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A conference in honor of Pierre DolbeaultOn the occasion of his 90th birthday anniversary

Cohomologies on complex manifolds

Daniele Angella

Istituto Nazionale di Alta Matematica(Dipartimento di Matematica e Informatica, Università di Parma)

June 04, 2014

Introduction, iIt was 50s. . . , i

Introduction, iiIt was 50s. . . , ii

Complex geometry encoded in global invariants:

Introduction, iiiIt was 50s. . . , iii

What informations in ∂-cohomology?

# Algebraic struct induced by di�erential algebra(∧•,•X , ∂, ∧

).

J. Neisendorfer, L. Taylor, Dolbeault homotopy theory, Trans. Amer. Math. Soc. 245

(1978), 183�210.

# Relation with topological informations.

A. Weil, Introduction à l'etude des variétés kählériennes, Hermann, Paris, 1958.

Introduction, ivIt was 50s. . . , iv



).


(1978), 183�210.



Introduction, vIt was 50s. . . , v



).


(1978), 183�210.



Introduction, viIt was 50s. . . , vi



).


(1978), 183�210.



Introduction, viiIt was 50s. . . , vii

On a complex (possibly non-Kähler) manifold:

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

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

Introduction, viiiIt was 50s. . . , viii

On a complex (possibly non-Kähler) manifold:



Introduction, ixIt was 50s. . . , ix

Interest on non-Kähler manifold since 70s:

K. Kodaira, On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86 (1964),

751�798.

W. P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55

(1976), no. 2, 467�468.

Introduction, xIt was 50s. . . , x

Interest on non-Kähler manifold since 70s:

K. Kodaira, On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86 (1964),

751�798.

W. P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55

(1976), no. 2, 467�468.

Introduction, xiIt was 50s. . . , xi

Bott-Chern and Aeppli cohomologies for complex manifolds:

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

holomorphic sections, Acta Math. 114 (1965), 71�112.

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

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

they provide bridges between de Rham and Dolbeault cohomologies,

allowing their comparison

Introduction, xiiIt was 50s. . . , xii

Bott-Chern and Aeppli cohomologies for complex manifolds:


holomorphic sections, Acta Math. 114 (1965), 71�112.

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

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

they provide bridges between de Rham and Dolbeault cohomologies,

allowing their comparison

Introduction, xiiisummary, i

Aim:study the algebra of Bott-Chern cohomology, and its relation withde Rham cohomology:

use Bott-Chern as degree of �non-Kählerness�. . .

. . . in order to characterize ∂∂-Lemma;

develop techniques for computations on special classes of manifolds.

Introduction, xivsummary, ii





Introduction, xvsummary, iii





Introduction, xvisummary, iv





Cohomologies of complex manifolds, idouble complex of forms, i

Consider the double

complex(∧•,•X , ∂, ∂

)associated to

a cplx mfd X

p− 2 p− 1 p p+ 1 p+ 2

q − 2

q − 1

q

q + 1

q + 2

Cohomologies of complex manifolds, iidouble complex of forms, ii

Consider the double

complex(∧•,•X , ∂, ∂

)associated to

a cplx mfd X

p− 2 p− 1 p p+ 1 p+ 2

q − 2

q − 1

q

q + 1

q + 2

Cohomologies of complex manifolds, iiiDolbeault cohomology, i

H•,•∂

(X ) :=ker ∂

im ∂

p− 2 p− 1 p p+ 1 p+ 2

q − 2

q − 1

q

q + 1

q + 2

P. Dolbeault, Sur la cohomologie des variétés analytiques complexes, C. R. Acad. Sci. Paris 236

(1953), 175�177.

Cohomologies of complex manifolds, ivDolbeault cohomology, ii

H•,•∂ (X ) :=

ker ∂

im ∂

p− 2 p− 1 p p+ 1 p+ 2

q − 2

q − 1

q

q + 1

q + 2

Cohomologies of complex manifolds, vDolbeault cohomology, iii

In the Frölicher spectral sequence

H•,•∂

(X ) +3 H•dR(X ;C)

the Dolbeault cohom plays the role of approximation of de Rham.

As a consequence, the Frölicher inequality holds:∑p+q=k

dimC Hp,q

∂(X ) ≥ dimC Hk

dR(X ;C) .



Cohomologies of complex manifolds, viDolbeault cohomology, iv

In the Frölicher spectral sequence

H•,•∂

(X ) +3 H•dR(X ;C)

the Dolbeault cohom plays the role of approximation of de Rham.

As a consequence, the Frölicher inequality holds:∑p+q=k

dimC Hp,q

∂(X ) ≥ dimC Hk

dR(X ;C) .



Cohomologies of complex manifolds, viiBott-Chern and Aeppli cohomologies, i

In general, there is no natural map between Dolb and de Rham:

we would like to have a �bridge� between them.

N

��xx &&H•,•∂ (X )

&&

H•dR(X ;C)

��

H•,•∂

(X )

xxH

The bridges are provided by Bott-Chern and Aeppli cohomologies.

Cohomologies of complex manifolds, viiiBott-Chern and Aeppli cohomologies, ii



N

��xx &&H•,•∂ (X )

&&

H•dR(X ;C)

��

H•,•∂

(X )

xxH


Cohomologies of complex manifolds, ixBott-Chern and Aeppli cohomologies, iii



N

��xx &&H•,•∂ (X )

&&

H•dR(X ;C)

��

H•,•∂

(X )

xxH


Cohomologies of complex manifolds, xBott-Chern and Aeppli cohomologies, iv

H•,•BC (X ) :=

ker ∂ ∩ ker ∂

im ∂∂

p− 2 p− 1 p p+ 1 p+ 2

q − 2

q − 1

q

q + 1

q + 2


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

Cohomologies of complex manifolds, xiBott-Chern and Aeppli cohomologies, v

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.

Cohomological properties of non-Kähler manifolds, icohomologies of complex manifolds, i

On cplx mfds, identity induces natural maps

H•,•BC (X )

��yy %%H

•,•∂ (X )

%%

H•dR(X ;C)

��

H•,•∂

(X )

yyH

•,•A (X )

Cohomological properties of non-Kähler manifolds, iicohomologies of complex manifolds, ii

By def, a cpt cplx mfd satis�es ∂∂-Lemma

if every ∂-closed ∂-closed d-exact form is

∂∂-exact too

, equivalently, if all the above

maps are isomorphisms.

H•,•BC (X )

��H•dR(X ;C)

While compact Kähler mfds satisfy the ∂∂-Lemma, . . .

. . . Bott-Chern cohomology may supply further informations

on the geometry of non-Kähler manifolds.

Cohomological properties of non-Kähler manifolds, iiicohomologies of complex manifolds, iii



∂∂-exact too, equivalently, if all the above


H•,•BC (X )

��yy %%H•,•∂ (X )

%%

H•dR(X ;C)

��

H•,•∂

(X )

yyH•,•A (X )




Cohomological properties of non-Kähler manifolds, ivcohomologies of complex manifolds, iv





H•,•BC (X )

��yy %%H•,•∂ (X )

%%

H•dR(X ;C)

��

H•,•∂

(X )

yyH•,•A (X )




Cohomological properties of non-Kähler manifolds, vcohomologies of complex manifolds, v





H•,•BC (X )

��yy %%H•,•∂ (X )

%%

H•dR(X ;C)

��

H•,•∂

(X )

yyH•,•A (X )




Cohomological properties of non-Kähler manifolds, viinequality à la Frölicher for Bott-Chern cohomology, i

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 Hp,qBC (X ) + dimC H

p,qA (X )) ≥ 2 dimC H

kdR(X ;C) .

Furthermore, the equality characterizes the ∂∂-Lemma.

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

no. 1, 71�81.

Cohomological properties of non-Kähler manifolds, viiinequality à la Frölicher for Bott-Chern cohomology, ii

Dolbeault cohomology cares only about horizon-tal arrows, as Bott-Chern cares only about ingo-ing arrows, and, dually, Aeppli cares only aboutoutgoing arrows.



one gets:

0

0

1

1

2

2

3

3





kdR(X ;C) .



no. 1, 71�81.

Cohomological properties of non-Kähler manifolds, viiiinequality à la Frölicher for Bott-Chern cohomology, iii




one gets:

0

0

1

1

2

2

3

3





kdR(X ;C) .



no. 1, 71�81.

Cohomological properties of non-Kähler manifolds, ixinequality à la Frölicher for Bott-Chern cohomology, iv




one gets:

0

0

1

1

2

2

3

3





kdR(X ;C) .



no. 1, 71�81.

Cohomological properties of non-Kähler manifolds, xinequality à la Frölicher for Bott-Chern cohomology, v

Dolbeault cohomology cares only about horizon-tal arrows, as Bott-Chern cares only about ingo-ing arrows, and, dually, Aeppli cares only aboutoutgoing arrows.Since

] {ingoing} + ] {outgoing}


one gets: 0

0

1

1

2

2

3

3





kdR(X ;C) .



no. 1, 71�81.

Cohomological properties of non-Kähler manifolds, xiinequality à la Frölicher for Bott-Chern cohomology, vi




one gets: 0

0

1

1

2

2

3

3





kdR(X ;C) .



no. 1, 71�81.

Cohomological properties of non-Kähler manifolds, xiiinequality à la Frölicher for Bott-Chern cohomology, vii




one gets: 0

0

1

1

2

2

3

3





kdR(X ;C) .



no. 1, 71�81.

Cohomological properties of non-Kähler manifolds, xiiiinequality à la Frölicher for Bott-Chern cohomology, viii

For cpt cplx mfd:

∆k = 0 for any k ⇔ ∂∂-Lemma (= cohomologically-Kähler)

(where: ∆k := hkBC + hkA − 2 bk ∈ N).

For cpt cplx surfaces:

Kähler ⇔ b1 even ⇔ cohom-Kähler

hence ∆1 and ∆2 measure just Kählerness.

In fact, non-Kählerness is measured by just 12

∆2 ∈ N.

�, G. Dloussky, A. Tomassini, On Bott-Chern cohomology of compact complex surfaces,

arXiv:1402.2408 [math.DG].

Cohomological properties of non-Kähler manifolds, xivinequality à la Frölicher for Bott-Chern cohomology, ix

For cpt cplx mfd:







∆2 ∈ N.



Cohomological properties of non-Kähler manifolds, xvinequality à la Frölicher for Bott-Chern cohomology, x

For cpt cplx mfd:







∆2 ∈ N.



Cohomological properties of non-Kähler manifolds, xviinequality à la Frölicher for Bott-Chern cohomology, xi

For cpt cplx mfd:







∆2 ∈ N.



Cohomological properties of non-Kähler manifolds, xvii∂∂-Lemma and deformations � part I, i

By Hodge theory, dimC Hp,qBC and dimC Hp,q

A are upper-semi-continuous fordeformations of the complex structure.

Hence the equality∑p+q=k

(dimC Hp,qBC (X ) + dimC Hp,q

A (X )) = 2 dimC HkdR(X ;C)

is stable for small deformations. Then:

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

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

deformations.

Cohomological properties of non-Kähler manifolds, xviii∂∂-Lemma and deformations � part I, ii


A are upper-semi-continuous fordeformations of the complex structure. Hence the equality∑

p+q=k



is stable for small deformations.

Then:



deformations.

Cohomological properties of non-Kähler manifolds, xix∂∂-Lemma and deformations � part I, iii


A are upper-semi-continuous fordeformations of the complex structure. Hence the equality∑

p+q=k



is stable for small deformations. Then:



deformations.

Cohomological properties of non-Kähler manifolds, xx∂∂-Lemma and deformations � part I, iv

Problem:what happens for limits?

If Jt satis�es ∂∂-Lem for any t 6= 0, does J0 satisfy ∂∂-Lem, too?

We need tools for investigating explicit examples. . .

Cohomological properties of non-Kähler manifolds, xxitechniques of computations � nilmanifolds, i

X compact cplx mfd. We want to compute H•,•BC (X ).

Hodge theory reduces the probl to a pde system: �xed g Hermitianmetric, there is a 4th order elliptic di�erential operator ∆BC s.t.

H•,•BC (X ) ' ker∆BC =

{u ∈ ∧p,qX : ∂u = ∂u =

(∂∂)∗

u}.

For some classes of homogeneous mfds, the solutions of this system

may have further symmetries, which reduce to the study of ∆BC on

a smaller space. If this space is �nite-dim, we are reduced to solve a

linear system.

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

Cohomological properties of non-Kähler manifolds, xxiitechniques of computations � nilmanifolds, ii


Hodge theory reduces the probl to a pde system

: �xed g Hermitianmetric, there is a 4th order elliptic di�erential operator ∆BC s.t.


{u ∈ ∧p,qX : ∂u = ∂u =

(∂∂)∗

u}.




linear system.


Cohomological properties of non-Kähler manifolds, xxiiitechniques of computations � nilmanifolds, iii




{u ∈ ∧p,qX : ∂u = ∂u =

(∂∂)∗

u}.




linear system.


Cohomological properties of non-Kähler manifolds, xxivtechniques of computations � nilmanifolds, iv




{u ∈ ∧p,qX : ∂u = ∂u =

(∂∂)∗

u}.



a smaller space.

If this space is �nite-dim, we are reduced to solve a

linear system.


Cohomological properties of non-Kähler manifolds, xxvtechniques of computations � nilmanifolds, v




{u ∈ ∧p,qX : ∂u = ∂u =

(∂∂)∗

u}.




linear system.


Cohomological properties of non-Kähler manifolds, xxvitechniques of computations � nilmanifolds, vi

In other words, we would like to reduce the study to a H]-model,

that is, a sub-algebra

ι :(M•,•, ∂, ∂

)↪→(∧•,•X , ∂, ∂

)such that H](ι) isomorphism, where ] ∈

{dR, ∂, ∂, BC , A

}.

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

H]-model being �nite-dimensional as a vector space.

Cohomological properties of non-Kähler manifolds, xxviitechniques of computations � nilmanifolds, vii

In other words, we would like to reduce the study to a H]-model,

that is, a sub-algebra

ι :(M•,•, ∂, ∂

)↪→(∧•,•X , ∂, ∂

)such that H](ι) isomorphism, where ] ∈

{dR, ∂, ∂, BC , A

}.

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

H]-model being �nite-dimensional as a vector space.

Cohomological properties of non-Kähler manifolds, xxviiitechniques of computations � nilmanifolds, viii

Thm (Nomizu)

X = Γ\G nilmanifold (compact quotients of

connected simply-connected nilpotent Lie groups G by

co-compact discrete subgroups Γ).

Then it is HdR -computable.

More precisely, the �nite-dimensional sub-space

of forms being invariant for the left-action

G y X is a HdR -model.

Tj0 �� // X = Γ\G

��Tj1 �� // X1

��...

��Tjk �� // Xk

��Tjk+1

Cohomological properties of non-Kähler manifolds, xxixtechniques of computations � nilmanifolds, ix

Thm (Nomizu)








Tj0 �� // X = Γ\G

��Tj1 �� // X1

��...

��Tjk �� // Xk

��Tjk+1

Cohomological properties of non-Kähler manifolds, xxxtechniques of computations � nilmanifolds, x

Thm (Nomizu)








Tj0 �� // X = Γ\G

��Tj1 �� // X1

��...

��Tjk �� // Xk

��Tjk+1

Cohomological properties of non-Kähler manifolds, xxxitechniques of computations � nilmanifolds, xi

Thm (Nomizu; Console and Fino; �; et al.)

X = Γ\G nilmanifold

, endowed with a �suitable� left-invariant cplx

structure.

Then:

de Rham cohom (Nomizu)

Dolbeault cohom (Sakane, Cordero, Fernández, Gray, Ugarte, Console, Fino, Rollenske)

Bott-Chern cohom (�)

can be computed by considering only left-invariant forms.

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

Math. (2) 59 (1954), no. 3, 531�538.

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

no. 2, 111�124.

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

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

Cohomological properties of non-Kähler manifolds, xxxiitechniques of computations � nilmanifolds, xii


X = Γ\G nilmanifold, endowed with a �suitable� left-invariant cplx

structure.

Then:






Math. (2) 59 (1954), no. 3, 531�538.


no. 2, 111�124.


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

Cohomological properties of non-Kähler manifolds, xxxiiitechniques of computations � nilmanifolds, xiii


X = Γ\G nilmanifold, endowed with a �suitable� left-invariant cplx

structure.

Then:






Math. (2) 59 (1954), no. 3, 531�538.


no. 2, 111�124.


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

Cohomological properties of non-Kähler manifolds, xxxivIwasawa manifold, i

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

Cohomological properties of non-Kähler manifolds, xxxvIwasawa manifold, ii

Iwasawa manifold:

I3 := (Z [i])3∖

1 z1 z3

0 1 z2

0 0 1

∈ GL(C3)



)∗:{

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


dϕ2 = 0

dϕ3 = −ϕ1 ∧ ϕ2

Cohomological properties of non-Kähler manifolds, xxxviIwasawa manifold, iii

Iwasawa manifold:

I3 := (Z [i])3∖

1 z1 z3

0 1 z2

0 0 1

∈ GL(C3)



)∗:{

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


dϕ2 = 0

dϕ3 = −ϕ1 ∧ ϕ2

Cohomological properties of non-Kähler manifolds, xxxviiIwasawa manifold, iv

Iwasawa manifold:

I3 := (Z [i])3∖

1 z1 z3

0 1 z2

0 0 1

∈ GL(C3)



)∗:{

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


dϕ2 = 0

dϕ3 = −ϕ1 ∧ ϕ2

Cohomological properties of non-Kähler manifolds, xxxviiiIwasawa manifold, v

0

0

1

1

2

2

3

3

Left-invariant forms

provide a �nite-dim

cohomological-model

for the Iwasawa

manifold.

Cohomological properties of non-Kähler manifolds, xxxixIwasawa manifold, vi

Thm (Nakamura)

There exists a locally complete complex-analytic family of complex

structures, deformations of I3, depending on six parameters. They can be

divided into three classes according to their Hodge numbers

Bott-Chern yields a �ner classi�cation of Kuranishi space of I3 (�).

class h1∂

h1BC

h2∂

h2BC

h3∂

h3BC

h4∂

h4BC

h5∂

h5BC

(i) 5 4 11 10 14 14 11 12 5 6

(ii.a) 4 4 9 8 12 14 9 11 4 6(ii.b) 4 4 9 8 12 14 9 10 4 6

(iii.a) 4 4 8 6 10 14 8 11 4 6(iii.b) 4 4 8 6 10 14 8 10 4 6

b1 = 4 b2 = 8 b3 = 10 b4 = 8 b5 = 4

I. Nakamura, Complex parallelisable manifolds and their small deformations, J. Di�er. Geom. 10

(1975), no. 1, 85�112.

Cohomological properties of non-Kähler manifolds, xlIwasawa manifold, vii

Thm (Nakamura)

There exists a locally complete complex-analytic family of complex

structures, deformations of I3, depending on six parameters. They can be

divided into three classes according to their Hodge numbers

Bott-Chern yields a �ner classi�cation of Kuranishi space of I3 (�).

class h1∂

h1BC

h2∂

h2BC

h3∂

h3BC

h4∂

h4BC

h5∂

h5BC

(i) 5 4 11 10 14 14 11 12 5 6

(ii.a) 4 4 9 8 12 14 9 11 4 6(ii.b) 4 4 9 8 12 14 9 10 4 6

(iii.a) 4 4 8 6 10 14 8 11 4 6(iii.b) 4 4 8 6 10 14 8 10 4 6

b1 = 4 b2 = 8 b3 = 10 b4 = 8 b5 = 4

I. Nakamura, Complex parallelisable manifolds and their small deformations, J. Di�er. Geom. 10

(1975), no. 1, 85�112.

Cohomological properties of non-Kähler manifolds, xliIwasawa manifold, viii

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,


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

nilmanifolds, arXiv:1210.0395 [math.DG].

Cohomological properties of non-Kähler manifolds, xliitechniques of computations � solvmanifolds, i

Problem:

what about closedness of ∂∂-

Lemma under limits?

But:

non-tori nilmanifolds never satisfy ∂∂-Lemma (Hasegawa);

tori are closed (Andreotti and Grauert and Stoll).

Therefore:

consider solvmanifolds (compact quotients of connected simply-connected

solvable Lie groups by co-compact discrete subgroups).

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

A. Andreotti, W. Stoll, Extension of holomorphic maps, Ann. of Math. (2) 72 (1960), no. 2,

312�349.

Cohomological properties of non-Kähler manifolds, xliiitechniques of computations � solvmanifolds, ii

Problem:


Lemma under limits?

But:



Therefore:





312�349.

Cohomological properties of non-Kähler manifolds, xlivtechniques of computations � solvmanifolds, iii

Problem:


Lemma under limits?

But:



Therefore:





312�349.

Cohomological properties of non-Kähler manifolds, xlvtechniques of computations � solvmanifolds, iv

Several tools have been developed for computing cohomologies of

solvmanifolds with left-inv cplx structure

, and of their deformations

(H. Kasuya; �, H. Kasuya).

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.

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

(2013), no. 1-2, 437�447.

Cohomological properties of non-Kähler manifolds, xlvitechniques of computations � solvmanifolds, v

Several tools have been developed for computing cohomologies of

solvmanifolds with left-inv cplx structure, and of their deformations

(H. Kasuya; �, H. Kasuya).

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.

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

(2013), no. 1-2, 437�447.

Cohomological properties of non-Kähler manifolds, xlvii∂∂-Lemma and deformations � part II, i

Thanks to these tools:

Thm (�, H. Kasuya)

The property of satisfying the ∂∂-Lemma is non-closed under

deformations.

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

arXiv:1305.6709 [math.CV].

Cohomological properties of non-Kähler manifolds, xlviii∂∂-Lemma and deformations � part II, ii

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.

Cohomological properties of non-Kähler manifolds, xlix∂∂-Lemma and deformations � part II, iii

The Lie group


(ez 0

0 e−z

).



t∂

∂z1⊗ d z̄1 .

��



Cohomological properties of non-Kähler manifolds, l∂∂-Lemma and deformations � part II, iv

The Lie group


(ez 0

0 e−z

).



t∂

∂z1⊗ d z̄1 .

��



Cohomological properties of non-Kähler manifolds, li∂∂-Lemma and deformations � part II, v

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

Generalized-complex geometry, igeneralized-complex structures, i

Cplx structure:

J : TX'→ TX satisfying an algebraic condition (J2 = − idTX )

and an analytic condition (integrability in order to have

holomorphic coordinates).

Sympl structure:

ω : TX'→ T ∗X satisfying an algebraic condition (ω non-deg

2-form) and an analytic condition (dω = 0).

Generalized-complex geometry, iigeneralized-complex structures, ii

Cplx structure:

J : TX'→ TX satisfying an algebraic condition (J2 = − idTX )

and an analytic condition (integrability in order to have

holomorphic coordinates).

Sympl structure:

ω : TX'→ T ∗X satisfying an algebraic condition (ω non-deg

2-form) and an analytic condition (dω = 0).

Generalized-complex geometry, iiigeneralized-complex structures, iii

Hence, consider the bundle TX ⊕ T ∗X .

Note that it admits a natural

bilinear pairing: 〈X + ξ|Y + η〉 = 1

2(ιXη + ιY ξ).

Mimicking the def of cplx and sympl structures:'

&

$

%

a generalized-complex structure on a 2n-dim mfd X is a

J : TX ⊕ T ∗X → TX ⊕ T ∗X

such that J 2 = − idTX⊕T∗X , being orthogonal wrt 〈−|=〉, andsatisfying an integrability condition.

N. J. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281�308.

M. Gualtieri, Generalized complex geometry, Oxford University DPhil thesis, arXiv:math/0401221

[math.DG].

G. R. Cavalcanti, New aspects of the ddc -lemma, Oxford University D. Phil thesis,

arXiv:math/0501406 [math.DG].

Generalized-complex geometry, ivgeneralized-complex structures, iv

Hence, consider the bundle TX ⊕ T ∗X . Note that it admits a natural


2(ιXη + ιY ξ).


&

$

%


J : TX ⊕ T ∗X → TX ⊕ T ∗X




[math.DG].



Generalized-complex geometry, vgeneralized-complex structures, v



2(ιXη + ιY ξ).

Mimicking the def of cplx and sympl structures:

'

&

$

%


J : TX ⊕ T ∗X → TX ⊕ T ∗X




[math.DG].



Generalized-complex geometry, vigeneralized-complex structures, vi



2(ιXη + ιY ξ).


&

$

%


J : TX ⊕ T ∗X → TX ⊕ T ∗X




[math.DG].



Generalized-complex geometry, viigeneralized-complex structures, vii

Generalized-cplx geom uni�es cplx geom and sympl geom:

J cplx struct: then

J =

(−J 0

0 J∗

)is generalized-complex;

ω sympl struct: then

J =

(0 −ω−1ω 0

)is generalized-complex.

Generalized-complex geometry, viiigeneralized-complex structures, viii


J cplx struct: then

J =

(−J 0

0 J∗



J =

(0 −ω−1ω 0


Generalized-complex geometry, ixgeneralized-complex structures, ix


J cplx struct: then

J =

(−J 0

0 J∗



J =

(0 −ω−1ω 0


Generalized-complex geometry, xcohomological properties of symplectic manifolds, i

This explains the parallel between the cplx and sympl contexts:

e.g.,

for a symplectic manifold, consider the operators

d : ∧• X → ∧•+1X and dΛ := [d,−ιω−1 ] : ∧• X → ∧•−1X

as the counterpart of ∂ and ∂ in complex geometry.

De�ne the cohomologies

H•BC ,ω(X ) :=ker d∩ ker dΛ

im d dΛand H•A,ω(X ) :=

ker d dΛ

im d+ im dΛ.

L.-S. Tseng, S.-T. Yau, Cohomology and Hodge Theory on Symplectic Manifolds: I. II, J. Di�er.

Geom. 91 (2012), no. 3, 383�416, 417�443.

L.-S. Tseng, S.-T. Yau, Generalized Cohomologies and Supersymmetry, Comm. Math. Phys. 326

(2014), no. 3, 875�885.

Generalized-complex geometry, xicohomological properties of symplectic manifolds, ii

This explains the parallel between the cplx and sympl contexts: e.g.,


d : ∧• X → ∧•+1X and dΛ := [d,−ιω−1 ] : ∧• X → ∧•−1X





ker d dΛ

im d+ im dΛ.


Geom. 91 (2012), no. 3, 383�416, 417�443.


(2014), no. 3, 875�885.

Generalized-complex geometry, xiicohomological properties of symplectic manifolds, iii

This explains the parallel between the cplx and sympl contexts: e.g.,


d : ∧• X → ∧•+1X and dΛ := [d,−ιω−1 ] : ∧• X → ∧•−1X





ker d dΛ

im d+ im dΛ.


Geom. 91 (2012), no. 3, 383�416, 417�443.


(2014), no. 3, 875�885.

Generalized-complex geometry, xivcohomological properties of symplectic manifolds, v

Thm (Merkulov; Guillemin; Cavalcanti; �, A. Tomassini)

Let X be a 2n-dim cpt symplectic mfd.

Then, for any k,

dimR HkBC ,ω(X ) + dimR Hk

A,ω ≥ 2 dimR HkdR(X ;R) .

Furthermore, the following are equivalent:

X satis�es d dΛ-Lemma (i.e., Bott-Chern and de Rham cohom are natur isom);

X satis�es Hard Lefschetz Cond (i.e., [ωk ] : Hn−kdR

(X )→ Hn+kdR

(X ) isom ∀k);

equality dimHkBC ,ω(X ) + dimHk

A,ω = 2 dimHkdR(X ;R) holds for any k.

S. A. Merkulov, Formality of canonical symplectic complexes and Frobenius manifolds, Int. Math.

Res. Not. 1998 (1998), no. 14, 727�733.

V. Guillemin, Symplectic Hodge theory and the d δ-Lemma, preprint, Massachusetts Insitute of

Technology, 2001.

D. Angella, A. Tomassini, Inequalities à la Frölicher and cohomological decompositions, to appear

in J. Noncommut. Geom..

K. Chan, Y.-H. Suen, A Frölicher-type inequality for generalized complex manifolds,


Generalized-complex geometry, xvcohomological properties of symplectic manifolds, vi


Let X be a 2n-dim cpt symplectic mfd. Then, for any k,






(X )→ Hn+kdR

(X ) isom ∀k);




Res. Not. 1998 (1998), no. 14, 727�733.


Technology, 2001.





Generalized-complex geometry, xvicohomological properties of symplectic manifolds, vii


Let X be a 2n-dim cpt symplectic mfd. Then, for any k,






(X )→ Hn+kdR

(X ) isom ∀k);




Res. Not. 1998 (1998), no. 14, 727�733.


Technology, 2001.





Joint work with: Adriano Tomassini, Hisashi Kasuya, Federico A. Rossi, Maria Giovanna Franzini,Simone Calamai, Weiyi Zhang, Georges Dloussky.

And with the fundamental contribution of: Serena, Maria Beatrice and Luca, Alessandra, MariaRosaria, Francesco, Andrea, Matteo, Jasmin, Carlo, Junyan, Michele, Chiara, Simone, Eridano,Laura, Paolo, Marco, Cristiano, Amedeo, Daniele, Matteo, . . .

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