : fr] in homogeneous coordinates.

Blowups are characterized by the following universal property.

Lemma 8.1. The scheme-theoretic inverse image E = µ−1(Z) is a Cartier di-visor, i.e. a · OX′ is locally principal ideal sheaf. Further, −E is µ-very ample,and

µ∗OX′(−mE) = am

near any given compact subset of X for all m 1. Finally, a proper bimeromor-phic morphism ρ : Y → X with Y reduced factors through µ if and only if ρ−1(Z)is a Cartier divisor.

Assume now that the complex space X is normal. The normalized blowup

π : X → X of X along a coherent ideal a ⊂ OX is defined as the normalization of

the blowup Bla(X), i.e. π = µ ν with ν : X → X ′ the normalization morphism.Again, π is a projective bimeromorphic morphism, and an isomorphism overX \ Z. Normalized blowups are again characterized by a universal property:

Lemma 8.2. The scheme-theoretic inverse image D = π−1(Z) is a Cartier di-visor, and a proper bimeromorphic morphism ρ : Y → X with Y normal factorsthrough π if and only if ρ−1(Z) is a Cartier divisor. Further, −D is π-ample andπ-globally generated.

The meaning of this last property is that the evaluation map

π?(π∗OX(−D)

)⊗ O

X→ O

X(−D)

is surjective. Since X is normal, π∗OX = OX (i.e. , π has connected fiber), byZariski’s ’main theorem’. We can thus interpret

b := π∗OX(−D) ⊂ π∗OX = OX

as an ideal sheaf, which satisfies b · OX

= OX

(−D). As we shall next see, b canbe understood algebraically as the integral closure of a.


8.2. The integral closure of an ideal. If I ⊂ A is an ideal in an integrallyclosed domain, the integral closure I of I is defined as the set of f ∈ A satisfyinga polynomial equation of the form

(8.1) fd +

d∑j=1

ajfd−j = 0

with aj ∈ Ij . Using the usual determinant trick, one shows that f ∈ A belongsto I iff there exists a finitely generated nonzero ideal J ⊂ A such that fJ ⊂ IJ .This implies in particular that I is an ideal of A, and we say of course that I isintegrally closed if I = I.

Example 8.3. If I ⊂ A is either principal or reduced, then I is integrally closed.

If now a ⊂ OX is a coherent ideal sheaf on a normal complex space X, theintegral closure of a is the ideal sheaf a ⊂ OX defined by taking the integral closurestalkwise. Coherence of a is non-obvious, and a consequence of the followinggeometric characterization of a.

Theorem 8.4. Let π : X → X be the normalized blowup of the normal complexX along a coherent ideal sheaf a ⊂ OX , and denote by D the effective Cartier

divisor it determines on X. Then

π∗OX(−D) = a

Remark 8.5. For each m ≥ 1, X is also the normalized blowup of am, andhence π∗OX(−mD) = am. In more algebraic terms, the result thus means that⊕

m∈N am is the integral closure of the graded domain⊕

m∈N am.

As a consequence of Theorem 8.4, we get the following neat analytic charac-terization of the integral closure.

Corollary 8.6. Let a be an ideal with local generators (fi). Then f ∈ OX belongsto a if and only if |f | ≤ O(maxi |fi|).

Proof of Theorem 8.4. Recall that b := π∗OX(−D) is a coherent ideal sheaf onX, and

a · OX

= b · OX

= OX

(−D).

If f ∈ OX is integral over a, then f π is integral over the principal ideal b ·OX

= OX

(−D). By normality of X, it follows that f π ∈ OX

(−D), i.e. f ∈π∗OX(−D) = b, and we have proved that a ⊂ b. The converse inclusion is moredifficult, and we reproduce the elegant geometric argument of [Laz, II.11.1.7]. Thestatement being local over X, we may choose a system of generators (f1, . . . , fr)

for a. This defines a surjection O⊕rX → a, which induces, after pulling-back to Xand twisting by −mD, a surjection

OX

(−mD)⊕r → a · OX

(−mD) = OX

(−(m+ 1)D)

for any m ≥ 1. Set bm := π∗OX(−mD), so that b = b1. Since −D is π-ample,Serre vanishing implies that the induced map

b⊕rm = π∗OX (−mD)⊕r → bm+1 = π∗OX (−(m+ 1)D)


is surjective for m 1, i.e. a ·bm = bm+1. Since b ·bm ⊂ bm+1, b thus acts on thenonzero ideal bm by multiplication with a, and hence b ⊂ a by the determinantaltrick recalled above.

Remark 8.7. The above proof shows that a · am = a · am = am+1 for all m 1.

Proof of Corollary 8.6. If f ∈ OX satisfies an integral equation fd+∑d

i=1 aifd−i =

0 with ai ∈ ai, we have at each point |f |d ≤ d|ai||f |d−i for some i, and hence

|f | ≤ maxi

(d|ai|)1/i = O(maxi|fi|).

If conversely |f | = O(maxi |fi|), then |f π| = O(|fD|) locally on X, with fD a

local equation of D. By normality of X, it follows that f π ∈ OX

(−D), i.e.f ∈ π∗OX(−D) = a.

Example 8.8. If ϕ is a quasi-psh function on a complex manifold X, then themultiplier ideal sheaf J(ϕ) is integrally closed. Indeed, if f ∈ OX satisfies |f | =O(maxi |fi|) for a set of generators (fi) of J(ϕ), then |fi|e−ϕ ∈ L2

loc implies|f |e−ϕ ∈ L2

loc, i.e. f ∈ J(ϕ).

8.3. Valuative ideals. In what follows, X denotes a normal complex space, acondition which implies in particular that the singular locus Xsing has codimen-sion at least 2.

Let Z ⊂ X be an irreducible closed analytic subset, and assume that Z is notentirely contained in Xsing. For each f ∈ OX ,

ordZ(f) := infz∈Z∩Xreg

ordz(f)

satisfies ordZ(f) = ordz(f) for a general point z ∈ Z (i.e. outside a strict closedanalytic subset), and ordZ thus defines a valuation on OX .

The condition Z 6⊂ Xsing is in particular satisfied if Z has codimension 1,i.e. is a prime divisor. More generally, consider a modification (i.e. a proper,bimeromorphic morphism) µ : Y → X with Y a complex manifold, and a primedivisor E on Y . We then say that E is a divisor over X, and attach to it avaluation ordE on germs f ∈ OX by setting

ordE(f) := ordE(f µ) ∈ N.The center of E on X is the irreducible closed analytic subset

cX(E) := µ(E) ⊂ X.If ρ : Y ′ → Y is another modification and E′ is the strict transform, then cX(E) =cX′(E

′) and ordE = ordE′ as functions on OX .Given a coherent ideal a ⊂ OX , we set

ordE(a) := inff∈ax

ordE(f)

for a general point x in µ(E). It is easy to see that the infimum is achievedamong any given set of local generators of ax. For any two coherent ideals a, b,the valuation property of ordE yields

ordE(a · b) = ordE(a) + ordE(b)


andordE(a + b) = min ordE(a), ordE(b) .

Definition 8.9. A coherent ideal a ⊂ OX is valuative if

a = f ∈ OX | ordEα(f) ≥ cα for all αfor a (possibly infinite) family of prime divisors Eα over X and constants cα ∈ R.

Clearly, the constants can always be taken to be cα = ordEα(a). The nextresult is simply a reformulation of Theorem 8.4. The next result is basically areformulation of Theorem 8.4.

Theorem 8.10. Let a ⊂ OX be a coherent ideal, and denote by Eα the irreducible

components of the divisor D lying over a in the normalized blowup π : X → Xalong a. Then

a = f ∈ OX | ordEα(f) ≥ ordEα(a) .The divisors Eα are known as the Rees divisors of a, and the corresponding

valuations ordEα are the Rees valuations.

Proof. By definition, we have

ordEα(a) = ordEα(a · OX

) = ordEα(OX

(−D)) = ordEα(D),

and hence D =∑

α ordEα(a)Eα. But then f ∈ OX belongs to the ideal a =π∗OX(−D) if and only if ordEα(f) = ordEα(f π) ≥ ordEα(a) for all α.

Corollary 8.11. A coherent ideal is valuative if and only if it is integrally closed.

Proof. Theorem 10.13 implies that any integrally closed ideal is valuative. Theconverse is elementary, and similar to the easy direction in Corollary 8.6. Assumethat a = f ∈ OX | ordEα(f) ≥ ordEα(f) for all α for some family (Eα) of prime

divisors over X, and pick f ∈ OX such that fd +∑d

i=1 aifd−i = 0 with ai ∈ ai.

Since ordEα is a valuation, we get

d ordEα(f) = ordEα(fd) ≥ min1≤i≤d

(ordEα(ai) + (d− i) ordEα(f)).

It we pick i achieving the minimum, we get

i ordEα(f) ≥ ordEα(ai) ≥ ordEα(ai) = i ordEα(a),

and hence f ∈ a.

Example 8.12. Let Y ⊂ X be an irreducible closed analytic subset. For eachm ∈ N, the ideal sheaf

I〈m〉Y := f ∈ OX | ordY (f) ≥ m

is called the m-th symbolic power of IY . If we denote by µ : X → X the normal-ized blowup of X along Y , the restriction of µ over the manifold Xreg \Ysing is theblowup along the smooth connected submanifold Xreg∩Yreg, and µ−1(Xreg∩Yreg)is thus irreducible. As a consequence, there exists a unique Rees divisor Eα of Y

with cX(Eα) meeting Xreg ∩ Yreg, and then ordY = ordEα . This shows that I〈m〉Y

is (coherent and) valuative, and hence integrally closed. In particular, ImY ⊂ I〈m〉Y ,

the inclusion being strict in general, even when X is smooth.


For later use, we now discuss an alternative valuative characterization of inte-gral closure. The data of a germ of holomorphic curve γ : (C, 0)→ (X,x) definesa (semi)valuation ordγ on OX,x by setting for f ∈ OX,x

ordγ(f) := ord0(f γ) ∈ N ∪ ∞

For a coherent ideal sheaf a, we set as above

ordγ(a) := inff∈ax

ordγ(f),

which is again achieved among any given set of generators of ax.

Proposition 8.13. Let a ⊂ OX be a coherent ideal sheaf. A germ f ∈ OXbelongs to a if and only if ordγ(f) ≥ ordγ(a) for all germs γ : (C, 0)→ X.

In view of Corollary 8.6, the result says that for any function f ∈ OX thatdoes not satisfy |f | = O(

∑i |fi|) near x ∈ X, there exists γ : (C, 0) → (X,x)

such that f γ is not in the ideal of OC,0 generated by the fi γ, a result knownas the curve selection lemma.

Proof. Since ordγ is a (semi)valuation, f ∈ a implies ordγ(f) ≥ ordγ(a) just asin the proof of Corollary 10.15. Assume conversely that this condition holds forall γ : (C, 0)→ X, and denote by (Eα) the Rees divisors of a, i.e. the irreducible

components of D on the normalized blowup π : X → X. At a general point ofEα, X is smooth, and Eα is a smooth hypersurface. It is thus easy to find a local

gern γ : (C, 0)→ X transverse to Eα at a general point of X with ordγ(f π) =ordEα(f) and ordγ(a · O

X) = ordEα(a · O

X). Setting γ := π γ : (C, 0)→ X, we

get

ordEα(f) = ordγ(f) ≥ ordγ(a) = ordEα(a).

This is true for all α, and hence f ∈ a.

9. Fundamental properties of multiplier ideals

9.1. Singularities of psh functions. Given two quasi-psh functions ϕ,ψ, wesay that ϕ is more singular than ψ if ϕ ≤ ψ +O(1) locally on X.

Definition 9.1. A quasi-psh function ϕ on a complex manifold is said to haveanalytic singularities if there exists c > 0 and a coherent ideal sheaf a ⊂ OX suchthat we locally have

ϕ = c log

∑j

|fj |

+O(1)

for some (and hence, any) choice of local generators (fj) of a.

We sometimes say more precisely that ϕ has analytic singularities of type ac.If ϕ is less singular than a quasi-psh function ψ, we simply say that ac is lesssingular than ψ.


9.2. Multiplier ideals. In what follows, X denotes a complex manifold.

Definition 9.2. The multiplier ideal (sheaf) of a quasi-psh function ϕ on a com-plex manifold X is the ideal sheaf J(ϕ) ⊂ OX of germs of holomorphic functionf with |f |e−ϕ ∈ L2

loc.

As a first key property, we have:

Theorem 9.3. Multiplier ideal sheaves are coherent.

A we shall see later, a similar result actually holds for all Lploc with 1 ≤ p <∞,cf. Theorem 10.20.

Lemma 9.4. If νx(ϕ) ≥ n+ k with k ∈ N, then I(ϕ)x ⊂ mk+1x .

A partial converse to this elementary result is provided by Corollary 9.18 below.

Proof. Arguing in local coordinates, we may assume that (X,x) = (Cn, 0). By(10.1), we have ϕ(z) ≤ (n+k) log |z|+O(1), and hence J(ϕ)0 ⊂ J((n+k) log |z|)0,i.e. ˆ

|f(z)|2

|z|2(n+k)< +∞

on a small enough polydisc. By Lemma 9.5 below, we haveˆ ∏i |zi|2αi

(∑

i |zi|2)n+k)< +∞

for each monomial zα occuring in the Taylor expansion of f , and it is easy to seethat this implies |α| =

∑i αi ≥ k + 1 (a special case of Theorem 9.9 below).

Lemma 9.5. Let f =∑

α∈Nn fαzα be a holomorphic function on Dn, and assume

that h ≥ 0 is measurable on Dn and radial, i.e. h(z1, . . . , zn) only depends on |zi|.Then ˆ

Dn|f |2h =

∑α∈Nn

|fα|2ˆDn|zα|2h(z).

Proof. Pass to polar coordinates and use the Parseval formula for the L2-normof a Fourier series.

Lemma 9.6. Let A be an arbitrary set of holomorphic functions, and a ⊂ OXthe ideal sheaf locally generated by A. Then a is coherent.

Proof. Let I be the directed set of finite subsets of A. For each i ∈ I, thecorresponding (globally) finitely generated sheaf ai is coherent. We then havea =

⋃i ai, and for each compact set K ⊂ X, the strong noetherian property of

coherent sheaves shows that a = ai near K for some i, hence the result.

Proof of Theorem 9.3. Arguing locally, we may argue in the ball B ⊂ Cn, andassume that ϕ is psh. Let H = H(B,ϕ) be the Bergman space of holomorphicfunctions f ∈ O(B) such that

´B |f |

2e−ϕ < +∞. By Lemma 9.6, the ideal sheafa ⊂ OB generated by H is coherent, and it is of course contained in J(ϕ). It willthus be enough to show that J(ϕ) = a. By the Rees lemma, it will be enough toshow that J(ϕ)x ⊂ ax+mk

x for any given x ∈ B and k ∈ N. Pick f ∈ J(ϕ)x, and let


U ⊂ B be an open neighborhood of x such that f ∈ O(U) andÚ |f |

2e−2ϕ < +∞.Let χ ∈ C∞c (U) be a cut-off function with χ ≡ 1 near x. For C 1, the function

ψ(z) := ϕ(z) + (n+ k)χ(z) log |z − x|+ C|z|2

is satisfies i∂∂ψ+ Ric(ω) = i∂∂ψ ≥ εω, where ω = i2∂∂|z|

2 is the Kahler form ofCn. Note that

v := ∂(χf) = f∂χ

is L2 with respect to ψ. By Theorem 6.5, we may thus find a function u suchthat

´B |u|

2e−2ψdV < +∞ and ∂u = v. Note that g := u − χf belongs to H.

Since u is holomorphic near x and νx(ψ) ≥ n+k, Lemma 9.4 shows that u ∈ mkx,

and we are done.

9.3. An example: the toric case. The results in this section are due to Gue-nancia [Gue12]. A function defined on the polydisc Dn is toric if it invariantunder the compact (S1)n, i.e. its value at (z1, . . . , zn) only depends on |zi|. ByLemma 2.3, a toric function ϕ on Dn is psh if and only if

ϕ(z1, . . . , zn) = χ(log |z1|, . . . , log |zn|)where χ : Rn− → R is convex and non-decreasing in each variable.

Definition 9.7. The Newton set P (χ) of a convex function χ : Rn− → R is theset of α ∈ Rn such that 〈α, ·〉 ≤ χ+O(1) on Rn−.

It is easy to see that P (χ) is convex and invariant under positive translationP (χ) + Rn+ ⊂ P (χ). It is however neither open nor closed in general, and we

denote by P (χ) and P (χ) its closure and interior. By definition, P (χ) is thedomain of the Legendre transform

χ?(α) = supx∈Rn−

(〈α, x〉 − χ(x)) ,

and hence

(9.1) χ(x) = supα∈P (χ)

(〈α, x〉 − χ?(α)),

by Legendre duality. In particular, χ is non-decreasing in each variable if andonly if P (χ) ⊂ Rn+.

Example 9.8. If χ is convex and piecewise affine, i.e. χ = maxi (〈αi, ·〉+ ci)with αi ∈ Rn+ and ci ∈ R, then P (χ) is the Newton polyhedron of the αi, i.e. theconvex envelope of

⋃i(αi +Rn+). For ϕ(z) = log |z| = maxi log |zi|+O(1), we get

for instance

P (log |z|) =

α ∈ Rn+ | |α| :=

∑i

αi ≥ 1

.

Set 1 := (1, . . . , 1) ∈ Rn+.

Theorem 9.9. If ϕ = χ(log |z1|, . . . , log |zn|) is a toric psh function on Cn, then

J(ϕ)0 is generated by monomials, and zβ ∈ J(ϕ)0 if and only iff β + 1 ∈ P (ϕ).

As a consequence, we recover the following description of algebraic multiplierideals in the toric case, originally due to Howald.


Corollary 9.10. If a is a monomial ideal generated by a (possibly infinite) familyof monomials (zα)α∈A with A ⊂ Nn and c > 0, then zβ ∈ J(ac)0 if and only

β + 1 ∈ c P , with P the Newton polyhedron of A, i.e. the convex envelope of⋃α∈A(α+ Rn+).

Example 9.11. We have J(c log |z|)0 = mdce−n+10 .

As we shall see, passing to polar coordinates easily reduces Theorem 9.9 tothe following convex-analytic discussion. Define the homogeneization of χ as theconvex, positively homogeneous function

χhom(x) := supα∈P (χ)

〈α, x〉.

Using (9.1), one easily checks that for each a, x ∈ Rn−,

χhom(x) = limt→+∞

χ(a+ tx)

t

computes the slope at infinity of χ along the ray a+ R+x.

Proposition 9.12. If χ : Rn− → R is convex, the following conditions are equiv-alent.

(i) e−χ is integrable;

(ii) 0 ∈ P (ϕ);(iii) χ decays at least linearly at infinity, i.e. χ(x) ≤ −δ|x| + C for some

δ, C > 0;(iv) χhom(x) < 0 for each nonzero x ∈ Rn+.

Proof. It is clear that (ii)⇐⇒(iii)=⇒(i). Assume that e−χ ∈ L1, and pick x ∈Rn−. By Fubini-Study, there exists a ∈ Rn− such that the restriction of e−χ tothe ray R = a + R+x is integrable. By convexity of the one-variable functionχ|R, this clearly implies that the slope at infinity χhom(x) is negative, and weinfer (i)=⇒(iv). By homogeneity and compactness of

x ∈ Rn− | |x| = 1

, (iv) is

equivalent to χhom(x) ≤ −δ|x| for some δ > 0, which is in turn equivalent to(iii).

9.4. The Nadel vanishing theorem.

Theorem 9.13 (Nadel vanishing). Let (X,ω) be Kahler manifold, and assumethat X is weakly pseudonvex and contains a Stein Zariski open subset (e.g. Xprojective). Let L be a Hermitian holomorphic line bundle on X with curvatureform θ, and let also ϕ be a quasi-psh function such that θ + i∂∂ϕ ≥ εω for someε > 0. Then

Hq(X,O(KX + L)⊗ J(ϕ)) = 0

for all q ≥ 1.

Proof. For each q ≥ 0, denote by Aq the sheaf of germs u of measurable sectionsof Λn,qT ?X ⊗ L such that both ue−ϕ and ∂ue−ϕ are L2

loc. The ∂ operator inducesa complex of sheaves

0→ O(KX + L)⊗ J(ϕ)→ A0 → · · · → An.


An application of Theorem 6.5 to a small ball in X shows that this complex isexact. Since each Aq is a C∞-module, the de Rham-Weil theorem implies thatHq(X,O(KX+L)⊗J(ϕ)) is isomorphic to the q-th cohomology group of the com-plex of global sections of Aq. Let v be a ∂-closed global section on X of Aq, q ≥ 1,denote by U the Stein Zariski open subset of X, and pick a (strictly) psh exhaus-tion function ψ : X → R+. For each t ≥ 0,

´ψ<t |v|

2e−2ϕdV is finite, and we may

thus choose χ : R+ → R+ convex, increasing such that´X |v|

2e−2(ϕ+χψ)dV <

+∞. By Theorem 6.5, there exists u ∈ L2loc(X,Λ

n,q−1T ?X ⊗ L) such that ∂u = v

and´X |u|

2e−2(ϕ+χψ)dV < +∞. In particular, u defines a global section of Aq−1

on X, and the result follows.

9.5. Restriction property and subadditivity. A fairly direct application ofthe ’local’ Ohsawa–Takegoshi theorem yields the following qualitative counterpartof it.

Theorem 9.14. If ϕ is quasi-psh on a complex manifold X, then J(ϕ|Y ) ⊂J(ϕ)|Y for each complex submanifold Y ⊂ X.

Applying this to the diagonal in X × X, one can show (see [DEL, Theorem2.6]):

Theorem 9.15. For any two quasi-psh functions ϕ,ψ, we have J(ϕ + ψ) ⊂J(ϕ) · J(ψ).

Corollary 9.16. If k divides m, then J(mϕ)1/m is more singular than J(kϕ)1/k.

9.6. Demailly’s and Skoda’s theorem. The following result is a ’qualitative’version of a famous regularization theorem due to Demailly [Dem92].

Theorem 9.17. Let ϕ be a quasi-psh function on a complex manifold. Then themultiplier ideal J(ϕ) is less singular than ϕ.

Proof. Ohsawa–Takegoshi extension from a point.

Corollary 9.18. If νx(ϕ) < 1, then J(ϕ)x = OX,x.

Proof. By Theorem 9.17, we have νx(ϕ) ≥ ordx J(ϕ), and hence ordx J(ϕ) = 0since the latter is an integer.

9.7. Openness property. Let ϕ be a quasi-psh function on a complex manifoldX, and set

J+(ϕ) :=⋃ε>0

J((1 + ε)ϕ) ⊂ OX .

Since the singularities of (1 + ε)ϕ decrease as ε 0, the strong noetherianproperty shows that J+(ϕ) = J(((1 + ε)ϕ) for all 0 < ε 1 near any givencompact subset of X.

The following result is known as the ’strong openness conjecture’ of Demailly–Kollar [DK01], and was established by Guan–Zhou [GZ15].


Theorem 9.19. For each quasi-psh function ϕ on a complex manifold, we haveJ+(ϕ) = J(ϕ). Equivalently, for any holomorphic function f with |f |e−ϕ L2 in a

neighborhood of a compact K ⊂ X, there exists ε > 0 such that |f |e−(1+ε)ϕ is L2

in a neighborhood of K.

The proof presented below, which is basically that of Guan and Zhou, relieson the following two elementary facts.

Lemma 9.20. If g is an integrable function near 0 ∈ Rd, then there exists asequence xj → 0 in Rd such that |g(xj)| = o

(|xj |−d

).

Proof. Applying the Fubini-Study in polar coordinates yields v ∈ Sd−1 such thath(r) := rd−1g(rv) is integrable near 0 ∈ R, and we are thus reduced to the cased = 1. If the result fails, then |h(r)| ≥ c/r near 0 for some c > 0, contradictingthe integrability at 0.

Lemma 9.21. Let a, b be holomorphic functions near D ⊂ C such that ord0(a) >ord0(b), and assume that b is nowhere zero on D\0. Assume also given t0 ∈ D?with a = b on all k-th roots of t0. Then

|t0| sup∂D|a| ≥ inf

∂D|b| > 0.

Proof. After replacing a with a(tk)/b(tk), we can assume that b = 1 and k = 1.By assumption, a vanishes as 0 and satisfies a(t0) = 1. Applying the maximumprinciple to the holomorphic function a(t)/t, we get as desired

|t0|−1 = |t−10 a(t0)| ≤ sup

∂D|t−1a(t)| = sup

∂D|a|.

Proof of Theorem 9.19. The result being local, we can assume that ϕ is psh andϕ ≤ 0. Pick a germ f such that |f |e−ϕ is L2

loc at a point x ∈ X. Since J+(ϕ)locally coincides with J((1 + ε)ϕ) for 0 < ε 1, it is integrally closed (cf. Ex-ample 8.8). By Proposition 8.13, it is thus enough to show that ordγ(f) ≥ordγ(J+(ϕ)) for each germ of curve γ : (C, 0) → (X,x). Arguing by contradic-tion, we assume ordγ(f) < ordγ(J+(ϕ)), which implies in particular that f γhas isolated zeroes.

Pick local coordinates z = (z1, . . . , zn) at x. After permuting the indices,we can assume that the image of γ is not entirely contained in z1 = 0.Reparametrizing γ and scaling the coordinates, we then arrange the following:

(i) f ∈ O(Dn) satisfies´Dn |f |

2e−2ϕ < +∞;

(ii) γ : U → Dn is defined on a neighborhood U of D;(iii) γ1(s) := z1 γ(s) = sk for some k ∈ N?;(iv) f γ is nowhere zero on D \ 0.

By (i) and Lemma 9.20, there exists a sequence tj ∈ D converging to 0 such thatˆz1=tj∩Dn

|f |2e−2ϕ = o(|tj |−2

).


Since f γ is nonzero on any k-th root of tj , f |z1=tj∩Dn is not identically zero,

and hence´z1=tj∩Dn |f |

2e−2ϕ > 0. Pick 0 < r < 1 such that γ(D) ⊂ Dnr .

Arguing by induction on the dimension, we get εj > 0 such thatˆz1=tj∩Dnr

|f |2e−2(1+εj)ϕ ≤ 2

ˆz1=tj∩Dn

|f |2e−2ϕ = o(|tj |−2

).

By the ’local version’ of the Ohsawa–Takegoshi extension theorem, there existsgj ∈ O(Dnr ) such that gj |z1=tj∩Dnr = f |z1=tj∩Dnr and

ˆDnr|gj |2e−2(1+εj)ϕ ≤ C

ˆz1=tj∩Dnr

|f |2e−2(1+εj)ϕ = o(|tj |−2

)for some uniform constant C > 0. By (iii), we have (gj γ)(s) = (f γ)(s) for all

s ∈ D such that γ1(s) = sk = tj . Also gj ∈ J+(ϕ), and hence

ord0(gj γ) ≥ ordγ(J+(ϕ)) > ord0(f γ).

By Lemma 9.21, it follows that

|tj | sup∂D|gj γ| ≥ inf

∂D|f γ| > 0,

and hence sup∂D |gj γ| ≥ c|tj |−1. On the other hand, since γ(∂D) is compact inDnr , the mean value inequality gives

sup∂D|gj γ|2 ≤ C

ˆDnr|gj |2 ≤ C

ˆDnr|gj |2e−2(1+εj)ϕ = o

(|tj |−2

),

contradiction.

Corollary 9.22. Let ϕ,ψ be quasi-psh functions, and assume that ψ has analyticsingularities. If eψ−ϕ is L2 on a neighborhood a compact K ⊂ X, then there existsε > 0 such that eψ−(1+ε)ϕ is L2 on a neighborhood of K.

Proof. After passing to a log resolution, we may assume that X = Dn and ψ =∑ni=1 ci log |zi| with ci ∈ R+. Writing ci = mi + ri with mi ∈ N, we see that

eψ−ϕ ∈ L2loc iff f :=

∏i zmii belongs to J(ϕ+

∑i ri log |zi|). By openness, we get

ε > 0 such that

f ∈ J((1 + ε)(ϕ+∑i

ri log |zi|)) ⊂ J((1 + ε)ϕ+∑i

ri log |zi|),

i.e. eψ−(1+ε)ϕ ∈ L2loc.

10. Valuative description of multiplier ideals

10.1. Analytic semicontinuity of Lelong numbers. Recall from Section 2.3that the Lelong number of a psh function u near 0 ∈ Cn satisfies

(10.1) ν0(u) = maxγ ∈ R+ | u(z) ≤ γ log |z|+O(1) near 0.

Let now ϕ be a quasi-psh function on a complex manifold X. Near each pointx ∈ X, we have by definition ϕ = u+f with u psh and f smooth. Choosing local


coordinates z centered at x ∈ X, we can view u as a psh function near 0 ∈ Cnand we set νx(ϕ) := ν0(u). Clearly,

νx(ϕ) = maxγ ∈ R+ | ϕ ≤ γ log |z|+O(1) near x,and the definition is thus independent of the choice of local coordinates at x, anytwo choices local coordinates being Lipschitz equivalent.

Example 10.1. If ϕ has analytic singularities of type ac, then νx(ϕ) = c ordx(a).

By Lemma 2.6, νx(ϕ) is a usc function of x ∈ X, and also a usc function of ϕwith respect to the L1

loc topology. The following deep result was originally provedby Siu [Siu74]. The much simpler argument we present here is due to Demailly.

Theorem 10.2. If ϕ is a quasi-psh function on a complex manifold X, then x 7→νx(ϕ) is usc in the analytic Zariski topology. In other words, x ∈ X | νx(ϕ) ≥ cis a closed analytic subset of X for each c ∈ R.

The key point is the following consequence of the Ohsawa–Takegoshi theorem.

Lemma 10.3. For each x ∈ X we have νx(ϕ) ≥ ordx J(ϕ) ≥ νx(ϕ)− n.

Proof. By Theorem 9.17, ϕ is more singular than J(ϕ), and hence νx(ϕ) ≥ordx J(ϕ). To prove the converse, pick coordinates z at x, and set γ := νx(ϕ).Since ϕ ≤ γ log |z|+ C near x, we have

J(ϕ)x ⊂ J(γ log |z|)x = mdγe−n+1x ,

and hence ordx J(ϕ) ≥ νx(ϕ)− n.

Proof of Theorem 10.2. When ϕ has analytic singularities, the result amounts tothe fact that x ∈ X | ordx(a) ≥ c is closed analytic for a coherent ideal sheafa ⊂ OX , which is easy to see. Applying Lemma 10.3 to kϕ with k ∈ N?, we get

|νx(ϕ)− k−1 ordx J(kϕ)| ≤ n/k,which proves that x 7→ νx(ϕ) is the uniform limit of x 7→ k−1 ordx J(kϕ). Bycoherence of J(kϕ), x 7→ k−1 ordx J(kϕ) is usc in the analytic Zariski topologyfor each k, and hence so is their uniform limit x 7→ νx(ϕ).

10.2. Generic Lelong numbers. As above, ϕ is a quasi-psh function on acomplex manifold X.

Definition 10.4. The generic Lelong number of ϕ along an irreducible closedanalytic subset Y ⊂ X is defined as

νY (ϕ) := infy∈Y

νy(ϕ).

The terminology is justified by the following consequence of the analytic semi-continuity theorem.

Lemma 10.5. We have νy(ϕ) = νY (ϕ) for a very general point y ∈ Y , i.e. anypoint outside a countable union of strict closed analytic subsets Zj of Y .

Proof. For each j ≥ 1, Zj := y ∈ Y | νy(ϕ) ≥ νY (ϕ) + 1/j is a strict closedanaytic subset of Y , and νy(ϕ) = νY (ϕ) for each x ∈ Y \

⋃j Zj .


As a consequence, for any two quasi-psh functions ϕ,ψ we get the valuativeformulae

νY (ϕ+ ψ) = νY (ϕ) + νY (ψ)

andνY (maxϕ,ψ) = minνY (ϕ), νY (ψ).

Proposition 10.6. For each regular point y ∈ Yreg, νY (ϕ) is the largest c ∈ R+

such that ϕ is more singular then IcY near y. When Y has codimension 1, this isalso true when y is a singular point of Y .

Proof. If ϕ is more singular than IcY near y, then νy′(ϕ) ≥ c ordy′ IY ≥ c forall y′ ∈ Y near y. Since νy′(ϕ) = νY (ϕ) for a very general y′ ∈ Y , we inferνY (ϕ) ≥ c. Conversely, pick y ∈ Yreg a regular point, and choose ocal coordinatesz = (z1, . . . , zn) at x such that z1, . . . , zp generate IY near y. By (2.4), thereexists C > 0 such that

ϕ(z) ≤ νy(ϕ) log |z − y|+ C ≤ νY (ϕ) log |z − y|+ C

for all z, y near x = 0. Choosing y = (0, . . . , 0, zp+1, . . . , zn) ∈ E, we get ϕ ≤νY (ϕ) log(|z1|2 + · · ·+ |zp|2)1/2 +C near x, i.e. ϕ is more singular then I

νY (ϕ)Y near

y.Assume now that Y is a prime divisor. Since X is smooth, Y admits a local

equation fY near any given y ∈ Y . Arguing locally, we can assume that ϕ is psh.Since log |fY | is pluriharmonic outside Y , ϕ − νY (ϕ) log |fY | is psh outside Y .Being locally bounded above near each point of Yreg, it extends to a psh functionψ outside Ysing. But the latter has codimension at least 2 in X, and ψ is thusautomatically bounded above near near Ysing. In particular, ϕ ≤ νY (ϕ) log |fY |+O(1) near y.

Remark 10.7. The second part of Proposition 10.6 fails in general when Yhas codimension at least 2. Indeed, a holomorphic function f ∈ OX satisfies

ordY (f) = νY (log |f |) ≥ m if and only f belongs to the symbolic power I〈m〉Y . On

the other hand, log |f | is more singular than ImY if and only if f belongs to the

integral closure ImY , which is strictly contained in I〈m〉Y is general.

10.3. The case of analytic singularities. We start with a few preliminaryfacts. Let µ : Y → X be a modificatin with Y smooth. The Jacobian of µ, locallydefined in coordinate charts, yields a global section of the relative canonical bundleKY/X := KY − µ?KX . The corresponding divisor is identified with KY/X , andits support coincides with the exceptional locus Exc(µ). We may then show thatA := µ(Exc(µ)) has codimension at least 2 in X, and µ induces an isomorphismover X \A.

The log discrepancy of a prime divisor E ⊂ Y is defined as

AX(E) := 1 + ordE(KY/X).

If Y ′ is a modification of Y , the strict transform E′ of E on Y ′, then AX(E) =AX(E′).

Example 10.8. Let µ : Y → X be the blowup of a submanifold Z ⊂ X. ThenExc(µ) = E = P(NZ/X), and KX′/X = (codimZ − 1)E, i.e. AX(E) = codimZ.


Given a psh function ϕ on X, we define νE(ϕ) as the generic Lelong numberof ϕ µ along E.

Example 10.9. If ϕ has analytic singularities of type ac, then we have for eachprime divisor E over X

νE(ϕ) = ordE(ac) := cminf∈a

ordE(f µ),

Theorem 10.10. Let ϕ,ψ be two quasi-psh functions on a complex manifold X,and assume that ψ has analytic singularities. For each compact subset K ⊂ X,we then have

inf

νE(ϕ)

νE(ψ)| νE(ψ) > 0, cX(E) ∩K 6= ∅

= maxγ ≥ R+ | ϕ ≤ γψ+O(1) near K,

where the left-hand infimum ranges over all prime divisors E over X. Further,the infimum on the left-hand is achieved among the Rees valuations of ψ.

Definition 10.11. A log resolution of a coherent ideal sheaf a ⊂ OX is a properbimeromorphic morphism µ : X ′ → X with X ′ a complex manifold, such that

(i) a · OX′ = OX′(−D) with D an effective divisor on X ′;(ii) D +KX′/X has snc1 support.

The existence of a log resolution is guaranteed by Hironaka, later simplfiied byBierstone-Milman, Wlodarczyk, etc...

Proposition 10.12. Assume that ϕ has analytic singularities of type ac, and letµ : Y → X be a log resolution of a, with snc divisor

∑iEi. Then

J(ϕ) = J(ac) = f ∈ OX | ordEi(f) > ordEi(ac)−AX(ordEi) for all i .

Proof. Change of variable. J(ϕ) = µ∗OY (KY/X − bDc).

10.4. Valuative characterization of integrability. Let X be a complex man-ifold.

Theorem 10.13. Let ϕ,ψ be psh functions near 0 ∈ X.

(i) If eψ−ϕ ∈ L2loc near 0 ∈ X, then νE(ψ) ≥ νE(ϕ) − AX(E) for all prime

divisors E over X such that 0 ∈ cX(E).(ii) Assume conversely that there exists ε > 0 such that

νE(ψ) ≥ (1 + ε)νE(ϕ)−AX(E)

for all prime divisors E over X such that 0 ∈ cX(E). Then eψ−ϕ ∈ L2loc

at 0.

Remark 10.14. Arguing by approximation as in [BFJ08, Proposition 3.7], oneshows that it is enough to check (ii) for prime divisors E such that cX(E) = 0.

Corollary 10.15. If ψ further has analytic singularities, then eψ−ϕ ∈ L2loc iff

there exists ε > 0 such that νE(ψ) ≥ (1 + ε)νE(ϕ)−AX(E) for all prime divisorsE over X such that 0 ∈ cX(E).

1Short for simple normal crossing.


Proof. When ψ has analytic singularities, the condition eψ−ϕ ∈ L2loc implies

eψ−(1+ε)ϕ ∈ L2loc for some ε > 0, by Corollary 9.22. The result is now a di-

rect consequence of Theorem 10.13.

As the next example shows, the assumption on analytic singularities in Corol-lary 10.15 cannot be dispensed with.

Example 10.16. Take ψ(z) = − log(− log |z|) and ϕ = log |z| on the unit disc

X = D. Thené2(ψ−ϕ) =

´ |dz|2|z|2(log |z|)2 converges near 0, but ν0(ψ) = 0 and

ν0(ϕ) = AX(0) = 1.

Lemma 10.17. Let ϕ,ψ be psh functions near 0 ∈ Cn, and assume that eψ−ϕ ∈L2

loc at 0. Then νH(ψ) ≥ νH(ϕ)− 1 with H := (z1 = 0).

Proof. Set c := νH(ϕ). Since ϕ is more singular than c log |z1|, we may assumewlog that ϕ = c log |z1|. After scaling, we may also assume that eψ−ϕ = eψ|z1|−cis L2 on the unit polydisc Dn.

Assume first that n = 1, and hence H = 0. For each t ∈ R large enough setχ(t) :=

fflS(0,e−t) ψ. By Lemma 2.8,

limt→+∞

χ(t)/t = −ν0(ψ),

and we thus need to show that limt→+∞ χ(t)/t ≤ 1 − c. By Jensen’s inequality,fflS(0,e−t) e

2ψ ≥ e2χ(t), and hence

+∞ >1

2π

ˆDe2ψ|z|−2c|dz|2 =

ˆ +∞

0e2(c−1)tdt

S(0,e−t)

e2ψ ≥ˆ +∞

0e2(χ(t)+(c−1)t)dt.

We may thus find a sequence ti → +∞ such that χ(ti) + (c− 1)ti ≤ 0, and hencelimt→+∞ χ(t)/t = limχ(ti)/ti ≤ 1− c.

In the general case, the Fubini–Torelli theorem yieldsˆDe2ψ(z1,z′)|z1|−2c|dz1|2 < +∞

for a.e. z′ ∈ Cn−1 near 0. By the one-variable case, we thus have ν0(ψ(·, z′)) ≥c− 1 for a.e. z′, and (2.4) thus yields

ψ(z1, z′) ≤ (c− 1)(log |z1| − log 2) + sup

|z1|≤1/2ψ(z1, z

′) ≤ (c− 1) log |z1|+ C

for some uniform constant C and a.e. z′. As a result, this holds for all z′ near 0,and we conclude as desired that νH(ψ) ≥ c− 1.

Proof of Theorem 10.13. Assume that eψ−ϕ ∈ L2loc near 0 ∈ X. Let µ : Y → X

be a modification and E a prime divisor over X with 0 ∈ cX(E) = µ(E). Chooselocal coordinates (z1, . . . , zn) on Y near a regular point of E close to a preimageof 0. The change of variable formula yields

eψµ−ϕµ|z1|a ∈ L2loc

with a := ordE(KY/X) = AX(E)− 1. By Lemma 10.17, we infer

νE(ψ µ) + a ≥ νE(ϕ µ)− 1,


i.e. νE(ψ) ≥ µE(ϕ)−AX(E), which proves (i).Conversely, assume given ε > 0 such that νE(ψ) ≥ (1 + ε)νE(ϕ) − AX(E)

for all prime divisors E over X with cX(E) = 0. Assume first that ϕ,ψ haveanalytic singularities. We can then find a modification ψ : X ′ → X and ansnc divisor D =

∑iEi supporting KX′/X and such that ϕ and ψ have divisorial

singularities alongD. The change of variable formula then shows that eψ−ϕ ∈ L2loc

iff νEi(ψ) > νEi(ϕ)−AX(Ei) for all i, and we are done.Assume now that ϕ,ψ are arbitrary psh functions. For each k ∈ Z>0, let

ϕk (resp. ψk) be a psh function with analytic singularities of type J(kϕ)1/k

(resp. J(kψ)1/k). By Theorem 9.17, ϕk (resp. ψk) is less singular then ϕ (resp. ψ).For each prime E over X we thus have

νE(ϕ) ≥ νE(ϕk) ≥ νE(ϕ)− k−1AX(E),

νE(ψ) ≥ νE(ψk) ≥ νE(ψ)− k−1AX(E).

Thus

νE(ψk) ≥ (1 + ε)νE(ϕk)− (1 + k−1)AX(E),

and hence

νE(ψk) ≥1

1 + k−1νE(ψk) ≥

1 + ε

1 + k−1νE(ϕk)−AX(E) ≥ (1+ε/2)νE(ϕk)−AX(E)

for all k 1. By the case of functions with analytic singularities, we infereψk−(1+εk)ϕk ∈ L2

loc for any εk ∈ [0, ε/2), hence also eψ−(1+εk)ϕk ∈ L2loc, since ψ is

more singular than ψk. To conclude the proof of (i), it remains to note, followingDemailly, that

eψ−(1+εk)ϕk ∈ L2loc =⇒ eψ−ϕ ∈ L2

loc

for any εk ≥ 1k−1 . To see this, note that ek(ϕk−ϕ) ∈ L2

loc, by definition of the

multiplier ideal J(kϕ). Since ψ is locally bounded above, we haveê2(ψ−ϕ) ≤

ˆϕ≥(1+εk)ϕk

e2(ψ−(1+εk)ϕk) + C

ˆϕ≤(1+εk)ϕk

e−2ϕ

≤ê2(ψ−(1+εk)ϕk) + C

ˆϕ≤(1+εk)ϕk

e2(−ϕ+k(ϕ−ϕk))e2k(ϕk−ϕ)

≤ê2(ψ−(1+εk)ϕk) + C ′

ê2k(ϕk−ϕ) <∞,

since ϕ+ k(ϕ−ϕk) is bounded above when ϕ ≤ (1 + εk)ϕk ≤ kk−1ϕk +O(1).

Corollary 10.18. A function f ∈ OX belongs to J(ϕ) iff there exists ε > 0 suchthat ordE(f) ≥ (1 + ε)νE(ϕ)−AX(E) for all prime divisors E over X.

Remark 10.19. Interestingly, the ideal sheaf of germs f ∈ OX such that ordE(f) ≥νE(ϕ)−AX(E) for all prime divisors E over X is not coherent in general. Indeed,Corollary 10.18 implies that this ideal coincides with ∩δ>0J((1 − δ)ϕ), which isnot always coherent, by [Wu19, Proposition 2].

The following result is based on [Cao, Lemma 3.4] and [BFJ08].


Theorem 10.20. For each p ∈ (0,∞), the sheaf

Lp(ϕ) :=f ∈ OX | |f |e−ϕ ∈ Lploc

is coherent. Further, a germ f ∈ OX is in Lp(ϕ) iff there exists ε > 0 such that

ordE(f) ≥ (1 + ε)νE(ϕ)− p−1AX(E)

for all prime divisors E over X.

Lemma 10.21. If ϕk has analytic singularities of type J(kϕ)1/k, then there existsa sequence εk → 0 such that

Lp(ϕ) =⋃k

Lp((1 + εk)ϕk).

Proof. Note that e2k(ϕk−ϕ) ∈ L1loc. As in [Dem12, (16.12)], this impliesˆ

|f |pe−pϕ =

ˆϕ≥(1+εk)ϕk

|f |pe−pϕ +

ˆϕ<(1+εk)ϕk

|f |pe2k(ϕk−ϕ)e(2k−p)ϕ−2kϕk

≤ˆ|f |pe−(1+εk)ϕk + C

ê2k(ϕ−ϕk)

whenever εk ≥ p2k−p , i.e. (1 + εk) ≥ 2k

2k−p , because (2k − p)ϕ − 2kϕk is then

bounded above on ϕ < (1 + εk)ϕk. This proves

Lp((1 + εk)ϕk) ⊂ Lp(ϕ).

In the other direction, pick f ∈ Lp(ϕ). Pick a positive integer m such that2m > p, and note thatˆ

|f |pe−pϕ =

ˆ|f |2me−(pϕ+(2m−p) log |f |) < +∞,

i.e. fd ∈ J(pϕ + (2m − p) log |f |). The openness property of multiplier idealsyields 0 < ε 1 such that

fd ∈ J((1 + ε)pϕ+ (2m− p) log |f |),

i.e.´|f |pe−(1+ε)pϕ < +∞. It follows that f ∈ Lp((1 + ε)ϕ) ⊂ Lp((1 + εk)ϕk) for

k 1, which concludes the proof.

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Documents

Contents - sebastien.boucksom.perso.math.cnrs.frsebastien.boucksom.perso.math.cnrs.fr/notes/L2.pdf · 7. The Ohsawa-Takegoshi extension theorem 26 8. Blow-ups and valuations 30 9