Cohomological Aspects on Complex and Symplectic Manifolds

Abstract

We discuss how quantitative cohomological informations could provide qualitative properties on complex and symplectic manifolds. In particular we focus on the Bott-Chern and the Aeppli cohomology groups in both cases, since they represent useful tools in studying non Kähler geometry. We give an overview on the comparisons among the dimensions of the cohomology groups that can be defined and we show how we reach the ∂∂-lemma in complex geometry and the Hard-Lefschetz condition in symplectic geometry. For more details we refer to [6] and [29].

Bigraded notions of formality and Aeppli-Bott-Chern-Massey products

“…For the first equality of part ( 2 Remark 3.11. The same kind of arguments (with more cumbersome notation) show that Proposition 3.10 remains valid for quadruple ABC-Massey products, for Tardini's [Ta17] Massey products, and also for usual Massey products and quasi-isomorphisms.…”

“…This is in contrast with the more common ad hoc definitions of (triple ABC-) Massey products, which do not need an augmentation, as for instance in [M58, Section 2], [K66] (resp. [AT15], [Ta17]). The ad hoc version always starts with a pure tensor of classes as input and outputs a subset of cohomology, which, apart from the triple product case, is generally not the coset of a linear subspace.…”

“…We thank her and Scott Wilson for many useful discussions and comments that improved the presentation of the material. We further thank Michael Albanese and Leo Zoller for stimulating conversations, and Nicoletta Tardini for pointing us to [Ta17].…”

Symplectic cohomologies and deformations

“…Indeed, in [27] the authors construct an explicit example of a compact symplectic 4-manifold with ∆ 2 s = 4. For a more detailed comparison between the complex and symplectic settings see [26]. Remark 2.3.…”

Some Morse-type inequalities for symplectic manifolds

“…Upper bounds for the d + d Λ and dd Λ groups have been given by Angella and Tardini, who give inequalities bounding certain combinations of dimensions of the H k d+d Λ and H k d+d Λ groups in terms of Betti numbers [1,25]. Tseng and Yau also defined related cohomological invariants [29] (see also Ref.…”