A conference in honor of Pierre Dolbeault On the occasion of his 90th birthday anniversary Cohomologies on complex manifolds Daniele Angella Istituto Nazionale di Alta Matematica (Dipartimento di Matematica e Informatica, Universit di Parma) June 04, 2014
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A conference in honor of Pierre DolbeaultOn the occasion of his 90th birthday anniversary
Cohomologies on complex manifolds
Daniele Angella
Istituto Nazionale di Alta Matematica(Dipartimento di Matematica e Informatica, Università di Parma)
June 04, 2014
Introduction, iIt was 50s. . . , i
Introduction, iiIt was 50s. . . , ii
Complex geometry encoded in global invariants:
Introduction, iiiIt was 50s. . . , iii
What informations in ∂-cohomology?
# Algebraic struct induced by di�erential algebra(∧•,•X , ∂, ∧
).
J. Neisendorfer, L. Taylor, Dolbeault homotopy theory, Trans. Amer. Math. Soc. 245
(1978), 183�210.
# Relation with topological informations.
A. Weil, Introduction à l'etude des variétés kählériennes, Hermann, Paris, 1958.
Introduction, ivIt was 50s. . . , iv
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
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