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
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, vIt was 50s. . . , v
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, viIt was 50s. . . , vi
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, 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:
A. Frölicher, Relations between the cohomology groups of Dolbeault and topological invariants,
Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 641�644.
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:
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, 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
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, xvsummary, iii
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, xvisummary, iv
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.
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) .
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 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) .
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 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
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, ixBott-Chern and Aeppli cohomologies, iii
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, 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
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 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
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 )
��yy %%H•,•∂ (X )
%%
H•dR(X ;C)
��
H•,•∂
(X )
yyH•,•A (X )
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, ivcohomologies of complex manifolds, iv
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 )
��yy %%H•,•∂ (X )
%%
H•dR(X ;C)
��
H•,•∂
(X )
yyH•,•A (X )
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, vcohomologies of complex manifolds, v
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 )
��yy %%H•,•∂ (X )
%%
H•dR(X ;C)
��
H•,•∂
(X )
yyH•,•A (X )
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, 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.
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, viiiinequality à la Frölicher for Bott-Chern cohomology, iii
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 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, ixinequality à la Frölicher for Bott-Chern cohomology, iv
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 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, 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}
≥ ] {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, xiinequality à la Frölicher for Bott-Chern cohomology, vi
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 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, xiiinequality à la Frölicher for Bott-Chern cohomology, vii
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 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, 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:
∆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, xvinequality à la Frölicher for Bott-Chern cohomology, x
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, xviinequality à la Frölicher for Bott-Chern cohomology, xi
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, 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
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, xix∂∂-Lemma and deformations � part I, iii
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, 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
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, xxiiitechniques of computations � nilmanifolds, iii
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, xxivtechniques of computations � nilmanifolds, iv
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, xxvtechniques of computations � nilmanifolds, v
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, 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)
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, xxxtechniques of computations � nilmanifolds, x
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, 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
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, xxxiiitechniques of computations � nilmanifolds, xiii
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, 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)
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, xxxviIwasawa manifold, iii
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, xxxviiIwasawa manifold, iv
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, 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,
arXiv:1210.0406 [math.DG].
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:
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, xlivtechniques of computations � solvmanifolds, iii
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, 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
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, l∂∂-Lemma and deformations � part II, iv
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, 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
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, vgeneralized-complex structures, v
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, vigeneralized-complex structures, vi
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, 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
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, ixgeneralized-complex structures, ix
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, 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.,
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, xiicohomological properties of symplectic manifolds, iii
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, 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,
arXiv:1403.1682 [math.DG].
Generalized-complex geometry, xvcohomological properties of symplectic manifolds, vi
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, xvicohomological properties of symplectic manifolds, vii
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, MariaRosaria, Francesco, Andrea, Matteo, Jasmin, Carlo, Junyan, Michele, Chiara, Simone, Eridano,Laura, Paolo, Marco, Cristiano, Amedeo, Daniele, Matteo, . . .