Cohomologies on complex manifolds...A conference in honor of Pierre Dolbeault On the occasion of his...
Transcript of Cohomologies on complex manifolds...A conference in honor of Pierre Dolbeault On the occasion of his...
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, . . .