on bott-chern cohomology for complex manifolds · (dipartimento di matematica e informatica,...

Komplex Analysis Winter School and WorkshopCIRM, Marseille, March 24�29, 2014

On Bott-Chern cohomology for complex manifolds

Daniele Angella

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

March 28, 2014

Introduction, isummary, i

On a complex manifold:

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

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

A natural map from Bott-Chern to de Rham relates holomorphicand topological aspects, . . .. . . and its non-injectivity measures the non-Kählerness.



several cohomological invariants can be de�ned

(de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies).





several cohomological invariants can be de�ned(de Rham

, Dolbeault, Bott-Chern, Aeppli cohomologies).





several cohomological invariants can be de�ned(de Rham, Dolbeault

, Bott-Chern, Aeppli cohomologies).






On Kähler mfds, they are all isomorphic, . . .

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






A natural map from Bott-Chern to de Rham relates holomorphicand topological aspects, . . .

. . . and its non-injectivity measures the non-Kählerness.

Introduction, iisummary, ii

We are interested in:

using Bott-Chern as degree of �non-Kähler� (ineq à la Frölicher);

using Bott-Chern to characterize ∂∂-Lemma;

developing techniques for the study of Bott-Chern cohomology onspecial classes of manifolds.

Cohomologies of a complex manifold, ithe double complex of forms

The double complex(∧•,•X , ∂, ∂

)is the �rst algebraic

object associated to a

cplx mfd

p− 2 p− 1 p p+ 1 p+ 2

q − 2

q − 1

q

q + 1

q + 2

Cohomologies of a complex manifold, iiDolbeault cohomology

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 a complex manifold, iiiconjugate Dolbeault 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, ivFrölicher inequality

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, one has the Frölicher inequality:∑p+q=k

dimC Hp,q

∂(X ) ≥ dimC Hk

dR(X ;C) .

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

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

Cohomologies of a complex manifold, vwhy to study further cohomologies � the Bott-Chern �bridge�

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

We would like to have a �bridge� between Dolbeault and de Rham.

That is, a way to read complex structure in de Rham cohom.

N

��xxqqqqqqqqqqqq

&&MMMMMMMMMMM

H•,•∂ (X )

&&MMMMMMMMMMMMH•dR(X ;C)

��

H•,•∂

(X )

xxqqqqqqqqqqq

H

The bridges are provided by Bott-Chern cohomology and by its

dual, Aeppli cohomology.

Cohomologies of a complex manifold, viBott-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, viiAeppli 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, viiicohomological properties of non-Kähler manifolds

On cplx mfds, there are natural maps

H•,•BC (X )

��yyrrrrrrrrrr

%%LLLLLLLLLL

H•,•∂ (X )

%%LLLLLLLLLLH•dR(X ;C)

��

H•,•∂

(X )

yyrrrrrrrrrr

H•,•A (X )

When the ∂∂-Lemma property (namely, every ∂-closed ∂-closed d-exact

form is ∂∂-exact too) holds, then all the above maps are isomorphisms.

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

. . . Bott-Chern cohomology may supply further informations

on the geometry of non-Kähler manifolds.



H•,•BC (X )

��yyrrrrrrrrrr

%%LLLLLLLLLL

H•,•∂ (X )


��

H•,•∂

(X )

yyrrrrrrrrrr

H•,•A (X )


form is ∂∂-exact too) holds,

then all the above maps are isomorphisms.






H•,•BC (X )

��yyrrrrrrrrrr

%%LLLLLLLLLL

H•,•∂ (X )


��

H•,•∂

(X )

yyrrrrrrrrrr

H•,•A (X )


form is ∂∂-exact too) holds, then all the above maps are isomorphisms.




Cohomological properties of non-Kähler manifolds, iinequality à la Frölicher for Bott-Chern

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 Hp,q

A (X )) ≥ 2 dimC HkdR(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.


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







dR(X ;C)).


no. 1, 71�81.


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







dR(X ;C)).


no. 1, 71�81.

Cohomological properties of non-Kähler manifolds, iiopenness of ∂∂-Lemma under deformations

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


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.



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

p+q=k



is stable for small deformations.

Then:



deformations.



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, iiitechniques of computations, 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:0709.3528v1 [math.AG].



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.






{u ∈ ∧p,qX : ∂u = ∂u =

(∂∂)∗

u}.




linear system.






{u ∈ ∧p,qX : ∂u = ∂u =

(∂∂)∗

u}.



a smaller space.

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

linear system.






{u ∈ ∧p,qX : ∂u = ∂u =

(∂∂)∗

u}.




linear system.


Cohomological properties of non-Kähler manifolds, ivtechniques of computations, ii

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

X = Γ\G nilmanifold (compact quotients of connected simply-connected

nilpotent Lie groups G by co-compact discrete subgroups Γ)

, 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 considereing only the �nite-dimensional sub-space offorms being invariant for the left-action G y X.

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.




nilpotent Lie groups G by co-compact discrete subgroups Γ), 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.




nilpotent Lie groups G by co-compact discrete subgroups Γ), 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, vnon-closedness of ∂∂-Lemma under deformations

Several tools have been developed for computing Dolbeault and

Bott-Chern cohomology of solvmanifolds (compact quotients of connected

simply-connected solvable Lie groups by co-compact discrete subgroups) with

left-inv cplx structure

, and of their deformations

(H. Kasuya; �, H. Kasuya).

Thanks to these tools:

Thm (�, H. Kasuya)

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

deformations.

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

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

arXiv:1305.6709v1 [math.CV].

Cohomological properties of non-Kähler manifolds, vnon-closedness of ∂∂-Lemma under deformations

Several tools have been developed for computing Dolbeault and

Bott-Chern cohomology of solvmanifolds (compact quotients of connected

simply-connected solvable Lie groups by co-compact discrete subgroups) with

left-inv cplx structure, and of their deformations

(H. Kasuya; �, H. Kasuya).

Thanks to these tools:

Thm (�, H. Kasuya)

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

deformations.

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

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

arXiv:1305.6709v1 [math.CV].

Cohomological properties of compact complex surfaces, inon-Kählerness degree

We proved that, for cpt cplx mfd:

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

(recall: ∆k := dimC TotkH•,•BC (X ) + dimC Tot

kH•,•A (X )− 2 dimC HkdR

(X ;C)).

For cpt cplx surfaces:

Kähler ⇔ b1 even ⇔ cohom-Kähler

hence ∆1 and ∆2 measure just Kählerness.

Thm (�, G. Dloussky, A. Tomassini)

For compact complex surfaces, non-Kählerness is measured by just

1

2∆2 =

∑p+q=2

dimC Hp,qBC (X )− b2 ∈ N .

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

arXiv:1402.2408v1 [math.DG].

Cohomological properties of compact complex surfaces, iisurfaces di�eomorphic to solvmfds and class VII

Thm (�, G. Dloussky, A. Tomassini)

Bott-Chern cohomology of complex surfaces di�eomorphic to

solvmanifolds (primary Kodaira, secondary Kodaira, Inoue SM ,

Inoue S±: Hasegawa) can be explicitly computed by using

left-inv forms.

The Bott-Chern numbers of class VII surfaces are:

h0,0BC = 1

h1,0BC = 0 h0,1BC = 0

h2,0BC = 0 h1,1BC = b2 + 1 h0,2BC = 0

h2,1BC = 1 h1,2BC = 1

h2,2BC = 1

that is, 1

2∆2 = 1.

Generalized-complex geometry, icomplex and symplectic structures

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, i � de�nition

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/0401221v1 [math.DG].

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



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


2(ιXη + ιY ξ).


&

$

%


J : TX ⊕ T ∗X → TX ⊕ T ∗X










2(ιXη + ιY ξ).

Mimicking the def of cplx and sympl structures:

'

&

$

%


J : TX ⊕ T ∗X → TX ⊕ T ∗X










2(ιXη + ιY ξ).


&

$

%


J : TX ⊕ T ∗X → TX ⊕ T ∗X







Generalized-complex geometry, iiigeneralized-complex structures, ii � examples

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, ivcohomological 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, ivcohomological 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, vcohomological properties of symplectic manifolds, ii

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,

arXiv:1403.1682 [math.DG].

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

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,

arXiv:1403.1682 [math.DG].

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, Andrea,Maria Rosaria, Matteo, Jasmin, Carlo, Junyan, Michele, Chiara, Simone, Eridano, Laura, Paolo,Marco, Cristiano, Amedeo, Daniele, Matteo, . . .

on bott-chern cohomology for complex manifolds · (dipartimento di matematica e informatica,...

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