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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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, 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, 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, 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.
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.
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.
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.
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.
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.
Cohomological properties of non-Kähler manifolds, iinequality à la Frölicher for Bott-Chern
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}
≥ ] {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.
Cohomological properties of non-Kähler manifolds, iinequality à la Frölicher for Bott-Chern
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}
≥ ] {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.
Cohomological properties of non-Kähler manifolds, iinequality à la Frölicher for Bott-Chern
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}
≥ ] {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.
Cohomological properties of non-Kähler manifolds, iinequality à la Frölicher for Bott-Chern
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}
≥ ] {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.
Cohomological properties of non-Kähler manifolds, iinequality à la Frölicher for Bott-Chern
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}
≥ ] {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.
Cohomological properties of non-Kähler manifolds, iinequality à la Frölicher for Bott-Chern
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}
≥ ] {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.
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
(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, 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
(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, 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
(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, 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].
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].
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].
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].
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].
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.
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.
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.
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 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, 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, 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, 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.
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.
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, 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,
arXiv:math/0501406v1 [math.DG].
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,
arXiv:math/0501406v1 [math.DG].
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,
arXiv:math/0501406v1 [math.DG].
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,
arXiv:math/0501406v1 [math.DG].
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, 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, 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, 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].
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, . . .