Seminario de Geometría y TopologíaFacultad de Ciencias Matemáticas, Universidad Complutense de Madrid

Cohomological aspects

in complex non-Kähler geometry

Daniele Angella

Istituto Nazionale di Alta Matematica

and

Dipartimento di Matematica, Università di Pisa

November 19, 2013

Introductionaim

Consider, e.g., a complex manifold:

several cohomological invariants can be de�ned(de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies).

For compact Kähler mfds, they are all isomorphic, . . .. . . but Bott-Chern cohom may give further inform for non-Kähler.

We are interested in studying how they are related each other. . .. . . by investigating explicit examples. . .. . . whose cohom is encoded in a suitable �cohomological-model�.

We focus on property �all cohomol are isomorphic� (∂∂-Lemma). . .. . . by studying its behaviour for deformations of cplx struct.

Introductionaim

Consider, e.g., a complex manifold:

several cohomological invariants can be de�ned(de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies).

For compact Kähler mfds, they are all isomorphic, . . .. . . but Bott-Chern cohom may give further inform for non-Kähler.

We are interested in studying how they are related each other. . .. . . by investigating explicit examples. . .. . . whose cohom is encoded in a suitable �cohomological-model�.

We focus on property �all cohomol are isomorphic� (∂∂-Lemma). . .. . . by studying its behaviour for deformations of cplx struct.

Introductionaim

Consider, e.g., a complex manifold:

several cohomological invariants can be de�ned(de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies).

For compact Kähler mfds, they are all isomorphic, . . .

. . . but Bott-Chern cohom may give further inform for non-Kähler.

We are interested in studying how they are related each other. . .. . . by investigating explicit examples. . .. . . whose cohom is encoded in a suitable �cohomological-model�.

We focus on property �all cohomol are isomorphic� (∂∂-Lemma). . .. . . by studying its behaviour for deformations of cplx struct.

Introductionaim

Consider, e.g., a complex manifold:

several cohomological invariants can be de�ned(de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies).

For compact Kähler mfds, they are all isomorphic, . . .. . . but Bott-Chern cohom may give further inform for non-Kähler.

We are interested in studying how they are related each other. . .. . . by investigating explicit examples. . .. . . whose cohom is encoded in a suitable �cohomological-model�.

We focus on property �all cohomol are isomorphic� (∂∂-Lemma). . .. . . by studying its behaviour for deformations of cplx struct.

Introductionaim

Consider, e.g., a complex manifold:

several cohomological invariants can be de�ned(de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies).

For compact Kähler mfds, they are all isomorphic, . . .. . . but Bott-Chern cohom may give further inform for non-Kähler.

We are interested in studying how they are related each other. . .

. . . by investigating explicit examples. . .

. . . whose cohom is encoded in a suitable �cohomological-model�.

We focus on property �all cohomol are isomorphic� (∂∂-Lemma). . .. . . by studying its behaviour for deformations of cplx struct.

Introductionaim

Consider, e.g., a complex manifold:

several cohomological invariants can be de�ned(de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies).

For compact Kähler mfds, they are all isomorphic, . . .. . . but Bott-Chern cohom may give further inform for non-Kähler.

We are interested in studying how they are related each other. . .. . . by investigating explicit examples. . .

. . . whose cohom is encoded in a suitable �cohomological-model�.

We focus on property �all cohomol are isomorphic� (∂∂-Lemma). . .. . . by studying its behaviour for deformations of cplx struct.

Introductionaim

Consider, e.g., a complex manifold:

several cohomological invariants can be de�ned(de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies).

For compact Kähler mfds, they are all isomorphic, . . .. . . but Bott-Chern cohom may give further inform for non-Kähler.

We are interested in studying how they are related each other. . .. . . by investigating explicit examples. . .. . . whose cohom is encoded in a suitable �cohomological-model�.

We focus on property �all cohomol are isomorphic� (∂∂-Lemma). . .. . . by studying its behaviour for deformations of cplx struct.

Introductionaim

Consider, e.g., a complex manifold:

several cohomological invariants can be de�ned(de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies).

For compact Kähler mfds, they are all isomorphic, . . .. . . but Bott-Chern cohom may give further inform for non-Kähler.

We are interested in studying how they are related each other. . .. . . by investigating explicit examples. . .. . . whose cohom is encoded in a suitable �cohomological-model�.

We focus on property �all cohomol are isomorphic� (∂∂-Lemma). . .

. . . by studying its behaviour for deformations of cplx struct.

Introductionaim

Consider, e.g., a complex manifold:

several cohomological invariants can be de�ned(de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies).

For compact Kähler mfds, they are all isomorphic, . . .. . . but Bott-Chern cohom may give further inform for non-Kähler.

We are interested in studying how they are related each other. . .. . . by investigating explicit examples. . .. . . whose cohom is encoded in a suitable �cohomological-model�.

We focus on property �all cohomol are isomorphic� (∂∂-Lemma). . .. . . by studying its behaviour for deformations of cplx struct.

Cohomologies of a complex manifold, ithe double complex of forms, i

X complex manifold

, i.e., locally modelled on Cn

The tangent spaces have a linear complex structure, yielding the

bundle decomposition

TX ⊗ C = T 1,0X ⊕ T 0,1X .

It moves to

∧•X ⊗ C =⊕

p+q=k

∧p,qX .

The exterior di�erential d : ∧• X → ∧•+1X splits as

d = ∂ + ∂ : ∧•,• X → ∧•+1,•X ⊕ ∧•,•+1X .

By d2 = 0, it follows ∂2 = ∂2

= ∂∂ + ∂∂ = 0, i.e.,(∧•,•X , ∂, ∂

)double complex .

Cohomologies of a complex manifold, ithe double complex of forms, i

X complex manifold, i.e., locally modelled on Cn

The tangent spaces have a linear complex structure, yielding the

bundle decomposition

TX ⊗ C = T 1,0X ⊕ T 0,1X .

It moves to

∧•X ⊗ C =⊕

p+q=k

∧p,qX .

The exterior di�erential d : ∧• X → ∧•+1X splits as

d = ∂ + ∂ : ∧•,• X → ∧•+1,•X ⊕ ∧•,•+1X .

By d2 = 0, it follows ∂2 = ∂2

= ∂∂ + ∂∂ = 0, i.e.,(∧•,•X , ∂, ∂

)double complex .

Cohomologies of a complex manifold, ithe double complex of forms, i

X complex manifold, i.e., locally modelled on Cn

The tangent spaces have a linear complex structure

, yielding the

bundle decomposition

TX ⊗ C = T 1,0X ⊕ T 0,1X .

It moves to

∧•X ⊗ C =⊕

p+q=k

∧p,qX .

The exterior di�erential d : ∧• X → ∧•+1X splits as

d = ∂ + ∂ : ∧•,• X → ∧•+1,•X ⊕ ∧•,•+1X .

By d2 = 0, it follows ∂2 = ∂2

= ∂∂ + ∂∂ = 0, i.e.,(∧•,•X , ∂, ∂

)double complex .

Cohomologies of a complex manifold, ithe double complex of forms, i

X complex manifold, i.e., locally modelled on Cn

The tangent spaces have a linear complex structure, yielding the

bundle decomposition

TX ⊗ C = T 1,0X ⊕ T 0,1X .

It moves to

∧•X ⊗ C =⊕

p+q=k

∧p,qX .

The exterior di�erential d : ∧• X → ∧•+1X splits as

d = ∂ + ∂ : ∧•,• X → ∧•+1,•X ⊕ ∧•,•+1X .

By d2 = 0, it follows ∂2 = ∂2

= ∂∂ + ∂∂ = 0, i.e.,(∧•,•X , ∂, ∂

)double complex .

Cohomologies of a complex manifold, ithe double complex of forms, i

X complex manifold, i.e., locally modelled on Cn

The tangent spaces have a linear complex structure, yielding the

bundle decomposition

TX ⊗ C = T 1,0X ⊕ T 0,1X .

It moves to

∧•X ⊗ C =⊕

p+q=k

∧p,qX .

The exterior di�erential d : ∧• X → ∧•+1X splits as

d = ∂ + ∂ : ∧•,• X → ∧•+1,•X ⊕ ∧•,•+1X .

By d2 = 0, it follows ∂2 = ∂2

= ∂∂ + ∂∂ = 0, i.e.,(∧•,•X , ∂, ∂

)double complex .

Cohomologies of a complex manifold, ithe double complex of forms, i

X complex manifold, i.e., locally modelled on Cn

The tangent spaces have a linear complex structure, yielding the

bundle decomposition

TX ⊗ C = T 1,0X ⊕ T 0,1X .

It moves to

∧•X ⊗ C =⊕

p+q=k

∧p,qX .

The exterior di�erential d : ∧• X → ∧•+1X splits as

d = ∂ + ∂ : ∧•,• X → ∧•+1,•X ⊕ ∧•,•+1X .

By d2 = 0, it follows ∂2 = ∂2

= ∂∂ + ∂∂ = 0, i.e.,(∧•,•X , ∂, ∂

)double complex .

Cohomologies of a complex manifold, ithe double complex of forms, i

X complex manifold, i.e., locally modelled on Cn

The tangent spaces have a linear complex structure, yielding the

bundle decomposition

TX ⊗ C = T 1,0X ⊕ T 0,1X .

It moves to

∧•X ⊗ C =⊕

p+q=k

∧p,qX .

The exterior di�erential d : ∧• X → ∧•+1X splits as

d = ∂ + ∂ : ∧•,• X → ∧•+1,•X ⊕ ∧•,•+1X .

By d2 = 0, it follows ∂2 = ∂2

= ∂∂ + ∂∂ = 0

, i.e.,(∧•,•X , ∂, ∂

)double complex .

Cohomologies of a complex manifold, ithe double complex of forms, i

X complex manifold, i.e., locally modelled on Cn

The tangent spaces have a linear complex structure, yielding the

bundle decomposition

TX ⊗ C = T 1,0X ⊕ T 0,1X .

It moves to

∧•X ⊗ C =⊕

p+q=k

∧p,qX .

The exterior di�erential d : ∧• X → ∧•+1X splits as

d = ∂ + ∂ : ∧•,• X → ∧•+1,•X ⊕ ∧•,•+1X .

By d2 = 0, it follows ∂2 = ∂2

= ∂∂ + ∂∂ = 0, i.e.,(∧•,•X , ∂, ∂

)double complex .

Cohomologies of a complex manifold, iithe double complex of forms, ii

(∧•,•X , ∂, ∂

)double complex

p− 2 p− 1 p p+ 1 p+ 2

q − 2

q − 1

q

q + 1

q + 2

Cohomologies of a complex manifold, iiiDolbeault cohomology

H•,•∂

(X ) :=ker ∂

im ∂

p− 2 p− 1 p p+ 1 p+ 2

q − 2

q − 1

q

q + 1

q + 2

Cohomologies of a complex manifold, ivBott-Chern cohomology

H•,•BC (X ) :=

ker ∂ ∩ ker ∂

im ∂∂

p− 2 p− 1 p p+ 1 p+ 2

q − 2

q − 1

q

q + 1

q + 2

R. Bott, S. S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their

holomorphic sections, Acta Math. 114 (1965), no. 1, 71�112.

Cohomologies of a complex manifold, vAeppli cohomology

H•,•A (X ) :=

ker ∂∂

im ∂ + im ∂

p− 2 p− 1 p p+ 1 p+ 2

q − 2

q − 1

q

q + 1

q + 2

A. Aeppli, On the cohomology structure of Stein manifolds, Proc. Conf. Complex Analysis

(Minneapolis, Minn., 1964), Springer, Berlin, 1965, pp. 58�70.

Cohomologies of a complex manifold, viwhy Bott-Chern and Aeppli cohomologies

On cplx mfds, there are natural mapsH•,•BC (X )

��yyrrrrrrrrrr

%%LLLLLLLLLL

H•,•∂ (X )

%%LLLLLLLLLLH•dR(X ;C)

��

H•,•∂

(X )

yyrrrrrrrrrr

H•,•A (X )

While, on compact Kähler mfds, they are all isomorphisms,

because of the ∂∂-Lemma, . . .

. . . Bott-Chern cohomology may supply further informations on

the geometry of non-Kähler manifolds.

Interest in Bott-Chern cohomology arises from: cycles on algebraic or

analytic mfds; special metrics on complex mfds; cohomology theory;

Strominger's equations in string theory; . . .

Cohomologies of a complex manifold, viwhy Bott-Chern and Aeppli cohomologies

On cplx mfds, there are natural mapsH•,•BC (X )

��yyrrrrrrrrrr

%%LLLLLLLLLL

H•,•∂ (X )

%%LLLLLLLLLLH•dR(X ;C)

��

H•,•∂

(X )

yyrrrrrrrrrr

H•,•A (X )

While, on compact Kähler mfds

, they are all isomorphisms,

because of the ∂∂-Lemma, . . .

. . . Bott-Chern cohomology may supply further informations on

the geometry of non-Kähler manifolds.

Interest in Bott-Chern cohomology arises from: cycles on algebraic or

analytic mfds; special metrics on complex mfds; cohomology theory;

Strominger's equations in string theory; . . .

Cohomologies of a complex manifold, viwhy Bott-Chern and Aeppli cohomologies

On cplx mfds, there are natural mapsH•,•BC (X )

��yyrrrrrrrrrr

%%LLLLLLLLLL

H•,•∂ (X )

%%LLLLLLLLLLH•dR(X ;C)

��

H•,•∂

(X )

yyrrrrrrrrrr

H•,•A (X )

While, on compact Kähler mfds, they are all isomorphisms,

because of the ∂∂-Lemma, . . .

. . . Bott-Chern cohomology may supply further informations on

the geometry of non-Kähler manifolds.

Interest in Bott-Chern cohomology arises from: cycles on algebraic or

analytic mfds; special metrics on complex mfds; cohomology theory;

Strominger's equations in string theory; . . .

Cohomologies of a complex manifold, viwhy Bott-Chern and Aeppli cohomologies

On cplx mfds, there are natural mapsH•,•BC (X )

��yyrrrrrrrrrr

%%LLLLLLLLLL

H•,•∂ (X )

%%LLLLLLLLLLH•dR(X ;C)

��

H•,•∂

(X )

yyrrrrrrrrrr

H•,•A (X )

While, on compact Kähler mfds, they are all isomorphisms,

because of the ∂∂-Lemma, . . .

. . . Bott-Chern cohomology may supply further informations on

the geometry of non-Kähler manifolds.

Interest in Bott-Chern cohomology arises from: cycles on algebraic or

analytic mfds; special metrics on complex mfds; cohomology theory;

Strominger's equations in string theory; . . .

Cohomologies of a complex manifold, viwhy Bott-Chern and Aeppli cohomologies

On cplx mfds, there are natural mapsH•,•BC (X )

��yyrrrrrrrrrr

%%LLLLLLLLLL

H•,•∂ (X )

%%LLLLLLLLLLH•dR(X ;C)

��

H•,•∂

(X )

yyrrrrrrrrrr

H•,•A (X )

While, on compact Kähler mfds, they are all isomorphisms,

because of the ∂∂-Lemma, . . .

. . . Bott-Chern cohomology may supply further informations on

the geometry of non-Kähler manifolds.

Interest in Bott-Chern cohomology arises from: cycles on algebraic or

analytic mfds; special metrics on complex mfds; cohomology theory;

Strominger's equations in string theory; . . .

Cohomologies of a complex manifold, viiinequality à la Frölicher for the Bott-Chern cohomology and ∂∂-Lemma

For a compact cplx mfd, one has the Frölicher inequality∑p+q=k

dimC Hp,q

∂(X ) ≥ dimC Hk

dR(X ;C) .

As for Bott-Chern cohom:

Thm (�, A. Tomassini)

X cpt cplx mfd. The following inequality à la Frölicher holds:∑p+q=k

(dimC H

p,qBC (X ) + dimC H

p,qA (X )

)≥ 2 dimC Hk

dR(X ;C) .

Furthermore, the equality characterizes the ∂∂-Lemma, namely, the

property that H•,•BC (X )

'→ H•dR(X ;C).

�, A. Tomassini, On the ∂∂-Lemma and Bott-Chern cohomology, Invent. Math. 192 (2013),

no. 1, 71�81.

Cohomologies of a complex manifold, viiinequality à la Frölicher for the Bott-Chern cohomology and ∂∂-Lemma

For a compact cplx mfd, one has the Frölicher inequality∑p+q=k

dimC Hp,q

∂(X ) ≥ dimC Hk

dR(X ;C) .

As for Bott-Chern cohom:

Thm (�, A. Tomassini)

X cpt cplx mfd. The following inequality à la Frölicher holds:∑p+q=k

(dimC H

p,qBC (X ) + dimC H

p,qA (X )

)≥ 2 dimC Hk

dR(X ;C) .

Furthermore, the equality characterizes the ∂∂-Lemma, namely, the

property that H•,•BC (X )

'→ H•dR(X ;C).

�, A. Tomassini, On the ∂∂-Lemma and Bott-Chern cohomology, Invent. Math. 192 (2013),

no. 1, 71�81.

Cohomologies of a complex manifold, viiinequality à la Frölicher for the Bott-Chern cohomology and ∂∂-Lemma

For a compact cplx mfd, one has the Frölicher inequality∑p+q=k

dimC Hp,q

∂(X ) ≥ dimC Hk

dR(X ;C) .

As for Bott-Chern cohom:

Thm (�, A. Tomassini)

X cpt cplx mfd. The following inequality à la Frölicher holds:∑p+q=k

(dimC H

p,qBC (X ) + dimC H

p,qA (X )

)≥ 2 dimC Hk

dR(X ;C) .

Furthermore, the equality characterizes the ∂∂-Lemma, namely, the

property that H•,•BC (X )

'→ H•dR(X ;C).�, A. Tomassini, On the ∂∂-Lemma and Bott-Chern cohomology, Invent. Math. 192 (2013),

no. 1, 71�81.

Cohomological models, imodels and computability

We would like to compute H] ∈{HdR , H∂ , H∂ , HBC , HA

}for a

compact complex manifold X .

Hodge theory assures �nite-dimensionality (Schweitzer).

It may turn out useful to restrict to a H]-model, that is, a subobject

ι :(M•,•, ∂, ∂

)↪→(∧•,•X , ∂, ∂

)giving the same H]-cohomology (i.e., H](ι) isom).

We are interested in H]-computable cplx mfds, that is, admitting a

�nite-dimensional H]-model (in the correct category).

M. Schweitzer, Autour de la cohomologie de Bott-Chern, arXiv:0709.3528v1 [math.AG].

Cohomological models, imodels and computability

We would like to compute H] ∈{HdR , H∂ , H∂ , HBC , HA

}for a

compact complex manifold X .

Hodge theory assures �nite-dimensionality (Schweitzer).

It may turn out useful to restrict to a H]-model, that is, a subobject

ι :(M•,•, ∂, ∂

)↪→(∧•,•X , ∂, ∂

)giving the same H]-cohomology (i.e., H](ι) isom).

We are interested in H]-computable cplx mfds, that is, admitting a

�nite-dimensional H]-model (in the correct category).

M. Schweitzer, Autour de la cohomologie de Bott-Chern, arXiv:0709.3528v1 [math.AG].

Cohomological models, imodels and computability

We would like to compute H] ∈{HdR , H∂ , H∂ , HBC , HA

}for a

compact complex manifold X .

Hodge theory assures �nite-dimensionality (Schweitzer).

It may turn out useful to restrict to a H]-model, that is, a subobject

ι :(M•,•, ∂, ∂

)↪→(∧•,•X , ∂, ∂

)giving the same H]-cohomology (i.e., H](ι) isom).

We are interested in H]-computable cplx mfds, that is, admitting a

�nite-dimensional H]-model (in the correct category).

M. Schweitzer, Autour de la cohomologie de Bott-Chern, arXiv:0709.3528v1 [math.AG].

Cohomological models, imodels and computability

We would like to compute H] ∈{HdR , H∂ , H∂ , HBC , HA

}for a

compact complex manifold X .

Hodge theory assures �nite-dimensionality (Schweitzer).

It may turn out useful to restrict to a H]-model, that is, a subobject

ι :(M•,•, ∂, ∂

)↪→(∧•,•X , ∂, ∂

)giving the same H]-cohomology (i.e., H](ι) isom).

We are interested in H]-computable cplx mfds, that is, admitting a

�nite-dimensional H]-model (in the correct category).

M. Schweitzer, Autour de la cohomologie de Bott-Chern, arXiv:0709.3528v1 [math.AG].

Cohomological models, iifrom Dolbeault to de Rham

Note that:

H∂-computable mfds are HdR -computable, too.

In fact, any H∂-model is a HdR -model, too, because of the spectral

sequence induced by the double complex structure.

A. Frölicher, Relations between the cohomology groups of Dolbeault and topological invariants,

Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 641�644.

Cohomological models, iifrom Dolbeault to de Rham

Note that:

H∂-computable mfds are HdR -computable, too.

In fact, any H∂-model is a HdR -model, too, because of the spectral

sequence induced by the double complex structure.

A. Frölicher, Relations between the cohomology groups of Dolbeault and topological invariants,

Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 641�644.

Cohomological models, iiifrom Dolbeault to Bott-Chern, i � surjectivity

Prop (�, Kasuya)

X cplx mfd. Consider ι :(M•,•, ∂, ∂

)↪→(∧•,•X , ∂, ∂

).

If

ι is a H∂-model,

ι is a H∂-model,

and ker d∩Mp,q

im d→ ker d∩∧p,qX

im dsurj,

then HBC (ι) surjective.

Proof.

0 // im d∩Mp,q

im ∂∂//

����

Hp,qBC (M•,•) //

��

ker d∩Mp,q

im d//

����

0

0 // im d∩∧p,qXim ∂∂

// Hp,qBC (X ) // ker d∩∧p,qX

im d// 0

�, H. Kasuya, Bott-Chern cohomology of solvmanifolds, arXiv:1212.5708v3 [math.DG].

Cohomological models, iiifrom Dolbeault to Bott-Chern, i � surjectivity

Prop (�, Kasuya)

X cplx mfd. Consider ι :(M•,•, ∂, ∂

)↪→(∧•,•X , ∂, ∂

). If

ι is a H∂-model,

ι is a H∂-model,

and ker d∩Mp,q

im d→ ker d∩∧p,qX

im dsurj,

then HBC (ι) surjective.

Proof.

0 // im d∩Mp,q

im ∂∂//

����

Hp,qBC (M•,•) //

��

ker d∩Mp,q

im d//

����

0

0 // im d∩∧p,qXim ∂∂

// Hp,qBC (X ) // ker d∩∧p,qX

im d// 0

�, H. Kasuya, Bott-Chern cohomology of solvmanifolds, arXiv:1212.5708v3 [math.DG].

Cohomological models, iiifrom Dolbeault to Bott-Chern, i � surjectivity

Prop (�, Kasuya)

X cplx mfd. Consider ι :(M•,•, ∂, ∂

)↪→(∧•,•X , ∂, ∂

). If

ι is a H∂-model,

ι is a H∂-model,

and ker d∩Mp,q

im d→ ker d∩∧p,qX

im dsurj,

then HBC (ι) surjective.

Proof.

0 // im d∩Mp,q

im ∂∂//

����

Hp,qBC (M•,•) //

��

ker d∩Mp,q

im d//

����

0

0 // im d∩∧p,qXim ∂∂

// Hp,qBC (X ) // ker d∩∧p,qX

im d// 0

�, H. Kasuya, Bott-Chern cohomology of solvmanifolds, arXiv:1212.5708v3 [math.DG].

Cohomological models, iiifrom Dolbeault to Bott-Chern, i � surjectivity

Prop (�, Kasuya)

X cplx mfd. Consider ι :(M•,•, ∂, ∂

)↪→(∧•,•X , ∂, ∂

). If

ι is a H∂-model,

ι is a H∂-model,

and ker d∩Mp,q

im d→ ker d∩∧p,qX

im dsurj,

then HBC (ι) surjective.

Proof.

0 // im d∩Mp,q

im ∂∂//

����

Hp,qBC (M•,•) //

��

ker d∩Mp,q

im d//

����

0

0 // im d∩∧p,qXim ∂∂

// Hp,qBC (X ) // ker d∩∧p,qX

im d// 0

�, H. Kasuya, Bott-Chern cohomology of solvmanifolds, arXiv:1212.5708v3 [math.DG].

Cohomological models, ivfrom Dolbeault to Bott-Chern, ii � injectivity

As for injectivity, use Hodge theory:

Prop (�, H. Kasuya)

X cplx mfd. Consider ι :(M•,•, ∂, ∂

)↪→(∧•,•X , ∂, ∂

).

If

dimM•,• < +∞,

and M•,• is closed under ∂∗ and ∂∗,

then HBC (ι) injective.

Cohomological models, ivfrom Dolbeault to Bott-Chern, ii � injectivity

As for injectivity, use Hodge theory:

Prop (�, H. Kasuya)

X cplx mfd. Consider ι :(M•,•, ∂, ∂

)↪→(∧•,•X , ∂, ∂

). If

dimM•,• < +∞,

and M•,• is closed under ∂∗ and ∂∗,

then HBC (ι) injective.

Cohomological models, ivfrom Dolbeault to Bott-Chern, ii � injectivity

As for injectivity, use Hodge theory:

Prop (�, H. Kasuya)

X cplx mfd. Consider ι :(M•,•, ∂, ∂

)↪→(∧•,•X , ∂, ∂

). If

dimM•,• < +∞,

and M•,• is closed under ∂∗ and ∂∗,

then HBC (ι) injective.

Nilmanifolds, ide Rham cohomology

�

�

�

�

G connected simply-connected

nilpotent Lie group

Γ discrete co-compact subgroup

X = Γ\G nilmanifold

Thm (Nomizu)

X = Γ\G nilmanifold. The inclusion of

left-invariant forms,

ι : (∧•g∗, d) ↪→ (∧•X , d) ,

is a HdR -model.

Tj0� � // X = Γ\G

��Tj1

� � // X1

��...

��Tjk

� � // Xk

��Tjk+1

K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of

Math. (2) 59 (1954), no. 3, 531�538.

Nilmanifolds, ide Rham cohomology

�

�

�

�

G connected simply-connected

nilpotent Lie group

Γ discrete co-compact subgroup

X = Γ\G nilmanifold

Thm (Nomizu)

X = Γ\G nilmanifold. The inclusion of

left-invariant forms,

ι : (∧•g∗, d) ↪→ (∧•X , d) ,

is a HdR -model.

Tj0� � // X = Γ\G

��Tj1

� � // X1

��...

��Tjk

� � // Xk

��Tjk+1

K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of

Math. (2) 59 (1954), no. 3, 531�538.

Nilmanifolds, ide Rham cohomology

�

�

�

�

G connected simply-connected

nilpotent Lie group

Γ discrete co-compact subgroup

X = Γ\G nilmanifold

Thm (Nomizu)

X = Γ\G nilmanifold. The inclusion of

left-invariant forms,

ι : (∧•g∗, d) ↪→ (∧•X , d) ,

is a HdR -model.

Tj0� � // X = Γ\G

��Tj1

� � // X1

��...

��Tjk

� � // Xk

��Tjk+1

K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of

Math. (2) 59 (1954), no. 3, 531�538.

Nilmanifolds, iiDolbeault cohomology

Thm (Sakane, Cordero, Fernández, Gray, Ugarte, Console, Fino, Rollenske)

X = Γ\G nilmfd with a left-inv �suitable� cplx struct.

The inclusion of left-invariant forms,

ι :(∧•,• (g⊗ C)∗ , ∂, ∂

)↪→(∧•,•X , ∂, ∂

),

is a H∂-model.

It is conjectured that any left-inv cplx structure on nilmanifolds is

�suitable� (Rollenske).

S. Console, A. Fino, Dolbeault cohomology of compact nilmanifolds, Transform. Groups 6 (2001),

no. 2, 111�124.

S. Rollenske, Dolbeault cohomology of nilmanifolds with left-invariant complex structure, in W.

Ebeling, K. Hulek, K. Smoczyk (eds.), Complex and Di�erential Geometry: Conference held atLeibniz Universität Hannover, September 14 � 18, 2009, Springer Proceedings in Mathematics 8,Springer, 2011, 369�392.

Nilmanifolds, iiDolbeault cohomology

Thm (Sakane, Cordero, Fernández, Gray, Ugarte, Console, Fino, Rollenske)

X = Γ\G nilmfd with a left-inv �suitable� cplx struct.

The inclusion of left-invariant forms,

ι :(∧•,• (g⊗ C)∗ , ∂, ∂

)↪→(∧•,•X , ∂, ∂

),

is a H∂-model.

It is conjectured that any left-inv cplx structure on nilmanifolds is

�suitable� (Rollenske).

S. Console, A. Fino, Dolbeault cohomology of compact nilmanifolds, Transform. Groups 6 (2001),

no. 2, 111�124.

S. Rollenske, Dolbeault cohomology of nilmanifolds with left-invariant complex structure, in W.

Ebeling, K. Hulek, K. Smoczyk (eds.), Complex and Di�erential Geometry: Conference held atLeibniz Universität Hannover, September 14 � 18, 2009, Springer Proceedings in Mathematics 8,Springer, 2011, 369�392.

Nilmanifolds, iiiBott-Chern cohomology

Thm (�)

X = Γ\G nilmfd with a left-inv �suitable� cplx struct.

The inclusion of left-invariant forms,

ι :(∧•,• (g⊗ C)∗ , ∂, ∂

)↪→(∧•,•X , ∂, ∂

),

is a HBC -model.

Summarizing:��

��

nilmanifolds with �suitable� left-inv cplx struct are cohomologically-computable, by means of the �nite-dim sub-complex of left-inv forms,(∧•,•g∗C, ∂, ∂

).

�, The cohomologies of the Iwasawa manifold and of its small deformations, J. Geom. Anal. 23

(2013), no. 3, 1355-1378.

Nilmanifolds, iiiBott-Chern cohomology

Thm (�)

X = Γ\G nilmfd with a left-inv �suitable� cplx struct.

The inclusion of left-invariant forms,

ι :(∧•,• (g⊗ C)∗ , ∂, ∂

)↪→(∧•,•X , ∂, ∂

),

is a HBC -model.

Summarizing:��

��

nilmanifolds with �suitable� left-inv cplx struct are cohomologically-computable, by means of the �nite-dim sub-complex of left-inv forms,(∧•,•g∗C, ∂, ∂

).

�, The cohomologies of the Iwasawa manifold and of its small deformations, J. Geom. Anal. 23

(2013), no. 3, 1355-1378.

Nilmanifolds, ivIwasawa manifold I3, i � de�nition

Iwasawa manifold:

I3 := (Z [i])3∖

1 z1 z3

0 1 z2

0 0 1

∈ GL(C3)

holomorphically-parallelizable nilmanifold

left-inv co-frame for(T 1,0I3

)∗:{

ϕ1 := d z1, ϕ2 := d z2, ϕ3 := d z3 − z1 d z2}

structure equations:dϕ1 = 0

dϕ2 = 0

dϕ3 = −ϕ1 ∧ ϕ2

Nilmanifolds, ivIwasawa manifold I3, i � de�nition

Iwasawa manifold:

I3 := (Z [i])3∖

1 z1 z3

0 1 z2

0 0 1

∈ GL(C3)

holomorphically-parallelizable nilmanifold

left-inv co-frame for(T 1,0I3

)∗:{

ϕ1 := d z1, ϕ2 := d z2, ϕ3 := d z3 − z1 d z2}

structure equations:dϕ1 = 0

dϕ2 = 0

dϕ3 = −ϕ1 ∧ ϕ2

Nilmanifolds, ivIwasawa manifold I3, i � de�nition

Iwasawa manifold:

I3 := (Z [i])3∖

1 z1 z3

0 1 z2

0 0 1

∈ GL(C3)

holomorphically-parallelizable nilmanifold

left-inv co-frame for(T 1,0I3

)∗:{

ϕ1 := d z1, ϕ2 := d z2, ϕ3 := d z3 − z1 d z2}

structure equations:dϕ1 = 0

dϕ2 = 0

dϕ3 = −ϕ1 ∧ ϕ2

Nilmanifolds, ivIwasawa manifold I3, i � de�nition

Iwasawa manifold:

I3 := (Z [i])3∖

1 z1 z3

0 1 z2

0 0 1

∈ GL(C3)

holomorphically-parallelizable nilmanifold

left-inv co-frame for(T 1,0I3

)∗:{

ϕ1 := d z1, ϕ2 := d z2, ϕ3 := d z3 − z1 d z2}

structure equations:dϕ1 = 0

dϕ2 = 0

dϕ3 = −ϕ1 ∧ ϕ2

Nilmanifolds, vIwasawa manifold I3, ii � double complex of left-invariant forms

0

0

1

1

2

2

3

3

Nilmanifolds, viIwasawa manifold I3, iii � Dolbeault cohomology

0

0

1

1

2

2

3

3

(p, q) dimC Hp,q

∂(I3)

(0, 0) 1

(1, 0) 3(0, 1) 2

(2, 0) 3(1, 1) 6(0, 2) 2

(3, 0) 1(2, 1) 6(1, 2) 6(0, 3) 1

(3, 1) 2(2, 2) 6(1, 3) 3

(3, 2) 2(2, 3) 3

(3, 3) 1

Nilmanifolds, viiIwasawa manifold I3, iv � Bott-Chern cohomology

0

0

1

1

2

2

3

3

(p, q) dimC Hp,qBC (I3)

(0, 0) 1

(1, 0) 2(0, 1) 2

(2, 0) 3(1, 1) 4(0, 2) 3

(3, 0) 1(2, 1) 6(1, 2) 6(0, 3) 1

(3, 1) 2(2, 2) 8(1, 3) 2

(3, 2) 3(2, 3) 3

(3, 3) 1

Nilmanifolds, viii6-dimensional nilmanifolds

More in general:

any left-invariant complex structure on a 6-dim nilmfd admits a

�nite-dim cohomological-model (except, perhaps, h7)

cohomol classi�cation of 6-dim nilmfds with left-inv cplx struct.

�, M. G. Franzini, F. A. Rossi, Degree of non-Kählerianity for 6-dimensional nilmanifolds,

arXiv:1210.0406 [math.DG].

A. Latorre, L. Ugarte, R. Villacampa, On the Bott-Chern cohomology and balanced Hermitian

nilmanifolds, arXiv:1210.0395v1 [math.DG].

Solvmanifolds, ide�nition and motivations

�

�

�

�G connected simply-connected solvable Lie group

Γ discrete co-compact subgroup

X = Γ\G solvmanifold

While nilmanifolds (except tori) are non-Kähler, non-formal,

non-∂∂-Lemma, non-HLC, . . .

(Benson and Gordon, Hasegawa)

. . . there exist non-Kähler solvmanifolds satisfying ∂∂-Lemma

(Kasuya).

Ch. Benson, C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988),

no. 4, 513�518.

K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65�71.

H. Kasuya, Hodge symmetry and decomposition on non-Kähler solvmanifolds, arXiv:1109.5929v5

[math.DG].

Solvmanifolds, ide�nition and motivations

�

�

�

�G connected simply-connected solvable Lie group

Γ discrete co-compact subgroup

X = Γ\G solvmanifold

While nilmanifolds (except tori) are non-Kähler, non-formal,

non-∂∂-Lemma, non-HLC, . . .

(Benson and Gordon, Hasegawa)

. . . there exist non-Kähler solvmanifolds satisfying ∂∂-Lemma

(Kasuya).

Ch. Benson, C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988),

no. 4, 513�518.

K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65�71.

H. Kasuya, Hodge symmetry and decomposition on non-Kähler solvmanifolds, arXiv:1109.5929v5

[math.DG].

Solvmanifolds, ide�nition and motivations

�

�

�

�G connected simply-connected solvable Lie group

Γ discrete co-compact subgroup

X = Γ\G solvmanifold

While nilmanifolds (except tori) are non-Kähler, non-formal,

non-∂∂-Lemma, non-HLC, . . .

(Benson and Gordon, Hasegawa)

. . . there exist non-Kähler solvmanifolds satisfying ∂∂-Lemma

(Kasuya).Ch. Benson, C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988),

no. 4, 513�518.

K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65�71.

H. Kasuya, Hodge symmetry and decomposition on non-Kähler solvmanifolds, arXiv:1109.5929v5

[math.DG].

Solvmanifolds, iide Rham cohomology

Thm (Hattori)

X = Γ\G solvmanifold. If G is completely-solvable, then the

inclusion of left-inv forms, ι : ∧• g∗ ↪→ ∧•X, is a HdR -model.

In general, de Rham cohom depends on Γ (de Bartolomeis, Tomassini):

Thm (Kasuya)

X = Γ\G solvmfd. There exists a �nite-dim HdR -model

(A•Γ, d) ↪→ (∧•X ⊗ C, d) .

A. Hattori, Spectral sequence in the de Rham cohomology of �bre bundles, J. Fac. Sci. Univ.

Tokyo Sect. I 8 (1960), no. 1960, 289�331.

P. de Bartolomeis, A. Tomassini, On solvable generalized Calabi-Yau manifolds, Ann. Inst. Fourier

(Grenoble) 56 (2006), no. 5, 1281�1296.

H. Kasuya, Minimal models, formality and hard Lefschetz properties of solvmanifolds with local

systems, J. Di�er. Geom. 93, (2013), 269�298.

Solvmanifolds, iide Rham cohomology

Thm (Hattori)

X = Γ\G solvmanifold. If G is completely-solvable, then the

inclusion of left-inv forms, ι : ∧• g∗ ↪→ ∧•X, is a HdR -model.

In general, de Rham cohom depends on Γ (de Bartolomeis, Tomassini):

Thm (Kasuya)

X = Γ\G solvmfd. There exists a �nite-dim HdR -model

(A•Γ, d) ↪→ (∧•X ⊗ C, d) .

A. Hattori, Spectral sequence in the de Rham cohomology of �bre bundles, J. Fac. Sci. Univ.

Tokyo Sect. I 8 (1960), no. 1960, 289�331.

P. de Bartolomeis, A. Tomassini, On solvable generalized Calabi-Yau manifolds, Ann. Inst. Fourier

(Grenoble) 56 (2006), no. 5, 1281�1296.

H. Kasuya, Minimal models, formality and hard Lefschetz properties of solvmanifolds with local

systems, J. Di�er. Geom. 93, (2013), 269�298.

Solvmanifolds, iide Rham cohomology

Thm (Hattori)

X = Γ\G solvmanifold. If G is completely-solvable, then the

inclusion of left-inv forms, ι : ∧• g∗ ↪→ ∧•X, is a HdR -model.

In general, de Rham cohom depends on Γ (de Bartolomeis, Tomassini):

Thm (Kasuya)

X = Γ\G solvmfd. There exists a �nite-dim HdR -model

(A•Γ, d) ↪→ (∧•X ⊗ C, d) .

A. Hattori, Spectral sequence in the de Rham cohomology of �bre bundles, J. Fac. Sci. Univ.

Tokyo Sect. I 8 (1960), no. 1960, 289�331.

P. de Bartolomeis, A. Tomassini, On solvable generalized Calabi-Yau manifolds, Ann. Inst. Fourier

(Grenoble) 56 (2006), no. 5, 1281�1296.

H. Kasuya, Minimal models, formality and hard Lefschetz properties of solvmanifolds with local

systems, J. Di�er. Geom. 93, (2013), 269�298.

Solvmanifolds, iiiDolbeault cohomology and Bott-Chern cohomology

X = Γ\G solvmfd with left-inv cplx struct of one of the following types:

holomorphically-parallelizable;

splitting-type: i.e., G = Cn nφ N, with N nilpotent with left-inv cplx

struct, φ semi-simple, holomorphic at every time; assume Γ′′\N has

left-inv forms as H∂-model.

Thm (Kasuya; �, Kasuya)

Then there exist:

a �nite-dim H∂-model(B•,•Γ , ∂, ∂

)↪→(∧•,•X , ∂, ∂

);

a �nite-dim HBC -model(C•,•Γ , ∂, ∂

)↪→(∧•,•X , ∂, ∂

).

H. Kasuya, De Rham and Dolbeault cohomology of solvmanifolds with local systems,

arXiv:1207.3988v3 [math.DG].

H. Kasuya, Techniques of computations of Dolbeault cohomology of solvmanifolds, Math. Z. 273

(2013), no. 1-2, 437�447.

Solvmanifolds, iiiDolbeault cohomology and Bott-Chern cohomology

X = Γ\G solvmfd with left-inv cplx struct of one of the following types:

holomorphically-parallelizable;

splitting-type: i.e., G = Cn nφ N, with N nilpotent with left-inv cplx

struct, φ semi-simple, holomorphic at every time; assume Γ′′\N has

left-inv forms as H∂-model.

Thm (Kasuya; �, Kasuya)

Then there exist:

a �nite-dim H∂-model(B•,•Γ , ∂, ∂

)↪→(∧•,•X , ∂, ∂

);

a �nite-dim HBC -model(C•,•Γ , ∂, ∂

)↪→(∧•,•X , ∂, ∂

).

H. Kasuya, De Rham and Dolbeault cohomology of solvmanifolds with local systems,

arXiv:1207.3988v3 [math.DG].

H. Kasuya, Techniques of computations of Dolbeault cohomology of solvmanifolds, Math. Z. 273

(2013), no. 1-2, 437�447.

Solvmanifolds, iiiDolbeault cohomology and Bott-Chern cohomology

X = Γ\G solvmfd with left-inv cplx struct of one of the following types:

holomorphically-parallelizable;

splitting-type: i.e., G = Cn nφ N, with N nilpotent with left-inv cplx

struct, φ semi-simple, holomorphic at every time; assume Γ′′\N has

left-inv forms as H∂-model.

Thm (Kasuya; �, Kasuya)

Then there exist:

a �nite-dim H∂-model(B•,•Γ , ∂, ∂

)↪→(∧•,•X , ∂, ∂

);

a �nite-dim HBC -model(C•,•Γ , ∂, ∂

)↪→(∧•,•X , ∂, ∂

).

H. Kasuya, De Rham and Dolbeault cohomology of solvmanifolds with local systems,

arXiv:1207.3988v3 [math.DG].

H. Kasuya, Techniques of computations of Dolbeault cohomology of solvmanifolds, Math. Z. 273

(2013), no. 1-2, 437�447.

Solvmanifolds, ivcohomological models for deformations

By using the spectral theory for families of elliptic di�erential operators:

Thm (�, Kasuya)

Consider a cohomologically-computable cplx manifold. Then

suitable small deformations are still cohomologically-computable.

As a corollary, one recovers:

Thm (Console, Fino)

On a nilmanifold, the set of cplx structures having left-inv forms as H∂-model

is open in the space of left-inv cplx structures.

�, H. Kasuya, Cohomologies of deformations of solvmanifolds and closedness of some properties,

arXiv:1305.6709v1 [math.CV].

Solvmanifolds, ivcohomological models for deformations

By using the spectral theory for families of elliptic di�erential operators:

Thm (�, Kasuya)

Consider a cohomologically-computable cplx manifold. Then

suitable small deformations are still cohomologically-computable.

As a corollary, one recovers:

Thm (Console, Fino)

On a nilmanifold, the set of cplx structures having left-inv forms as H∂-model

is open in the space of left-inv cplx structures.

�, H. Kasuya, Cohomologies of deformations of solvmanifolds and closedness of some properties,

arXiv:1305.6709v1 [math.CV].

∂∂-Lemma and deformations, i∂∂-Lemma and deformations

A compact cplx mfd X satis�es the ∂∂-Lemma i� the natural map

H•,•BC (X )

'→ H•dR(X ;C)

is an isomorphism: it is a cohomological-decomposition property.

��

� Aim: to study the behaviour of ∂∂-Lemma under deformations

of the cplx structure.

i.e., consider a family {Xt}t∈∆(0,ε) of cpt cplx mfds:

if X0 satis�es the ∂∂-Lemma, then does Xt for t small? (openness)

if Xt satis�es the ∂∂-Lemma for any t 6= 0, then does X0? (closedness)

∂∂-Lemma and deformations, i∂∂-Lemma and deformations

A compact cplx mfd X satis�es the ∂∂-Lemma i� the natural map

H•,•BC (X )

'→ H•dR(X ;C)

is an isomorphism: it is a cohomological-decomposition property.��

� Aim: to study the behaviour of ∂∂-Lemma under deformations

of the cplx structure.

i.e., consider a family {Xt}t∈∆(0,ε) of cpt cplx mfds:

if X0 satis�es the ∂∂-Lemma, then does Xt for t small? (openness)

if Xt satis�es the ∂∂-Lemma for any t 6= 0, then does X0? (closedness)

∂∂-Lemma and deformations, i∂∂-Lemma and deformations

A compact cplx mfd X satis�es the ∂∂-Lemma i� the natural map

H•,•BC (X )

'→ H•dR(X ;C)

is an isomorphism: it is a cohomological-decomposition property.��

� Aim: to study the behaviour of ∂∂-Lemma under deformations

of the cplx structure.

i.e., consider a family {Xt}t∈∆(0,ε) of cpt cplx mfds:

if X0 satis�es the ∂∂-Lemma, then does Xt for t small? (openness)

if Xt satis�es the ∂∂-Lemma for any t 6= 0, then does X0? (closedness)

∂∂-Lemma and deformations, i∂∂-Lemma and deformations

A compact cplx mfd X satis�es the ∂∂-Lemma i� the natural map

H•,•BC (X )

'→ H•dR(X ;C)

is an isomorphism: it is a cohomological-decomposition property.��

� Aim: to study the behaviour of ∂∂-Lemma under deformations

of the cplx structure.

i.e., consider a family {Xt}t∈∆(0,ε) of cpt cplx mfds:

if X0 satis�es the ∂∂-Lemma, then does Xt for t small? (openness)

if Xt satis�es the ∂∂-Lemma for any t 6= 0, then does X0? (closedness)

∂∂-Lemma and deformations, i∂∂-Lemma and deformations

A compact cplx mfd X satis�es the ∂∂-Lemma i� the natural map

H•,•BC (X )

'→ H•dR(X ;C)

is an isomorphism: it is a cohomological-decomposition property.��

� Aim: to study the behaviour of ∂∂-Lemma under deformations

of the cplx structure.

i.e., consider a family {Xt}t∈∆(0,ε) of cpt cplx mfds:

if X0 satis�es the ∂∂-Lemma, then does Xt for t small? (openness)

if Xt satis�es the ∂∂-Lemma for any t 6= 0, then does X0? (closedness)

∂∂-Lemma and deformations, iiIwasawa manifold, i � summary of cohomologies

Iwasawa manifold

I3 dR ∂ ∂ BC A

(0, 0) 1 1 1 1 1

(1, 0) 4 3 2 2 3(0, 1) 2 3 2 3

(2, 0) 8 3 2 3 2(1, 1) 6 6 4 8(0, 2) 2 3 3 2

(3, 0) 10 1 1 1 1(2, 1) 6 6 6 6(1, 2) 6 6 6 6(0, 3) 1 1 1 1

(3, 1) 8 2 3 2 3(2, 2) 6 6 8 4(1, 3) 3 2 2 3

(3, 2) 4 2 3 3 2(2, 3) 3 2 3 2

(3, 3) 1 1 1 1 1

I3 dR ∂ ∂ BC A

0 1 1 1 1 1

1 4 5 5 4 6

2 8 11 11 10 12

3 10 14 14 14 14

4 8 11 11 12 10

5 4 5 5 6 4

6 1 1 1 1 1

∂∂-Lemma and deformations, iiIwasawa manifold, i � summary of cohomologies

Iwasawa manifold

I3 dR ∂ ∂ BC A

(0, 0) 1 1 1 1 1

(1, 0) 4 3 2 2 3(0, 1) 2 3 2 3

(2, 0) 8 3 2 3 2(1, 1) 6 6 4 8(0, 2) 2 3 3 2

(3, 0) 10 1 1 1 1(2, 1) 6 6 6 6(1, 2) 6 6 6 6(0, 3) 1 1 1 1

(3, 1) 8 2 3 2 3(2, 2) 6 6 8 4(1, 3) 3 2 2 3

(3, 2) 4 2 3 3 2(2, 3) 3 2 3 2

(3, 3) 1 1 1 1 1

I3 dR ∂ ∂ BC A

0 1 1 1 1 1

1 4 5 5 4 6

2 8 11 11 10 12

3 10 14 14 14 14

4 8 11 11 12 10

5 4 5 5 6 4

6 1 1 1 1 1

∂∂-Lemma and deformations, iiiIwasawa manifold, ii � remarks on the cohomologies

Iwasawa manifold

hk∂≥ bk (Frölicher inequality)

but, in general, hkBC is not greaterthan or equal to hk

∂or bk .

Anyway, note thathkBC + hkA ≥ 2 bk in the example. . .. . . and this is a general result.

I3 dR ∂ ∂ BC A

0 1 1 1 1 1

1 4 5 5 4 6

2 8 11 11 10 12

3 10 14 14 14 14

4 8 11 11 12 10

5 4 5 5 6 4

6 1 1 1 1 1

∂∂-Lemma and deformations, ivinequality à la Frölicher for Bott-Chern, i � an heuristic argument

Dolbeault cohomology cares only about hor-izontal arrows, as Bott-Chern cares onlyabout ingoing arrows, and, dually, Aepplicares only about outgoing arrows.

Since

] {ingoing} + ] {outgoing}

≥ ] {horizontal} + ] {vertical}

one gets:

0

0

1

1

2

2

3

3

Thm (�, A. Tomassini)

X cpt cplx mfd. The following inequality à la Frölicher holds:∑p+q=k

(dimC H

p,qBC (X ) + dimC H

p,qA (X )

)≥ 2 dimC Hk

dR(X ;C) .

Furthermore, the equality characterizes the ∂∂-Lemma.

∂∂-Lemma and deformations, ivinequality à la Frölicher for Bott-Chern, i � an heuristic argument

Dolbeault cohomology cares only about hor-izontal arrows, as Bott-Chern cares onlyabout ingoing arrows, and, dually, Aepplicares only about outgoing arrows.

Since

] {ingoing} + ] {outgoing}

≥ ] {horizontal} + ] {vertical}

one gets:

0

0

1

1

2

2

3

3

Thm (�, A. Tomassini)

X cpt cplx mfd. The following inequality à la Frölicher holds:∑p+q=k

(dimC H

p,qBC (X ) + dimC H

p,qA (X )

)≥ 2 dimC Hk

dR(X ;C) .

Furthermore, the equality characterizes the ∂∂-Lemma.

∂∂-Lemma and deformations, ivinequality à la Frölicher for Bott-Chern, i � an heuristic argument

Dolbeault cohomology cares only about hor-izontal arrows, as Bott-Chern cares onlyabout ingoing arrows, and, dually, Aepplicares only about outgoing arrows.

Since

] {ingoing} + ] {outgoing}

≥ ] {horizontal} + ] {vertical}

one gets:

0

0

1

1

2

2

3

3

Thm (�, A. Tomassini)

X cpt cplx mfd. The following inequality à la Frölicher holds:∑p+q=k

(dimC H

p,qBC (X ) + dimC H

p,qA (X )

)≥ 2 dimC Hk

dR(X ;C) .

Furthermore, the equality characterizes the ∂∂-Lemma.

∂∂-Lemma and deformations, ivinequality à la Frölicher for Bott-Chern, i � an heuristic argument

Dolbeault cohomology cares only about hor-izontal arrows, as Bott-Chern cares onlyabout ingoing arrows, and, dually, Aepplicares only about outgoing arrows.Since

] {ingoing} + ] {outgoing}

≥ ] {horizontal} + ] {vertical}

one gets:0

0

1

1

2

2

3

3

Thm (�, A. Tomassini)

X cpt cplx mfd. The following inequality à la Frölicher holds:∑p+q=k

(dimC H

p,qBC (X ) + dimC H

p,qA (X )

)≥ 2 dimC Hk

dR(X ;C) .

Furthermore, the equality characterizes the ∂∂-Lemma.

∂∂-Lemma and deformations, ivinequality à la Frölicher for Bott-Chern, i � an heuristic argument

Dolbeault cohomology cares only about hor-izontal arrows, as Bott-Chern cares onlyabout ingoing arrows, and, dually, Aepplicares only about outgoing arrows.Since

] {ingoing} + ] {outgoing}

≥ ] {horizontal} + ] {vertical}

one gets:0

0

1

1

2

2

3

3

Thm (�, A. Tomassini)

X cpt cplx mfd. The following inequality à la Frölicher holds:∑p+q=k

(dimC H

p,qBC (X ) + dimC H

p,qA (X )

)≥ 2 dimC Hk

dR(X ;C) .

Furthermore, the equality characterizes the ∂∂-Lemma.

∂∂-Lemma and deformations, vinequality à la Frölicher for Bott-Chern, ii � openness of ∂∂-Lemma

By Hodge theory, dimC Hp,qBC and dimC H

p,qA are

upper-semi-continuous for deformations of the complex structure.

Hence the equality∑p+q=k

(dimC H

p,qBC (X ) + dimC H

p,qA (X )

)= 2 dimC Hk

dR(X ;C)

is stable for small deformations. It follows:

Cor (Voisin; Wu; Tomasiello; �, A. Tomassini)

The property of satisfying the ∂∂-Lemma is open under

deformations.

∂∂-Lemma and deformations, vinequality à la Frölicher for Bott-Chern, ii � openness of ∂∂-Lemma

By Hodge theory, dimC Hp,qBC and dimC H

p,qA are

upper-semi-continuous for deformations of the complex structure.

Hence the equality∑p+q=k

(dimC H

p,qBC (X ) + dimC H

p,qA (X )

)= 2 dimC Hk

dR(X ;C)

is stable for small deformations.

It follows:

Cor (Voisin; Wu; Tomasiello; �, A. Tomassini)

The property of satisfying the ∂∂-Lemma is open under

deformations.

∂∂-Lemma and deformations, vinequality à la Frölicher for Bott-Chern, ii � openness of ∂∂-Lemma

By Hodge theory, dimC Hp,qBC and dimC H

p,qA are

upper-semi-continuous for deformations of the complex structure.

Hence the equality∑p+q=k

(dimC H

p,qBC (X ) + dimC H

p,qA (X )

)= 2 dimC Hk

dR(X ;C)

is stable for small deformations. It follows:

Cor (Voisin; Wu; Tomasiello; �, A. Tomassini)

The property of satisfying the ∂∂-Lemma is open under

deformations.

∂∂-Lemma and deformations, viNakamura manifold and deformations, i

The Lie group

Cnφ C2 dove φ(z) =

(ez 0

0 e−z

).

admits a lattice: the quotient is called Nakamura manifold.

Consider the small deformations in the direction

t∂

∂z1⊗ d z̄1 .

��

� the previous theorems furnish �nite-dim sub-complexes to

compute Dolbeault and Bott-Chern cohomologies

∂∂-Lemma and deformations, viNakamura manifold and deformations, i

The Lie group

Cnφ C2 dove φ(z) =

(ez 0

0 e−z

).

admits a lattice: the quotient is called Nakamura manifold.

Consider the small deformations in the direction

t∂

∂z1⊗ d z̄1 .

��

� the previous theorems furnish �nite-dim sub-complexes to

compute Dolbeault and Bott-Chern cohomologies

∂∂-Lemma and deformations, viNakamura manifold and deformations, i

The Lie group

Cnφ C2 dove φ(z) =

(ez 0

0 e−z

).

admits a lattice: the quotient is called Nakamura manifold.

Consider the small deformations in the direction

t∂

∂z1⊗ d z̄1 .

��

� the previous theorems furnish �nite-dim sub-complexes to

compute Dolbeault and Bott-Chern cohomologies

∂∂-Lemma and deformations, viiNakamura manifold and deformations, ii � non-closedness of ∂∂-Lemma

dimC H•,•] Nakamura deformations

dR ∂̄ BC dR ∂̄ BC

(0, 0) 1 1 1 1 1 1

(1, 0)2

3 12

1 1

(0, 1) 3 1 1 1

(2, 0)5

3 3

5

1 1

(1, 1) 9 7 3 3

(0, 2) 3 3 1 1

(3, 0)

8

1 1

8

1 1

(2, 1) 9 9 3 3

(1, 2) 9 9 3 3

(0, 3) 1 1 1 1

(3, 1)5

3 3

5

1 1

(2, 2) 9 11 3 3

(1, 3) 3 3 1 1

(3, 2)2

3 52

1 1

(2, 3) 3 5 1 1

(3, 3) 1 1 1 1 1 1

Thm (�, Kasuya)

The property of satisfying the

∂∂-Lemma is not closed under

deformations.

∂∂-Lemma and deformations, viiNakamura manifold and deformations, ii � non-closedness of ∂∂-Lemma

dimC H•,•] Nakamura deformations

dR ∂̄ BC dR ∂̄ BC

(0, 0) 1 1 1 1 1 1

(1, 0)2

3 12

1 1

(0, 1) 3 1 1 1

(2, 0)5

3 3

5

1 1

(1, 1) 9 7 3 3

(0, 2) 3 3 1 1

(3, 0)

8

1 1

8

1 1

(2, 1) 9 9 3 3

(1, 2) 9 9 3 3

(0, 3) 1 1 1 1

(3, 1)5

3 3

5

1 1

(2, 2) 9 11 3 3

(1, 3) 3 3 1 1

(3, 2)2

3 52

1 1

(2, 3) 3 5 1 1

(3, 3) 1 1 1 1 1 1

Thm (�, Kasuya)

The property of satisfying the

∂∂-Lemma is not closed under

deformations.

Future work�Di�cult to see. Always in motion is the future.�

For the future:

construct a cohomological-model for (generalized-)cplx mfds,

maybe including geometric informations;

(Sullivan formality, Dolbeault homotopy theory, . . . )

compute it for an enlarged class of cohom-computable cplx

mfds. . .

(López de Medrano and Verjovsky mfds, cpt cplx surf, . . . )

. . . in order to investigate geometric properties.

(contractions of Fujiki class C or Mo��²hezon, . . . )

�, A. Tomassini.

�, S. Calamai.

�, H. Kasuya.

�, J. Raissy.

Joint work with: Adriano Tomassini, Hisashi Kasuya, Federico A. Rossi, Maria Giovanna Franzini, SimoneCalamai.

And with the essential contribution of: Serena, Maria Beatrice e Luca, Andrea, Maria Rosaria, Mat-teo, Jasmin, Carlo, Junyan, Michele, Chiara, Simone, Eridano, Laura, Paolo, Marco, Cristiano, Daniele,Matteo, . . .