# On the $J$-anti-invariant and symplectic cohomologies

##
Adriano Tomassini

created by daniele on 20 May 2019

28 may 2019
-- 09:30

Aula Tricerri, DiMaI, Firenze

**Abstract.**

We will present some recent results obtained in two joint papers with Nicoletta Tardini and Richard Hind, concerning the cohomolgy of symplectic and almost complex manifolds.

Let $J$ be an almost complex structure on a $2n$-dimensional manifold $M$. Then $J$ acts in a natural way on the bundle of $2$-forms $\Lambda^2(M)$, so that
$$
\Omega^{2}(M)=\Omega_{J}^{+}(M) \oplus \Omega_{J}^{}-(M),
$$
where $\Omega^2(M)$ is the space of smooth sections of $\Lambda^2(M)$ and $\Omega_J^-(M)$ denote the space of $J$-invariant and $J$-anti-invariant forms, respectively. Therefore, it is natural to consider the cohomology subgroups
$H^+_J(M)$, $H^-_J(M)$ of $H^2_{dR}(M;R)$ whose elements are de Rham classes represented by $J$-invariant respectively $J$-anti-invariant forms. \newline
We will describe some features of the anti-invariant cohomology. Furthermore, we discuss some properties of the Bott-Chern symplectic cohomology.