UNIVERSITÀ DI PISA
——————————————————————————
Dipartimento di Matematica e Informatica
Corso di Dottorato in Matematica
XXX ciclo
Tesi di dottorato
Cohomological aspects on
complex and symplectic manifolds
Nicoletta Tardini
Relatore: Prof. Adriano Tomassini
Settore Scientifico-Disciplinare: MAT/03 Geometria ——————————————————————————
Introduction 4
1 Preliminaries on (non-)Kähler geometry 19
1.1 (Almost-)complex geometry . . . 20
1.1.1 Almost-complex structures . . . 20
1.1.2 Complex structures and Dolbeault cohomology . . . . 21
1.1.3 Currents and de Rham homology . . . 27
1.2 Symplectic geometry . . . 30
1.3 Kähler geometry . . . 31
1.3.1 Kähler identities and Hodge theory . . . 34
1.3.2 Sullivan formality and the ddc-lemma . . . 36
1.4 Special metrics on complex manifolds . . . 40
1.5 Nilmanifolds and Solvmanifolds . . . 41
1.6 Deformations of complex structures . . . 46
2 Almost-complex and symplectic cohomologies and proper surjective maps 52 2.1 Comparisons of de Rham and Dolbeault cohomology groups under proper surjective maps . . . 53
2.2 Cohomology groups on almost-complex manifolds . . . 56
2.3 Behavior under proper surjective maps of the almost-complex cohomology groups . . . 59
2.3.1 Example . . . 64
2.4 Cohomology groups on symplectic manifolds . . . 66
2.4.1 Hodge theory and ddΛ-lemma . . . 69
2.5 Behavior under proper surjective maps of the symplectic cohomology groups . . . 72
3 Bott-Chern and Aeppli cohomologies 77 3.1 The Bott-Chern cohomology and the ∂∂-lemma . . . 78
CONTENTS 2
3.1.2 The ∂∂-lemma . . . 82
3.2 An upper bound on the dimension of Bott-Chern cohomology for compact complex manifolds . . . 88
3.3 A quantitative characterization of the ∂∂-Lemma and examples 91 3.4 A qualitative characterization of the ∂∂-Lemma . . . 94
3.5 An algebraic generalization with application to compact symplectic manifolds . . . 95
3.5.1 Algebraic upper bound . . . 95
3.5.2 Upper bound for symplectic Bott-Chern cohomology . 97 3.6 A quantitative characterization of the ddΛ-lemma . . . 98
3.7 Complex Orbifolds . . . 103
3.7.1 Special metrics and resolutions . . . 104
3.7.2 Complex orbifolds and the ∂∂-lemma . . . 107
4 Bott-Chern formality 113 4.1 Geometric Bott-Chern formality . . . 114
4.2 Aeppli-Bott-Chern-Massey triple products . . . 115
4.2.1 Relation with Sullivan formality . . . 116
4.3 Instability of Bott-Chern geometrical formality . . . 119
4.3.1 Bott-Chern geometric formality for complex surfaces . 119 4.3.2 Instability result . . . 121
4.4 Higher order Aeppli-Bott-Chern-Massey products . . . 131
4.4.1 Algebraic definition . . . 132
4.4.2 Application to complex geometry . . . 136
5 Locally conformally symplectic manifolds and cohomology 140 5.1 Locally conformally symplectic structures . . . 140
5.2 Bi-differential graded vector space for lcs structures and cohomology . . . 143
5.2.1 Elliptic Hodge theory for lcs cohomologies . . . 148
5.2.2 Symmetries in lcs cohomologies . . . 149
5.2.3 Hard Lefschetz Condition for lcs cohomologies . . . 151
5.3 Twisted cohomologies of solvmanifolds . . . 158
5.3.1 Hattori’s Theorem for completely-solvable solvmanifolds158 5.3.2 Mostow condition for solvmanifolds . . . 159
5.4 Examples . . . 162
5.4.1 Kodaira-Thurston surface . . . 162
5.4.2 Lie algebra d4 . . . 164
5.4.3 Inoue surfaces S0 . . . 165
5.4.4 Oeljeklaus-Toma manifolds with precisely one complex place . . . 168
Bibliography 178
Introduction
A very special class of complex manifolds is provided by Kähler manifolds, namely complex manifolds admitting a Kähler metric, that is a Hermitian metric whose associated fundamental 2-form is d-closed (equivalently, the metric is locally Euclidean up to 2nd order terms).
Any projective manifold (i.e., compact complex submanifolds of the complex projective space CPn) is Kähler but the converse is not true in
general; indeed Kodaira in [Kod54] proves that a compact Kähler manifold is projective (or equivalently projective algebraic by Chow’s theorem, see [Cho49]) if and only if it admits a positive line bundle. In this sense Kähler manifolds can be seen as a generalization of projective manifolds. For dimensional reasons every Riemann surface is Kähler but the situation in higher dimension is very different. In complex dimension two, Kählerness can be topologically characterized in terms of the first Betti number (see [Kod64], [Siu83], [Lam99], [Buc99]) but a similar result does not hold in dimension greater than two. Nevertheless there are many topological obstructions to the existence of a Kähler metric on a manifold, for instance on Kähler manifolds the odd Betti numbers are even and the even Betti numbers are positive. These results follow from the strong requests on the involved geometric structures and their deep relations; being more precise, Kähler manifolds carry a complex-analytic structure (i.e., a (1, 1)-tensor J whose square is −Id and satisfying an integrability condition), a metric structure and a symplectic structure (i.e., a non-degenerate d-closed 2-form) which are compatible to each other. On the other hand, in general neither complex manifolds nor symplectic manifolds are Kähler, for example the Iwasawa manifold (see e.g., [FG85]) is a complex manifold which is not Kähler; as regards symplectic manifolds McDuff in [McD84] shows the first example of a simply-connected compact symplectic non-Kähler manifold (in [FMS03] the authors show also explicit examples of compact symplectic manifolds which are cohomologically Kähler but that do not admit any Kähler metric).
metric and to weaken the geometric structures that are involved and/or to weaken their relations. One can consider for example almost-complex manifolds, generalizing the notion of complex structure requiring the non-integrability of the tensor J (see e.g., [NN57]) or one can look for "special" (non necessarily Kähler) metrics on complex manifolds, for instance balanced metrics, (strongly-)Gauduchon metrics, SKT metrics and so on (cf.[Mic83], [Gau77], [Pop13a], [Bis89]). On the other side one could ignore the (almost-)complex structure focusing the attention on the existence of a non-degenerate d-closed 2-form (i.e., a symplectic form) moving therefore to symplectic geometry. More generally, one could also consider a non-degenerate 2-form which is locally conformal to a symplectic structure, namely a locally conformally symplectic structure. An important global tool in studying (almost-)complex and (locally conformally) symplectic manifolds is furnished by cohomology, more precisely cohomology groups that provide invariants for the considered geometric structures.
Roughly speaking, suppose that X is a smooth manifold endowed with a geometric structure (in our case complex, symplectic, locally conformally symplectic; the almost-complex case will be different) to which we can associate a double complex (K●,●, δ
1, δ2) of K-vector spaces (K = R , C),
namely a Z2-graded (in the symplectic and locally conformally-symplectic
cases this will be just Z-graded, however it is always possible to associate a double complex to it) K-vector space K●,● endowed with δ
1 ∈ End1,0(K●,●)
and δ2 ∈ End0,1(K●,●) such that (δ1)2 = (δ2)2 = δ1δ2+ δ2δ1 = 0; then one can
consider the Dolbeault cohomologies: Hδ●,● 1 (K ●,●) ∶= Ker δ1 Im δ1 , Hδ●,● 2 (K ●,●) ∶= Ker δ2 Im δ2 ,
and the Bott-Chern and Aeppli cohomology groups defined respectively as Hδ●,● 1+δ2(K ●,●) ∶= Ker δ1∩ Ker δ2 Im δ1δ2 , Hδ●,● 1δ2(K ●,●) ∶= Ker δ1δ2 Im δ1+ Im δ2 . Denoting with HD●,●(Tot● K●,●) ∶= Ker D Im D
the cohomology associated to the single complex
(Tot●
K●,●∶= ⊕
6 natural maps Hδ●,● 1+δ2(K ●,●) vv (( Hδ●,● 1 (K ●,●) (( HD●,●(Tot● K●,●) Hδ●,● 2 (K ●,●) vv Hδ●,● 1δ2(K ●,●) ;
in general these maps are neither injective nor surjective. The bijectivity of these maps goes under the name of δ1δ2-lemma and it will play an
important role in this Thesis. We will see that these cohomologies can be studied from two different points of view: quantitative and qualitative. In particular, we discuss comparisons on the dimensions of these cohomology groups either of different manifolds related by structure-preserving, proper, surjective maps or on the same manifold discussing the relations with the δ1δ2-lemma. Moreover, we will study Hodge theory for such cohomology
groups focusing also, in the complex setting, on the algebraic structure of the space of harmonic forms associated to the Bott-Chern cohomology.
Being more precise, on a (compact) complex manifold X one can consider the space of (p, q)-forms A●,●(X) endowed with the two differential
operators ∂ and ∂ which are respectively the (1, 0)- and (0, 1)-components of the exterior derivative d; the integrability of the almost complex structure guarantees that the triple (A●,●(X) , ∂ , ∂) is a double complex.
One can define the classical de Rham cohomology H●
dR(X, C) ∶=
Ker d Im d , and Dolbeault cohomology
H●,●
∂ (X) ∶=
Ker ∂ Im ∂ ,
but it turns out that in complex non-Kähler geometry they do not suffice in studying a complex manifold (see e.g., [Ang13b]). Many informations are indeed contained in the Bott-Chern [BC65] and Aeppli [Aep64] cohomologies, defined, on a complex manifold X, respectively as
HBC●,●(X) ∶= Ker ∂∩ Ker ∂ Im ∂∂ , H ●,● A (X) ∶= Ker ∂∂ Im ∂+ Im ∂.
These two cohomologies represent a bridge between a topological invariant (the de Rham cohomology) and a complex invariant (the Dolbeault cohomology), indeed we have that the identity induces natural maps
HBC●,●(X) xx && H∂●,●(X) && H● dR(X, C) H●,● ∂ (X) xx HA●,●(X) ,
while there is no natural map between the Dolbeault and de Rham cohomologies. Generally such maps are neither injective nor surjective but when the map H●,●
BC(X) Ð→ H
●
dR(X) is injective, namely every ∂-closed,
∂-closed and d-exact form is ∂∂-exact, the manifold X is said to satisfy the ∂∂-lemma. Moreover, the injectivity of HBC●,●(X) Ð→ H●
dR(X) is equivalent
to all the maps in the diagram being isomorphisms [DGMS75, Lemma 5.15]. Every Kähler manifold satisfies the ∂∂-lemma [DGMS75] but the converse is not true. More generally, manifolds in class C of Fujiki and Moishezon manifolds (namely manifolds that can be respectively modified to a Kähler manifold and a projective manifold) satisfy the ∂∂-lemma. In Theorem 3.10 we show that the ∂∂-lemma is stable under the blow-up on a point. In §3.7 we discuss special Hermitian metrics and the notion of ∂∂-lemma on complex orbifolds providing a few examples starting from the smooth Iwasawa manifold and the completely solvable Nakamura manifold.
As a consequence of Hodge theory, all these cohomology groups on compact complex manifolds are finite dimensional vector spaces, in particular by [Sch] the Bott-Chern and Aeppli cohomologies are isomorphic to the kernel of two suitable 4th-order elliptic self-adjoint differential
operators, that will be denoted respectively by ∆BC and ∆A. Differently
from the Dolbeault cohomology, Hermitian duality does not preserve the Bott-Chern cohomology, in fact it realizes an isomorphism with the Aeppli cohomology. More precisely when a Hermitian metric is fixed on a compact complex manifold X of complex dimension n, the C-anti-linear Hodge-∗-operator induces an (un-natural) isomorphism between the Bott-Chern cohomology and the Aeppli cohomology, namely
∗ ∶ Hp,q
BC(X) Ð→ H n−p,n−q A (X)
is an isomorphism for any p, q ∈ Z; this means that we do not have a symmetry with respect to the center in the Bott-Chern (and Aeppli)
8
diamond.
As regards the algebraic structure, a very easy computation shows that the product induced by the wedge product on forms induces a structure of algebra for the Bott-Chern cohomology of a complex manifold and a structure of H●,●
BC(X)-module for the Aeppli cohomology. However, when
fixed a Hermitian metric g on X it turns out that in general Ker ∆BC does
not have a structure of algebra, namely the wedge product of harmonic forms can be non-harmonic. The "incompatibility of wedge products and harmonicity of forms" was first noticed by D. P. Sullivan in [Sul77] for harmonic forms with respect to the classical de Rham Laplacian and this motivated D. Kostchick to define in [Kot01] formal Riemannian metrics on compact differentiable Riemannian manifolds. Similarly, D. Angella and A. Tomassini introduce in [AT15b] a notion of geometric formality for the Bott-Chern cohomology. A compact complex manifold is called geometrically-HBC-formal if there exists a Hermitian metric on X such that
Ker ∆BC has a structure of algebra. On Kähler manifolds geometric
formality and geometric-HBC-formality are related (see Corollary 4.6);
while in Example 4.2.1 we show that on non-Kähler manifolds Sullivan formality and geometric-HBC-formality are not linked. One way to see it is
to consider the Aeppli-Bott-Chern-Massey triple products which represent an obstruction to the existence of such metrics. It turns out that these cohomological objects strongly depend on the complex structure. Indeed, motivated by stability results for compact Kähler manifolds ([KS60, Theorem 15]) and, more in general, for compact complex manifolds satisfying the ∂∂-lemma ([Voi02], [Wu96], [AT13]) under small deformations of the complex structure we study the behavior under small deformations of
the Aeppli-Bott-Chern-Massey triple products and
geometric-HBC-formality. In particular, we prove the following.
Theorem 0.1 (Theorem 4.8, Theorem 4.11, Corollary 4.12). The property of geometric-HBC-formality is not stable under small deformations of the
complex structure. However, let X be either a primary Kodaira surface or a secondary Kodaira surface or a Inoue surface of typeSM or a Inoue surface of
type S± or a Hopf surface. Then any small deformation of X is
geometrically-HBC-formal.
The instability is proven by showing that the Calabi-Eckmann manifold S3 × S3 is geometrically-HBC-formal and by constructing an explicit
deformation which is not, because a non-trivial Aeppli-Bott-Chern-Massey triple product appears. Further comments on special metrics and Frölicher spectral sequence of this manifold are done in Remarks 4.13, 4.14, discussing that S3× S3 is SKT (i.e., the fundamental form associated to the
fixed metric is ∂∂-closed) and the Frölicher spectral sequence degenerates at the second step (cf. [Pop16, Conjecture 1.3] where it is conjectured that the existence of a SKT metric implies the degeneration of the Frölicher spectral sequence at the second page).
Further obstructions to the existence of geometrically-HBC-formal metrics
are provided by higher order Aeppli-Bott-Chern Massey products, which we define as a generalization of triple products, inspired by the definition of a-Massey products in [CFM08] (see Lemmas 4.16, 4.17, Definition 4.18, Lemma 4.20 and Theorem 4.21). Some Examples of non-trivial Aeppli-Bott-Chern Massey products of order 5 will be presented (Examples 4.4.2, 4.4.2).
On the other side, many informations can be obtained investigating quantitative properties of the Bott-Chern and Aeppli cohomologies (namely, relations between their dimensions in terms of the Betti and Hodge numbers) towards the study of their qualitative properties (namely, their algebraic structure).
On compact Kähler manifolds the Hodge decomposition Theorem ([Wei58, Théorème IV.3]) states that the de Rham cohomology decomposes as a direct sum of the Dolbeault cohomology groups, namely
H●
dR(X, C) ≃ ∑ p+q=●
Hp,q
∂ (X) .
Such a decomposition does not hold in general, however a classical result by Frölicher [Fro55] states that the Hodge numbers are bounded below by the Betti numbers ∑ p+q=● dimCH p,q ∂ (X) ≥ b●(X)
as a consequence of the convergence to the de Rham cohomology of a spectral sequence starting at the first page with the Dolbeault cohomology. The Betti numbers bound from below also the Bott-Chern and Aeppli numbers, indeed in [AT13, Theorem A, Theorem B] it is proven that, in general it holds for every k, ∆k(X) ∶= ∑ p+q=k (dimCH p,q BC(X) + dimCH p,q A (X)) − 2 bk ≥ 0 .
Moreover, the equalities characterize the validity of the ∂∂-lemma on X, meaning that from a purely quantitative information we are able to understand if the manifold is cohomologically Kähler or not. In particular, if X is a compact complex surface by [Tel06] and [ADT16] one has that
10
∆1 = 0 and ∆2 ∈ {0 , 2}; notice that in complex dimension 2, ∆2 = 0 is
equivalent to Kählerness (cf. [Lam99], [Buc99]).
It turns out that the Aeppli and Bott-Chern numbers are not just bounded by Hodge numbers from below (as shown in the proof in [AT13]) but also from above. We prove in fact the following.
Theorem 0.2 (Theorem 3.15, Remark 3.16). Let X be a compact complex manifold of complex dimension n. Then, for any k∈ Z,
∑ p+q=k dimCHAp,q(X) (1) ≤ min{k + 1, (2n − k) + 1} ⋅⎛⎝ ∑ p+q=k dimCHp,q ∂ (X) + ∑ p+q=k+1 dimCHp,q ∂ (X) ⎞ ⎠ . These inequalities, valid dually for the Bott-Chern cohomology (cf. Remark 3.16), are shown to be sharp (see Table 3.1 where some explicit examples are collected and Remark 3.19). The algebraic nature of the proof of this result allows to generalize it to double complexes with some additional hypothesis of boundedness, in particular applications to symplectic geometry follow (cf. Subsections 3.5.1 and 3.5.2).
Moreover, we prove that the Bott-Chern cohomology is an invariant strong enough to characterize the ∂∂-lemma, by the following
Theorem 0.3 (Theorem 3.17, Remark 3.18). A compact complex manifold X of complex dimension n satisfies the ∂∂-Lemma if and only if, for any k∈ Z, there holds ∑ p+q=k (dimCH p,q BC(X) − dimCH n−p,n−q BC (X)) = 0 .
This means that a central simmetry in the Bott-Chern diamond forces the natural maps among all the cohomology groups to be isomorphisms. Moreover, a qualitative characterization of the ∂∂-lemma is provided asking that the composition map of the cup product on Bott-Chern cohomology and of the pairing with the fundamental class of X is non-degenerate; this is called qualitative Kodaira-Spencer-Schweitzer property.
Theorem 0.4 (Theorem 3.21). Let X be a compact complex manifold. Then X satisfies the qualitative Kodaira-Spencer-Schweitzer property, i.e., the natural pairing
HBC●,●(X) × HBC●,●(X) → C , ([α], [β]) ↦ ∫
X
α∧ β is non-degenerate if and only if X satisfies the ∂∂-Lemma.
If X is a compact smooth manifold endowed with a symplectic structure ω one can define the analogue of the Bott-Chern and Aeppli cohomologies. Indeed denoting with dΛ the symplectic adjoint of the exterior derivative d
([Bry88]), L.-S. Tseng and S.-T. Yau in [TY12a, TY12b] define the symplectic Bott-Chern and Aeppli cohomology groups respectively as
H● d+dΛ(X) ∶= Ker(d + dΛ) Im (ddΛ) , H ● ddΛ(X) ∶= Ker(ddΛ) Im d+ Im dΛ.
These cohomology groups are isomorphic to the kernel of suitable 4th-order
elliptic self-adjoint differential operators and consequently they are finite-dimensional vector spaces. In Remark 2.26 we show that these cohomologies can be similarly defined on symplectic orbifolds of global quotient type (i.e., quotients of the form Y/G where G is a finite group acting linearly on a compact smooth manifold Y of dimension 2n endowed with a G-invariant symplectic structure) and we study Hodge theory for them.
These are the appropriate cohomology groups to talk about "symplectic Hodge theory" on smooth symplectic manifolds, because the operator ddΛ+ dΛdis not elliptic, more precisely it is identically zero and it turns out
that there exists a both d-closed and dΛ-closed representative in every
de-Rham cohomology class if and only if (X, ω) satisfies the Hard Lefschetz condition, i.e., for every k ∈ N, the maps
Lk ∶= [ω]k⌣∶ HdRn−k(X, R) → HdRn+k(X, R)
are isomorphisms (cf. [Bry88, Conjecture 2.2.7], [Mat95, Corollary 2], [Mer98, Proposition 1.4], [Yan96, Theorem 0.1], [Cav, Theorem 5.4], [TY12a, Proposition 3.13]). Equivalently, (X, ω) satisfies the ddΛ-lemma,
i.e., the natural maps in the diagram H● d+dΛ(X) ww && H● dR(X, R) '' H● dΛ(X) xx H● ddΛ(X)
are isomorpisms. These equivalent properties can be studied from a quantitative point of view and, as a consequence of [AT15a] and Theorem 3.28, one can prove the following result.
12
Theorem 0.5 (Theorem 3.28, Theorem 3.30). Let (X4, ω) be a compact
symplectic 4-manifold, then it satisfies
Hard Lefschetz condition ⇐⇒ b2(X) = dim Hd+d2 Λ(X).
This means that on a 4-dimensional compact symplectic manifold the Hard Lefschetz condition can be studied by just studying the dependence of H2
d+dΛ(X) on the symplectic structure, since b2(X) is a topological invariant.
The symplectic analogue of the non-negative numbers ∆kdefined on complex
manifolds are defined in [AT15a] as ∆k
s(X) ∶= dim Hd+dk Λ(X) + dim HddkΛ(X) − 2 bk.
It turns out that they behave differently on manifolds of real dimension 4, namely we do not have the analogue results presented in [Tel06], [ADT16]; indeed, while on compact complex manifolds ∆1 = 0 only in real dimension
four, we prove that on compact symplectic manifolds ∆1
s = 0 in any
dimension (Theorem 3.28). However, while on compact complex surfaces ∆2 ∈ {0 , 2}, we construct in Theorem 3.31 an explicit symplectic example
with ∆2
s= 4, showing that we do not have the same restrictions presented in
[ADT16] and in [Tel06].
Since the Hard Lefschetz condition plays an important role in symplectic geometry we discuss in Chapter 5 a possible generalization to locally conformally symplectic (lcs for brevity) manifolds. They arise naturally as generalized phase spaces of Hamiltonian dynamical systems, since the form of the Hamilton equations is preserved by homothetic canonical transformations (cf. e.g., [Vai85]). More precisely, a lcs manifold ([Lib54], [Lee43]) is a smooth manifold X of dimension 2n endowed with a non-degenrate 2-form Ω such that
dΩ = ϑ ∧ Ω ,
where ϑ is a d-closed 1-form. This is equivalent to say that locally Ω is conformal to a symplectic form. Notice that if ϑ= 0 then dΩ = 0 and so we recover the symplectic case. In general, Ω is closed with respect to the twisted differential:
dϑ∶= d − ϑ ∧ - .
It is easy to show that d2
ϑ = 0 and so one can consider the cohomology of
the perturbed complex (A●(X) , d
ϑ), namely Hϑ●(X) which is called
Morse-Novikov cohomology of X with respect to ϑ (cf. [Nov81, Nov82, GL84]). The Morse-Novikov cohomology is not a topological invariant, it depends on
[ϑ] ∈ H1
dR(X; R) up to gauge equivalence. However it can happen that it
gives more information than the de Rham cohomology (cf. [Oti16]).
The operator dϑ is not a differential in the usual sense, in fact it does not
satisfy the Leibniz rule, moreover it does not commute with the Lefschetz operator L. For this reason, and others that will be better explained in the following, it seems natural to define the following operators for any k ∈ R, (compare [LV15, Section 2] and [TY12a]):
dk ∶= dkϑ ∶= d − (kϑ) ∧ -∶ A●X→ A●+1X ,
δk ∶= dk−1Λ− Λdk∶ A●X→ A●−1X .
We prove in Lemma 5.9 that (dk)2 = 0 , δkδk+1 = 0 , dk−1δk+ δkdk = 0 , for
every k∈ R; hence the following cohomologies are well-defined (cf. Definition 5.10) H● dk(X) ∶= Ker dk Im dk , H● δk(X) ∶= Ker δk Im δk+1 , H● dk+δk(X) ∶= Ker dk∩ Ker δk Im δk+1dk+1 , H● δkdk(X) ∶= Ker δkdk Im dk+ Im δk+1 . The space H●
dk+δk(X) is called lcs-Bott-Chern cohomology of weight k of X, and H●
δkdk(X) is called the lcs-Aeppli cohomology of weight k of X. Again the identity induces natural maps of Z-graded vector spaces:
H● dk+δk(X) xx && H● dk(X) && H● δk(X). xx H● δkdk(X) (2)
We say that the manifold X satisfies the lcs-Lemma (Definition 5.11) if the natural map H●
dk+δk(X) → H
●
δkdk(X) induced by the identity is injective for any k∈ R. In this case, all the above maps are isomorphisms, see [DGMS75, Lemma 5.15], adapted in [AT15a, Lemma 1.4] to the Z-graded case. These cohomolgies reduce to L.-S. Tseng and S.-T. Yau’s symplectic cohomologies when ϑ = 0. We study elliptic Hodge theory for the lcs cohomologies introduced above (Proposition 5.13); in fact, those cohomology groups can be computed by just computing the harmonic forms with respect to suitable elliptic self-adjoint differential operators and we discuss dualities among them. In particular, we prove the following
14
Theorem 0.6 (Proposition 5.14, Theorem 5.15, Theorem 5.17). Let X be a compact differentiable manifold of dimension 2n endowed with a locally conformal symplectic form Ω with Lee form ϑ. Then, for any weight k ∈ R, for any degree h∈ Z, the symplectic-⋆-operator induces the isomorphism
⋆∶ Hn−h dk (X)
≃
→ Hn+h δh+k(X) .
On the other side, once chosen an almost-Kähler structure (g, J, Ω) on X, for any k∈ R, h ∈ Z, the Hodge-∗-operator induces the isomorphisms
∗∶ Hn−h dk (X) ≃ → Hn+h d−k (X) , ∗∶ H n−h δ−k−h(X) ≃ → Hn+h δk+h(X) , ∗∶ Hn−h dk+δk(X) ≃ → Hn+h δ−kd−k(X) . Moreover, the Lefschetz operator induces isomorphisms
Lh∶ Hdn−h k+δk(X) ≃ → Hn+h dk+h+δk+h(X) , Lh∶ Hδn−h kdk(X) ≃ → Hn+h δk+hdk+h(X) .
We also introduce a locally conformal version of the Hard Lefschetz condition which turns out to be equivalent to the structure being symplectic (up to global conformal changes) by the following
Theorem 0.7 (Theorem 5.18). Let X be a compact manifold of dimension 2n endowed with a lcs structure Ω with Lee form ϑ. Then, the following conditions are equivalent:
1. it satisfies the lcs-Hard Lefschetz condition, that is, for any h∈ Z, for any k∈ R, the map
Lh∶ Hdn−h
k (X) → H
n+h dk+h(X) is an isomorphism;
2. it satisfies the lcs-lemma, equivalently, for any h∈ Z, for any k ∈ R, the map Hdh k+δk(X) → H h dk(X) is an isomorphism;
3. it is symplectic up to global conformal changes and it satisfies the Hard Lefschetz Condition.
Moreover, we will discuss other equivalent properties that can be thought as the twisted version of some results proven by [Mer98, Proposition 1.4], [Gui01], [Cav, Theorem 5.4].
Some explicit examples are presented on nilmanifolds (Kodaira-Thurston surface [Kod64, Thu76]) and solvmanifolds (Inoue surfaces of type S+
[Ino74], for which see also [Oti16], and Oeljeklaus-Toma manifolds [OT05]). For compact quotients of connected simply-connected completely solvable Lie groups by lattices (these manifolds are called completely solvable solvmanifold), Hattori’s Theorem [Hat60, Corollary 4.2] allows to reduce the computation of the Morse-Novikov cohomology at the linear level of the Lie algebra, and the same holds for lcs cohomologies, see Lemma 5.24. Indeed, in general on non completely solvable solvmanifolds the invariant forms are not enough in computing cohomologies. However, one situation when they suffice is when the manifold satisfies the Mostow condition ([Mos61]). We prove that this condition suffices also for the lcs cohomologies with respect to an invariant closed one-form, see Proposition 5.26. The case of Inoue surfaces is interesting because two subclasses, S±,
are completely-solvable, and so Hattori’s Theorem applies, but this is not the case of the subclass S0. We prove here that Inoue surfaces of type S0
and, more in general, certain Oeljeklaus-Toma manifolds of type (s, 1), also known in the literature as with one complex place, satisfy the Mostow condition, see Proposition 5.29 and Theorem 5.32 respectively. More precisely, here we assume an arithmetic condition on the number field associated to their construction, namely, that there is no totally real intermediate extension. This holds for example for the Inoue surface of type S0, that is, in the case (s, t) = (1, 1), see also Proposition 5.29. As we show
in Proposition 5.34, for any s there exists an Oeljeklaus-Toma manifold of type (s, 1) satisfying such a property.
We have seen that quantitative and/or qualitative cohomological informations can provide a better understanding of the manifold itself. On the other side, instead of comparing the dimensions of cohomology groups on a fixed manifold one can also study comparisons among cohomology groups of manifolds related by structure-preserving proper surjective maps. In this spirit, R. O. Wells in [Wel74, Theorem 3.3] shows that a surjective proper differentiable map of non-zero degree between two orientable, differentiable manifolds of the same dimension induces an injection on the de Rham cohomology. If the map is also holomorphic between two complex manifolds, then the same holds for the Dolbeault and Bott-Chern cohomologies (see [Wel74, Theorem 3.1], [Ang13a, Theorem 6]). If the two manifolds have different dimensions a similar result can not exist, however
16
if the manifold domain of the map admits a symplectic/Kähler structure then the same injectivity results can be recovered. In Propositions 2.23 and 2.24 and in Theorem 2.25 we study the behavior of the Hard-Lefschetz condition and of the L.S. Tseng and S.T. Yau symplectic cohomologies under proper, surjective, symplectic maps.
Proposition 0.8 (Proposition 2.23, Proposition 2.24, Theorem 2.25). Let π∶ ( ˜X2n, ˜ω) Ð→ (X2n, ω) be a smooth, proper, surjective map such that π∗ω= ˜ω.
If ˜X satisfies HLC then X satisfies HLC. Moreover, if n = 2 and b1( ˜X) =
b1(X) then the converse holds.
Moreover, if X2n satisfies HLC then
π∗∶ Hk
d+dΛ(X) Ð→ H
k
d+dΛ( ˜X) is injective for any k.
Similarly we discuss what happens on almost-complex manifold. First of all, notice that if J is a non-integrable almost-complex structure on a differentiable manifold X then ∂2≠ 0; in particular the Dolbeault cohomology groups can not be defined. An almost-complex generalization of them has been introduced and studied by T. J. Li and W. Zhang in [LZ09]. On a compact almost-complex manifold (X, J) we consider
HJ(p,q)(q,p)(X)R∶=⎧⎪⎪⎨⎪⎪ ⎩[α] ∈ H p+q dR (X; R) ∣ α ∈ ⎛ ⎝ ⊕(p,q) Ap,qJ (X)⎞ ⎠∩ Ap+q(X; R)⎫⎪⎪⎬⎪⎪⎭; In particular, for (p, q) = (1, 1) and (p, q)(q, p) = (2, 0)(0, 2) we obtain the J-invariant and J-anti-invariant cohomology groups, namely
HJ(1,1)(X)R= H+ J(X)R∶= {[α] ∈ H 2 dR(X, R) ∣ Jα = α} , HJ(2,0),(0,2)(X)R= H− J(X)R∶= {[α] ∈ H 2 dR(X, R) ∣ Jα = −α} .
In [Zha17, Proposition 4.3] W. Zhang shows that if π ∶ ( ˜X, ˜J) Ð→ (X, J) is a proper, surjective, pseudo-holomorphic map between two almost-complex manifolds of the same dimension, then dim H+
J(X) ≤ dim H + ˜ J( ˜X) and dim H− J(X) ≤ dim H − ˜
J( ˜X). We generalize this result to every p and q.
Theorem 0.9 (Theorem 2.7). Let π ∶ ( ˜X, ˜J) Ð→ (X, J) be a proper, surjective, pseudo-holomorphic map between two almost-complex manifolds of the same dimension. Then,
π∗ ∶ H(p,q),(q,p)
J (X)RÐ→ H
(p,q),(q,p) ˜
is injective for any p, q.
In particular, if ˜X and X are compact we have the inequalities dim HJ(p,q),(q,p)(X) ≤ dim H(p,q),(q,p)˜
J ( ˜X)
for any p, q.
If the two considered manifolds have different dimensions we show that an additional hypothesis is needed in order to get the same conclusion; in particular in Example 2.3.1 we discuss an explicit example of a holomorphically parallelizable complex nilmanifold ˜X of real dimension 10 which projects over the four-torus X and dim H−
J(X) > dim H −
˜
J( ˜X). If the
manifold ˜X has a ˜J-compatible symplectic structure we prove the following. Theorem 0.10 (Theorem 2.11). Let π∶ ( ˜X2m, ˜J) Ð→ (X2n, J) be a proper,
surjective, pseudo-holomorphic map between two almost-complex manifolds and suppose that (˜ω, ˜J) is an almost-Kähler structure on ˜X. Then,
π∗ ∶ H(p,q),(q,p)
J (X)RÐ→ H
(p,q),(q,p) ˜
J ( ˜X)R
is injective for any p, q.
In particular, if ˜X and X are compact we have the inequalities dim HJ(p,q),(q,p)(X) ≤ dim H(p,q),(q,p)˜
J ( ˜X)
for any p, q.
The thesis is organized as follows.
In Chapter 1, which does not contain original results, we recall some well-known facts about complex, symplectic and Kähler geometry, focusing on the Dolbeault cohomology and on the obstructions to the existence of Kähler metrics. Nilmanifolds and solvmanifolds are also discussed followed by the classical theory of small deformations of complex structure.
In Chapter 2 we study almost-complex and symplectic cohomologies focusing on the comparisons of these cohomology groups on manifolds related by proper, surjective, structure-preserving maps. We discuss also some relations with the Hard Lefschetz condition on symplectic manifolds. These new results are contained in [TT16]. Finally, some considerations on symplectic orbifolds are done, which have not been submitted yet.
18
In Chapter 3 we recall the definition and some classical results about the Bott-Chern and Aeppli cohomologies on complex and symplectic manifolds. We study quantitative and qualitative properties on them in relation with the ∂∂-lemma. An algebraic version of some inequalities will be presented with application to symplectic geometry, in particular a new quantitative characterization of the Hard Lefschetz condition in dimension 4 is provided. The original results of this Chapter are contained in [AT17], [TT16], [Tar17]. Finally, some considerations on complex orbifolds are done, which have not been submitted yet.
In Chapter 4 a geometric notion of formality for the Bott-Chern cohomology of complex manifolds is studied focusing in particular on the behavior under small deformations of the complex structure. A general instability result is proven, showing however a partial stability for complex surfaces. These new results are contained in [TT17]. A new notion of higher order Aeppli-Bott-Chern Massey products is presented and discussed. These final results have not been submitted yet.
In Chapter 5 we deal with locally conformally symplectic manifolds. We introduce new twisted cohomologies and discuss their properties like Hodge Theory, dualities, Hard-Lefschetz condition. We study Hattori’s Theorem and the Mostow condition for these twisted cohomologies. We prove that Oeljeklaus-Toma manifolds with precisely one complex place, and under an additional arithmetic condition, satisfy the Mostow property. This holds in particular for the Inoue surface of type S0. The new results in this Chapter
are contained in [AOT17].
Preliminaries on (non-)Kähler
geometry
In this Chapter we recall some well-known facts about complex (§1.1) and symplectic geometry (§1.2). Every projective manifold (i.e., compact complex submanifolds of the complex projective space CPn) is Kähler but
the converse is not true in general; indeed Kodaira in [Kod54] proves that projective manifolds (or equivalently, by Chow’s theorem [Cho49], projective algebraic manifolds) are exactly compact complex manifolds carrying a Kähler metric with integral de Rham cohomology class. In this sense Kähler manifolds can be seen as a generalization of projective manifolds; more precisely, Kähler manifolds carry a complex-analytic structure, a metric structure and a symplectic structure (i.e., a non-degenerate d-closed 2-form) which are compatible to each other (§1.3). On the other hand, in general neither complex manifolds nor symplectic manifolds are Kähler (§1.5). Therefore one is led to study obstructions to the existence of a Kähler metric on complex manifolds and to weaken the geometric structures that are involved. One can consider, for example, almost-complex manifolds, generalizing the notion of complex structure requiring the non-integrability of the tensor J (see e.g., [NN57]) or one can look for "special" (non necessarily Kähler) metrics on complex manifolds, for instance balanced metrics ([Mic83]), (strongly-)Gauduchon metrics ([Gau77], [Pop13a]), SKT metrics ([Bis89]) and so on (§1.4). Furthermore one can consider symplectic manifolds that do not admit a compatible complex structure, i.e., symplectic non-Kähler manifolds.
We will see that an important global tool in studying (almost-)complex and symplectic manifolds is furnished by cohomology.
1.1 (Almost-)complex geometry 20
1.1
(Almost-)complex geometry
1.1.1
Almost-complex structures
Let X be a differentiable manifold of real dimension 2n. We recall the definition of almost-complex structure given by C. Ehresmann in [Ehr49]. Definition 1.1. An almost-complex structure on X is the datum of a (1, 1)-tensor J on X, such that J2= −Id. The pair (X, J) is called almost-complex
manifold.
Every almost-complex manifold is orientable. The C-linearly extended almost complex structure J induces a decomposition of the complex tangent bundle, namely TCX∶= TX ⊗
RC, in the i− and (−i)-eigenspaces as
TCX= T1,0X⊕ T0,1X ,
where
- T1,0X∶= {v ∈ TCX ∣ Jv = iv},
- T0,1X∶= {v ∈ TCX ∣ Jv = −iv}.
Similarly, denoting again with J the dual of the almost-complex structure, the complex co-tangent bundle TC∗X ∶= T∗X⊗
RC decomposes as
TC∗X = T
1,0X⊕ T0,1X
where
- T1,0X ∶= {α ∈ TC∗X ∣ α(Jv) = iα(v), for any v ∈ TCX},
- T0,1X ∶= {α ∈ TC∗X ∣ α(Jv) = −iα(v), for any v ∈ TCX}.
We denote the bundle of complex k-forms as Λk(X, C) and the space of
k-forms on X, Ak(X, C) ∶= Γ (X, Λk(X, C)), is given by smooth global
sections of Λk(X, C).
The extension of the almost-complex structure J to Λ●(X, C) induces a
bi-graduation on the space of forms, namely, setting
Λp,qJ (X) ∶= ΛpT
1,0X∧ ΛqT0,1X and Ap,qJ (X) ∶= Γ (X, Λp,q(X)), the following
holds: Λ●(X, C) = ⊕ p+q=●Λp,qJ (X) , A ●(X, C) = ⊕ p+q=●Ap,qJ (X). An element in Ap,q
J (X) is called (p, q)-form on (X, J); when the
Denoting with d ∶ A●(X, C) → A●+1(X, C) the C-linear extension of the
exterior derivative, d acts on each bi-degree as
d∶ Ap,q(X) → Ap+2,q−1(X) ⊕ Ap+1,q(X) ⊕ Ap,q+1(X) ⊕ Ap−1,q+2(X). This follows easily by induction writing locally a (p, q)-form as a sum of decomposable differential forms and applying the Leibniz rule. In particular, d splits into 4 operators
d= A + ∂ + ∂ + A where
A∶ Ap,q(X) → Ap+2,q−1(X), ∂ ∶ Ap,q(X) → Ap+1,q(X),
∂ ∶ Ap,q(X) → Ap,q+1(X), A ∶ Ap,q(X) → Ap−1,q+2(X). Since d2 = 0, by bi-degree reasons, we get the following relations
- A2 = 0, - A∂+ ∂A = 0, - A∂+ ∂2+ ∂A = 0, - AA+ ∂∂ + ∂∂ + AA = 0, - ∂A+ ∂2+ A∂ = 0, - A∂+ ∂A = 0, - A2 = 0.
1.1.2
Complex structures and Dolbeault cohomology
Every complex manifold is endowed with a natural almost- complex structure; indeed, let X be a complex manifold of complex dimension n and denote with {z1,⋯, zn} local holomorphic coordinates on an open chart U ⊆ X. Writing
zj ∶= xj+iyj, one can define an almost-complex structure on U in the following
way: ⎧⎪⎪⎪⎪ ⎪⎨ ⎪⎪⎪⎪⎪ ⎩ J( ∂ ∂xj) ∶= ∂ ∂yj, J( ∂ ∂yj) ∶= − ∂ ∂xj.
1.1 (Almost-)complex geometry 22
Since the transition functions on a complex manifold are holomorphic, the definition does not depend on the local coordinates and gives a global complex structure J on X. The converse is not true in general; an almost-complex structure induced by the almost-complex structure of a almost-complex manifold is called integrable. In real dimension 2, every almost-complex structure is integrable but this fails in higher dimension. In [NN57] A. Newlander and L. Nirenberg give a characterization of the integrable almost-complex structures in terms of the (1, 2)-tensor
NJ(⋅, ⋅⋅) ∶= [J⋅, J ⋅ ⋅] − [⋅, ⋅⋅] − J[J⋅, ⋅⋅] − J[⋅, J ⋅ ⋅].
This tensor is called Nijenhuis tensor and its vanishing characterizes the integrability condition by the following
Theorem 1.2 ([NN57, Theorem 1.1]). Let X be a differentiable manifold endowed with an almost-complex structure J. Then, J is integrable if and only if NJ ≡ 0.
There are many other equivalent conditions for the integrability of an almost-complex structure J on a differentiable manifold X. Here we list some of them: a) If Z, W ∈ Γ (X, T1,0(X)) then [Z, W] ∈ Γ (X, T1,0(X)); b) If Z, W ∈ Γ (X, T0,1(X)) then [Z, W] ∈ Γ (X, T0,1(X)); c) d(A1,0(X)) ⊆ A2,0(X) ⊕ A1,1(X), d(A0,1(X)) ⊆ A1,1(X) ⊕ A0,2(X); d) d(Ap,q(X)) ⊆ Ap+1,q(X) ⊕ Ap,q+1(X); e) NJ ≡ 0; f) (X, J) is a complex manifold.
In particular, by condition d), the two components A and A of the exterior derivative d vanish and we get d= ∂ + ∂, or equivalently
⎧⎪⎪⎪⎪ ⎨⎪⎪⎪ ⎪⎩ ∂2 = 0, ∂2 = 0, ∂ ∂+ ∂ ∂ = 0.
This means that the triple (A●,●(X) , ∂ , ∂) is a double complex of C∞(X,
C)-modules. Therefore, on a complex manifold we can consider the cohomology associated to the differential complex
The cohomology groups
H●,●
∂ (X) ∶=
Ker ∂ Im ∂
are called Dolbeault cohomology groups and they are bi-graded C-vector spaces. Moreover, the wedge product on forms induces a product on H●,●
∂ (X) giving it a structure of bi-graded algebra. Roughly speaking, if we
draw a double complex as follows, for the Dolbeault cohomology we are looking at vertical arrows, since the operator ∂ changes the second degree of a (p, q)-form, and for its conjugate H●,●
∂ (X), defined replacing ∂ with ∂,
we are looking at horizontal arrows, since the operator ∂ changes the first degree of a (p, q)-form. For a more detailed explanation of the interpretation of a double complex as a sum of indecomposable objects as zig-zags, dots and squares we refer to [Ang15] and [DGMS75].
0 1 2 0 1 2 γ δ α β
For instance, in the above picture we mean that ∂α = β, ∂α = ∂β = 0, ∂γ = δ and ∂γ = ∂δ = 0. So α and β are representatives of two non-trivial classes in H●,●
∂ (X) and β represents the trivial class in H 0,2
∂ (X). Similarly
goes for δ and γ. Notice that we can not have two consecutive vertical (respectively horizontal) arrows because ∂2 = 0 (respectively ∂2 = 0).
Two different interpretation of the Dolbeault cohomology can be given. Sheaf-theoretic interpretation
Similarly to the complex de Rham cohomology that can be seen as the cohomology of X with values in the constant sheaf CX, i.e.,
H●
dR(X, C) ≃ H
●(X, C X) ,
also the Dolbeault cohomology has a sheaf-theoretic interpretation as cohomology with values in a sheaf. More precisely, denoting with Ap,q
1.1 (Almost-)complex geometry 24
sheaf of germs of (p, q)-forms on X and with Ωp
X the sheaf of germs of
holomorphic p-forms on X, namely the kernel sheaf of the map ∂ ∶ Ap,0X → Ap,1X , we have that
0Ð→ ΩpX Ð→ Ai p,0X Ð→ A∂ p,1X Ð→ . . .∂ where i denotes the inclusion, is a fine resolution of the sheaf Ωp
X. Therefore,
the following holds
Theorem 1.3 (Dolbeault Theorem, [Dol53]). Let X be a complex manifold. Then, for every p, q∈ N,
Hp,q
∂ (X) ≃ H
q(X, Ωp X).
Similar results can be stated for the conjugate Dolbeault cohomology that is defined as
H∂●,●(X) ∶= Ker ∂ Im ∂ . Analytic interpretation: Hodge theory
Let (X, J) be a compact complex manifold of complex dimension n and let g be a Hermitian metric on X. We denote by ω the associated fundamental (1, 1)-form, that is ω(⋅, ⋅⋅) ∶= g(J⋅, ⋅⋅), and the volume form is then Vol ∶= ωn
n!.
The Hermitian metric g induces a natural Hermitian inner product on the space of bi-graded forms, more precisely, given α , β ∈ Ap,q(X)
⟨α, β⟩ ∶= ∫X⟨αz, βz⟩gVol,
and the Hodge-∗-operator associated to g is the C-anti-linear map ∗g ∶ Ap,q(X) → An−p,n−q(X)
defined as follows: given β ∈ Ap,q(X), for every α ∈ Ap,q(X)
α∧ ∗gβ∶= ⟨α, β⟩ Vol.
When the metric g is understood, we will simply write ∗ instead of ∗g. Note
that ∗2
∣Ap,q(X)= (−1)p+qId. The adjoint operators of ∂ and ∂, with respect to ⟨⋅, ⋅⋅⟩, are
The ∂-Laplacian and ∂-Laplacian are respectively defined as ∆∂ ∶= ∂∂∗+ ∂∗∂ ∶ A●,●(X) → A●,●(X) ,
∆∂ ∶= ∂∂∗+ ∂∗∂ ∶ A●,●(X) → A●,●(X) .
They are elliptic self-adjoint differential operators of the 2nd-order; as a
matter of notation we will denote H●,●
∂ (X) ∶= Ker ∆∂∣A●,●(X) and H
●,●
∂ (X) ∶= Ker ∆∂∣A●,●(X) and the elements in H●,●
∂ (X), resp. H ●,●
∂ (X), will be called ∆∂-harmonic
forms, resp. ∆∂-harmonic forms.
The following holds (see [Kod05, Theorem 3.16] and [Hod89]).
Theorem 1.4 (Hodge Theorem). Let (X, g) be a compact Hermitian manifold. Then, for any p, q∈ N there are orthogonal decompositions
- Ap,q(X) = Hp,q ∂ (X) ⊕ ∂Ap−1,q(X) ⊕ ∂ ∗Ap+1,q(X), - Ap,q(X) = Hp,q ∂ (X) ⊕ ∂A p,q−1(X) ⊕ ∂∗Ap,q+1(X)
and hence isomorphisms
H∂p,q(X) ≃ Hp,q∂ (X) and Hp,q ∂ (X) ≃ H p,q ∂ (X). In particular, dimCH p,q ∂ (X) < +∞ and dimCH p,q ∂ (X) < +∞.
Notice that, for any (p, q)-form α
∆∂α= 0 ⇐⇒ { ∂ α = 0 ∂∗α = 0 , and similarly, ∆∂α= 0 ⇐⇒ { ∂ α = 0 ∂∗α = 0 .
Let (X, g) be a compact Hermitian manifold of complex dimension n, then Hp,q
∂ (X) × H n−p,n−q
∂ (X) → C
(α, β) → ∫Xα∧ β
is a non-degenerate pairing and hence there is an isomorphism Hp,q ∂ (X) ≃ (H n−p,n−q ∂ (X)) ∗ .
Therefore, by Hodge theory, this means that the Hermitian duality preserves the Dolbeault cohomology.
1.1 (Almost-)complex geometry 26
Theorem 1.5 (Kodaira-Serre duality, [Ser55, Théorème 4]). Let X be a compact complex manifold of complex dimension n endowed with a Hermitian metric g. Then Hp,q ∂ (X) ≃ (H n−p,n−q ∂ (X)) ∗ . In particular, for every p, q∈ N, dimCH
p,q
∂ (X) = dimCH n−p,n−q ∂ (X).
Fröhlicher spectral sequence
Since a ∂-closed form is not d-closed in general, there is no natural map between the Dolbeault and de Rham cohomology of a complex manifold. The bridge between the topological invariant represented by the de Rham cohomology and the holomorphic invariant represented by the Dolbeault cohomology will be given by the Bott-Chern and Aeppli cohomology groups (see Chapter 3). However, another way to reach the de Rham cohomology starting from the Dolbeault cohomology is provided by the Fröhlicher spectral sequence. We will recall that on compact Kähler manifolds, or more in general, on compact complex manifolds satisfying the ∂∂-lemma, the de Rham cohomology can be decomposed as a direct sum of the Dolbeault cohomology groups, that is the bi-graduation on forms passes in cohomology. But, in general, for compact complex manifolds this is not true and the Fröhlicher spectral sequence will give a better description of this. Let X be a complex manifold and consider the double complex (A●,●(X), ∂, ∂); this induces two natural filtrations of A●(X, C), for any
p, q, k ∈ N ′Fp(Ak(X, C)) ∶= ⊕ p′+q=k, p′≥p Ap′,q(X) and ′′Fq(Ak(X, C)) ∶= ⊕ p+q′=k, q′≥q Ap,q′(X). The spectral sequences associated to these filtrations
{(′ Er●,●, d′ r)}r∈N and {( ′′ Er●,●, d′′ r)}r∈N
are called Fröhlicher spectral sequences (see [Fro55], [GH94, Section 3.5]). At the first step these spectral sequences recover, respectively, the Dolbeault and conjugate-Dolbeault cohomologies and they converge to the de Rham cohomology, namely ′ E1●,●≃ H●,● ∂ (X) ⇒ H ● dR(X, C) and ′′ E1●,●≃ H∂●,●(X) ⇒ H● dR(X, C).
This leads to the following Fröhlicher inequalities.
Theorem 1.6 ([Fro55, Theorem 2]). Let X be a compact complex manifold, then for every k∈ N
∑ p+q=k dimCHp,q ∂ (X) ≥ dimCH k dR(X, C).
For simplicity, we will denote with hp,q
∂ (X) ∶= dimCH p,q
∂ (X) the Hodge
numbers of X and with bk(X) ∶= dimRHdRk (X, R) the Betti numbers of X.
The previous inequalities show that there is a topological lower bound for the Hodge numbers.
In [CFGU97, Theorem 1] the authors describe explicitly the spaces {′E●,● r }r
as follows: for every p, q, r∈ N
′ Erp,q≃ X p,q r Yrp,q where, X1p,q∶= {α ∈ Ap,q(X) ∣ ∂α = 0} , Yp,q 1 ∶= ∂ (Ap,q−1(X)) , and for r≥ 2
Xrp,q∶= {αp,q∈ Ap,q(X) ∣ ∂αp,q = 0 and there exist αp+i,q−i∈ Ap−i,q−i(X)
such that ∂αp+i−1,q−i+1+ ∂αp+i,q−i= 0, 1 ≤ i ≤ r − 1} ,
Yrp,q∶= {∂βp−1,q+ ∂βp,q−1∈ Ap,q(X) ∣ there exist βp−i,q+i−1∈ Ap−i,q+i−1(X),
2≤ i ≤ r − 1, satisfying ∂βp−i,q+i−1+ ∂βp−i+1,q+i−2= 0, ∂βp−r+1,q+r−2= 0} .
Moreover, by [CFGU97, Theorem 3], for r≥ 2 the map d′ r∶′E p,q r →′Ep+r,q−r+1 is given by d′ r[αp,q] = [∂αp+r−1,q−r+1] and ′ Er+1p,q = X p,q r+1 Yr+1p,q = Ker d′ r∣′Ep,q r d′ r(′E p−r,q−r+1 r ) .
1.1.3
Currents and de Rham homology
We recall here some basic definitions about currents in order to fix some notations; we refer to [Dem12] for further details. Let us start with an oriented differentiable manifold X of real dimension n. In order to define a topology on the vector space ΛkX consider U⊆ X a coordinate open set and ϕ ∈ ΛkX,
written as ϕ= ∑ϕIdxI on U, I being a multi-index. To every compact subset
L⊆ U and every integer s ∈ N, one can associate a seminorm psL(ϕ) = sup x∈L max ∣I∣=k,∣α∣≤s∣ D αϕ I(x) ∣, where α = (α1, . . . , αn) ∈ Nn and Dα = ∂ ∣α∣ ∂xα11 ...∂xαnn is a derivation of order ∣ α ∣= α1+ ⋯ + αn.
1.1 (Almost-)complex geometry 28
It is possible to equip ΛkX with the topology defined by all seminorms ps L
when s, L, U vary; ΛkX endowed with this topology becomes a Fréchet space.
If L ⊆ X is a compact subset, we will denote with Λk
0L the subspace of
elements in ΛkX with support contained in L, together with the induced
topology. Finally we put Λk
0X ∶= ⋃L⊆XcompactΛk0L, i.e., the set of all differential
k-forms with compact support in X.
The spaces of currents are defined as the topological duals of the above spaces. Namely, the space of currents of dimension k, or degree n− k, on X is the space of continuous linear functionals Λk
0XÐ→ R denoted with
Dk(X) ∶=∶ Dn−k(X) ∶= (Λk0X) ∗
.
We will set ⟨−, =⟩ ∶ Dk(X) × Λk0X Ð→ R for the duality pairing. The support
of T , denoted by Supp T , is the smallest closed subset A ⊆ X such that TΛk
0(X∖A) = 0. The terminology introduced is justified by the following examples.
Let Z be a closed oriented submanifold of X of class C1 and dimension k; Z
may have a boundary ∂Z. The current of integration over Z, denoted by [Z], is the current of dimension k defined by
⟨[Z] , ϕ⟩ ∶= ∫Zϕ, ϕ∈ Λk0X. It is clear that Supp[Z] = Z.
Given a differential form α∈ ΛkX with L1
loc coefficients, we can associate
to α the current Tα of degree k as follows
⟨Tα, ϕ⟩ ∶= ∫ X
α∧ ϕ, ϕ∈ Λn−k0 X.
Many of the operations available for differential forms can be extended to currents by duality arguments. The exterior derivative
d∶ Dk(X) Ð→ Dk+1(X) is defined by
⟨dT, ϕ⟩ ∶= (−1)k+1⟨T, dϕ⟩ , ϕ∈ Λn−k−1 0 X.
A current T is called d-closed if dT = 0, and d-exact if there exists a current S such that T = dS. Then we have the differential complex (D●(X), d)
Hk(D●(X), d) =∶ H
k(X, R).
Let [Z] ∈ Dk(X) be the current of integration on Z defined above. The
Stokes’ formula implies that, for all ϕ∈ Λk−1 0 X: ⟨d [Z] , ϕ⟩ = (−1)n−k+1⟨[Z] , dϕ⟩ = = (−1)n−k+1 ∫Zdϕ= (−1)n−k+1∫ ∂Z ϕ= (−1)n−k+1⟨[∂Z] , ϕ⟩ , hence d[Z] = (−1)n−k+1[∂Z].
Let Tα ∈ Dk(X) be the associated current to α ∈ ΛkX. Using Stokes’
formula, we get, for all ϕ∈ Λn−k−1 0 X: ⟨dTα, ϕ⟩ = (−1)k+1⟨Tα, dϕ⟩ = (−1)k+1∫ X α∧ dϕ = = − ∫Xd(α ∧ ϕ) + ∫ X dα∧ ϕ = ⟨Tdα, ϕ⟩ ,
hence dTα= Tdα, in particular Tα is closed if and only if α is closed.
For T ∈ Dk(X) and α ∈ ΛhX, the wedge product T∧α ∈ Dk+h(X) is defined
by
⟨T ∧ α, ϕ⟩ ∶= ⟨T, α ∧ ϕ⟩ , ϕ∈ Λn−k−h 0 X.
By definition we get
d(T ∧ α) = dT ∧ α + (−1)kT ∧ dα.
Let (x1, . . . , xn) be a coordinate system on an open set U ⊆ X. Every
current T ∈ Dk(X) can be written in a unique way
T = ∑
∣I∣=k
TIdxI, on U,
where TI are distributions on U, considered as currents of degree 0.
If X admits an almost-complex structure J, then J can be induced on currents by duality. Therefore J ∈ End(D●(X)) induces a bi-graduation on
currents, namely there is a decomposition Dk(X, C) = ⊕
p+q=k
DJ p,q(X).
1.2 Symplectic geometry 30
The space Dp,q(X) = Dp,qJ (X) ∶= D n−p,n−q
J (X) is called space of currents of
bidimension (p, q) and bidegree (n − p, n − q) on X and it is the topological dual of Λp,q(X) ∩ (Λ●
0(X) ⊗RC), for every p, q ∈ N.
The duality between the positive and strongly positive cones of forms can be used to define corresponding positive notions for currents.
A current T ∈ Dp,p(X) is said to be positive (resp. strongly positive) if ⟨T, ϕ⟩ ≥
0 for all test forms ϕ∈ Ap,p(X) that are strongly positive (resp. positive) at
each point.
1.2
Symplectic geometry
The initial motivation for studying symplectic geometry arises from Hamiltonian mechanics; in this Section we will recall some well-known facts about symplectic manifolds. Symplectic cohomologies will be discussed in Chapter 2.
Definition 1.7. Let X be a differentiable manifold of dimension 2n. A symplectic structure on X is a 2-form ω ∈ A2(X) that is d-closed and
non-degenerate, i.e., ωn≠ 0. The pair (X, ω) is called symplectic manifold.
Notice that ω induces an orientation on X. Locally, symplectic manifolds can not be distinguished, indeed they are all "isomorphic" by the following Theorem 1.8 (Darboux Theorem, [Dar82]). Let (X, ω) be a symplectic manifold of dimension 2n. Then, for every x ∈ X, there exists a coordinate chart (U, {x1, . . . , x2n}) centered in x such that
ω=loc n
∑
i=1
dx2i−1∧ dx2i.
This means that locally all symplectic manifolds of the same dimension are symplectomorphic to one another and, for this reason, there are no local invariants in symplectic geometry.
We recall that an almost complex structure J on a symplectic manifold(X, ω) is said to be
● tamed by ω if ω is positive on the complex lines, i.e., ω(⋅, J ⋅ ⋅) > 0; ● calibrated by ω or, equivalently, ω is said to be compatible with J if in
addition ω(J⋅, J ⋅ ⋅) = ω(⋅, ⋅⋅).
In this last case, setting g(⋅, ⋅⋅) ∶= ω(⋅, J ⋅ ⋅) for the associated J-Hermitian Riemannian metric, the triple (ω, J, g) is called almost-Kähler structure on
X or, equivalently, J is said to be almost-Kähler on X. If, moreover, J is integrable the triple (ω, J, g) is called Kähler structure on X. We will discuss this situation in §1.3.
It is a consequence of linear algebra that on every symplectic manifold (X, ω) there exists an almost complex structure J calibrated by ω. Nevertheless there exist examples of almost complex manifolds without any compatible symplectic form.
We recall that a symplectic manifold (X, ω) of dimension 2n is said to satisfy the Hard Lefschetz Condition (HLC for short) (cf. Definition 2.15) if, for every k ∈ N, the maps
Lk= [ω]k⌣∶ HdRn−k(X, R) → HdRn+k(X, R) are isomorphisms.
1.3
Kähler geometry
Kähler geometry represents the combination of the previous sections, indeed on a Kähler manifold there exist a complex structure and a symplectic structure which are compatible to each other. We will recall here some equivalent definitions and some obstructions focusing mainly on the cohomological properties.
Definition 1.9. Let X be a complex manifold and denote with J the complex structure. A Hermitian metric g on X is called Kähler if the associated fundamental (1, 1)-form ω ∶= g(J⋅, ⋅⋅) is d-closed.
In particular ω gives a symplectic structure on X. Therefore, Kähler manifolds carry a complex-analytic structure, a metric structure and a symplectic structure which are compatible to each other. On the other hand, in general neither complex manifolds nor symplectic manifolds are Kähler, for example the Iwasawa manifold (see e.g., [FG85]) admits a complex structure and a symplectic structure but it does not admit any Kähler metric; as regards symplectic manifolds McDuff in [McD84] shows the first example of a simply-connected compact symplectic non-Kähler manifold (in [FMS03] the authors show also explicit examples of compact symplectic manifolds which are cohomologically Kähler but that do not admit any Kähler metric).
1.3 Kähler geometry 32
Remark 1.10. Every Riemann surface is Kähler since any 2-form is d-closed. In complex dimension 2, Kählerness can be topologically characterized in terms of the first Betti number, that is a compact complex surface X is Kähler if and only if the first Betti number b1(X) is even ([Siu83], [Lam99], [Buc99]).
A similar characterization in higher dimension is not known, nevertheless we will see that there are topological obstructions to the existence of Kähler metrics on compact complex manifolds.
Easy examples of Kähler manifolds in higher dimension are provided by Cn
with the natural metric
ω0∶= i 2 n ∑ j=1 dzj∧ d¯zj
and the complex projective space CPn with the Fubini-Study metric:
ωF S∶= i 2π∂∂ log( n ∑ j=0 ∣zj∣2)
where [z0∶ ⋯ ∶ zn] denote the homogeneous coordinates.
Every complex submanifold of a Kähler manifold is still Kähler and this gives a large class of examples. In particular, by Chow’s Theorem [Cho49] complex algebraic manifolds are projective manifolds (namely complex manifolds embedded in a complex projective space) and so they are Kähler. The converse is not true, namely there exist Kähler manifolds which are not algebraic. More precisely, the Kodaira embedding Theorem [Kod54, Theorem 4] gives a characterization for Kähler manifolds to be algebraic, more precisely a compact complex manifold endowed with a Kähler metric ω such that [ω] ∈ Im (H2(X, Z) → H2
dR(X, R)) ∩ HdR2 (X, R) can be
embedded in a complex projective space CPN, for some N ∈ N. This is
equivalent to say that X admits a positive line bundle. In [CE53] the first known example of a compact, simply connected complex manifold which is not algebraic (even non Kähler) has been constructed considering the products of odd spheres S2p+1× S2q+1.
Many different characterizations for the Kähler condition can be provided; here we list some of them referring for instance to [Bal06], [GH94], [Huy05], [Mor07], [Dem12] for more details. Let (X, J, g, ω) be a Hermitian manifold (that is ω is an almost-symplectic structure) of complex dimension n. We denote with ∇LC the Levi-Civita connection. The
following facts are equivalent: - g is Kähler;
- J is parallel with respect to the Levi-Civita connection: ∇LCJ = 0;
- if {z1, . . . , zn} is a holomorphic local coordinate system and
g = ∑gαβdzαdzβ, locally we have for every α, β, γ∈ {1, ⋯, n}
∂gαβ ∂zγ =
∂gγβ ∂zα ;
- if {z1, . . . , zn} is a holomorphic local coordinate system and
g = ∑gαβdzαdzβ, locally we have for every α, β, γ∈ {1, ⋯, n}
∂gαβ ∂zγ =
∂gαγ
∂zβ ;
- the Chern connection (that is the only connection such that ∇Cg = 0,
∇CJ = 0, π
Λ0,1(X)∇C
∣C∞(X,C) = ∂∣C∞(X,C)) is torsion free;
- the Chern connection coincides with the Levi-Civita connection: ∇C = ∇LC;
- the metric has a local potential, namely for all p ∈ X there exists an open neighborhood U of p and a smooth function f ∈ C∞(U; R) such
that
ω∣U = i∂∂f;
- the metric g osculates to order 2 the standard Hermitian metric of Cn,
namely for all p∈ X there exists a holomorphic local coordinate system {z1, . . . , zn} with center in p, such that
g = ∑
i,j
(δij + o(∣z∣)) dzidzj,
when evaluated at p. In particular, on a Kähler manifold, the identities involving just first derivatives hold if and only if they hold on Cn endowed
with the flat metric. This fundamental remark allows to prove the Kähler identities.
1.3 Kähler geometry 34
1.3.1
Kähler identities and Hodge theory
On a Kähler manifold (X, J, ω, g) the complex, symplectic and metric structures are all compatible to each other and this lead to important relations involving the operators ∂, ∂, L, Λ, ∗, where
L∶ A●(X) → A●+2(X),
α↦ ω ∧ α and
Λ∶ A●(X) → A●−2(X),
α↦ −ıω−1α, ıω−1 denoting the interior product with ω−1∈ Λ2(TX) (cf. §2.4).
Theorem 1.11 (Kähler identities, [Wei58, Théorème II.1, II.2]). Let X be a compact Kähler manifold. Then, the following commutation relations hold:
- [∂ , L] = [∂ , L] = [∂∗, Λ] = [∂∗, Λ] = 0;
- [∂∗, L] = −i∂, [∂∗, L] = i∂, [Λ , ∂] = i∂∗ and [Λ , ∂] = −i∂∗.
Therefore, considering the three Laplacian operators ∆d = dd∗ + d∗d,,
∆∂= ∂∂∗+ ∂∗∂ and ∆∂= ∂∂ ∗
+ ∂∗
∂ one gets easily the following
Corollary 1.12 ([Wei58, Corollaire II.1]). Let X be a compact Kähler manifold. Then the following relations hold:
- [∆d,∗] = 0, [∆d, ∂] = [∆d, ∂] = [∆d, ∂∗] = [∆d, ∂ ∗ ] = 0, [∆d, L] = [∆d, Λ] = 0; - ∆d = 2∆∂ = 2∆∂.
In general, if X is just a compact complex manifold and ω is the fundamental form of an arbitrary Hermitian metric on X, the previous relations can be generalized involving some extra terms arising from the torsion of ω.
Theorem 1.13 ([Dem86]). Let X be a compact complex manifold endowed with a Hermitian metric ω. Let τ ∶= [Λ , ∂ω] be the torsion operator. Then the following relations hold:
- [∂∗, L] = −i(∂ + ¯τ) and [∂∗, L] = i(∂ + τ);
- [Λ , ∂] = i(∂∗
- ∆∂ = ∆∂+ [∂ , τ
∗] − [∂ , ¯τ∗];
- ∆d = ∆∂+ ∆∂− [∂ , ¯τ∗] − [∂ , τ∗].
In particular, when ω is a Kähler form, τ = 0 and the spaces of harmonic forms coincide; hence, a complex form will be harmonic if and only if its components are harmonic too. This lead to the following decomposition Theorem, stating that the de Rham cohomology can be decomposed as the direct sum of the Dolbeault cohomology groups, namely the bi-graduation at the level of forms passes at the cohomology level. Theorem 1.14 (Hodge decomposition Theorem, [Wei58, Théorème IV.3]). Let X be a compact Kähler manifold. Then, there exists a decomposition
H●
dR(X, C) ≃ ⊕ p+q=●
Hp,q
∂ (X),
and, for every p, q∈ N, there exist isomorphisms Hp,q ∂ (X) ≃ H q,p ∂ (X). In particular, b●(X) = ∑ p+q=● hp,q ∂ (X)
and, for every p, q∈ N,
hp,q
∂ (X) = h q,p ∂ (X).
Remark 1.15. As a consequence of the fact that ω is a symplectic form and of the Hodge decomposition Theorem we have that on a compact Kähler manifold X of complex dimension n, the following facts hold
- the Fröhlicher spectral sequence degenerates at the first step, i.e. E1 =
E∞;
- the odd Betti numbers are even, i.e., b2k+1(X) ≡ 0 mod 2, for every
k∈ N;
- the even Betti numbers are positive, i.e., b2k(X) > 0, for every k ∈ N;
- hp,p
∂ (X) > 0 for every p ∈ N;
1.3 Kähler geometry 36
Notice that the second and third properties represent topological obstructions to the existence of a Kähler structure on a smooth manifold.
Moreover, if we consider the Hodge diamond hn,n ∂ (X) hn,n−1 ∂ (X) h n−1,n ∂ (X) hn,n−2 ∂ (X) h n−1,n−1 ∂ (X) h n−2,n ∂ (X) ⋮ ⋮ ⋮ h2,0 ∂ (X) h 1,1 ∂ (X) h 0,2 ∂ (X) h1,0 ∂ (X) h 0,1 ∂ (X) h0,0 ∂ (X)
there are some symmetries: reflection about the horizontal axis (induced by the Hodge-∗-operator), reflection about the vertical axis (induced by conjugation) and central symmetry (due to Kodaira-Serre duality).
While the previous Theorem states a cohomological decomposition due to the complex structure, the following result gives a cohomological decomposition due to the symplectic structure.
Theorem 1.16 (Lefschetz Decomposition Theorem, [Wei58, Théorème IV.5]). Let X be a compact Kähler manifold of complex dimension n. Then there exists a decomposition
H●
dR(X, C) = ⊕ r∈N
Lr(Ker (Λ ∶ H●−2r
dR (X, C) → HdR●−2r−2(X, C))) ,
and, for every k∈ N, there exist isomorphisms
Lk∶ HdRn−k(X, C) → HdRn+k(X, C) .
In particular, the Kähler form satisfies the Hard Lefschetz Condition.
1.3.2
Sullivan formality and the dd
c-lemma
In this Subsection we recall another obstruction to the existence of Kähler metrics on a compact smooth manifold. This property, called formality, is actually a consequence of the validity of the ∂∂-lemma on compact Kähler manifolds. We refer to [Sul77] and [DGMS75] for more details; it turns out that the real homotopy type of a Kähler manifold is a formal consequence of its cohomology ring H●
dR(X, R). We start reminding the definition of the
category DGA whose objects are differential graded algebras and whose morphisms are dga-homomorphisms (see also [Huy05]).
Definition 1.17([DGMS75, Section 1]). A differential graded algebra or dga over a field K is a graded K-algebra A = ⊕i≥0Ai together with a K-linear
map d∶ A Ð→ A such that
1) the K-algebra structure of A is given by an inclusion K ↪ A0;
2) the multiplication is graded commutative, i.e., for α ∈ Ai and β ∈ Aj
one has α⋅ β = (−1)ijβ⋅ α ∈ Ai+j;
3) the Leibniz rule holds, i.e., d(α ⋅ β) = dα ⋅ β + (−1)iα⋅ dβ for α ∈ Ai;
4) the map d is a differential (d2 = 0).
Any graded K-algebra becomes a dga by taking d = 0. Any dga (A = ⊕i≥0Ai, d) gives rise to a complex of K-vector spaces
A0 d Ð→ A 1 d Ð→ A 2 d Ð→. . .
The i-th cohomology group of a dga (A = ⊕i≥0Ai, d) is defined as
Hi(A, d) ∶= Ker(d ∶ A
i Ð→ Ai+1)
Im(d ∶ Ai−1 Ð→ Ai) .
Notice that the cohomology of a dga (H●(A, d) = ⊕
i≥0Hi(A, d), d = 0) has
the structure of a dga. Clearly if X is a differentiable (resp. complex) manifold of dimension n. Then (A●(X), d) (resp. (A●(X, C), d)) is a dga
over the field R (resp. C) whose cohomology is (H●
dR(X, R), 0) (resp.
(H●
dR(X.C), 0)).
Let (A, dA) and (B, dB) be two dga’s. A dga-homomorphism between A
and B is a K-linear map f ∶ A Ð→ B such that i) f(Ai) ⊂ Bi;
ii) f(α ⋅ β) = f(α) ⋅ f(β); iii) dB○ f = f ○ dA.
It is easy to prove that any dga-homomorphism f ∶ (A, dA) Ð→ (B, dB)
induces a dga-homomorphism in cohomology H(f) ∶ (H●(A, d
A), 0) Ð→ (H ●(B, d
B), 0) .
A dga-homomorphism f ∶ (A, dA) Ð→ (B, dB) is called
1.3 Kähler geometry 38
Two dga’s (A, dA) and (B, dB) are said to be equivalent if there exists a
sequence of dga-quasi-isomorphisms of the following form:
(C1, dC1) ⋯ (Cn, dCn)
↙ ↘ ↙ ↘ ↙ ↘
(A, dA) (C2, dC2) ⋯ (B, dB).
A dga (A, dA) is called formal if (A, dA) is equivalent to a dga (B, dB = 0).
Clearly (A, dA) is formal if and only if it is equivalent to its cohomology
dga (H●(A, d
A), d = 0).
In particular, a differentiable manifold X is called formal if its de Rham algebra (A●(X), d) is a formal dga.
Examples of formal manifolds are provided by Kähler manifolds, but actually something weaker is enough. Denoting with dc∶= i (∂ − ∂) we recall
the following
Theorem 1.18 (ddc-lemma for compact Kähler manifolds, [DGMS75,
Lemma 5.11]). Let X be a compact Kähler manifold. Then, X satisfies the ddc-lemma, namely every d-closed, dc-closed, d-exact form is also ddc-exact.
We will discuss later this property and we will see that it can be equivalently formulated in terms of the differential operators ∂ and ∂. One has the following
Theorem 1.19. Let X be a compact complex manifold which satisfies the ddc-lemma, then X is formal. In particular, compact Kähler manifolds are
formal.
This follows proving that the ddc-lemma implies that the maps in the
diagram (Ker dc, d ∣Ker dc) i vv p (( (A●(X), d) (ker dc Im dc, 0)
are dga-quasi-isomorphisms. Therefore, the de Rham complex (A●(X), d) is
equivalent to a dga with trivial differential. Hence, by definition, X is formal.
There are examples of formal non-Kähler manifolds (see e.g., [DBT01]). Further cohomological obstructions to Kählerianity (even to formality) are furnished by Massey products for which we recall the definition.
