Max LX - Miniworkshop on Non-Kählerian Geometry

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

Adriano Tomassini

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 $$ \Omega²(M)=\Omega_J⁺(M) \oplus \Omega_J-(M), $$ 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.