Complex symplectic structures and the $\partial \bar{\partial}$-lemma

TL;DR

This paper investigates complex symplectic manifolds, focusing on the smoothness and irreducibility of the Beauville-Bogomolov-Fujiki quadric in relation to the $ar{ ext{d}}$-lemma and Hodge numbers.

Contribution

It establishes conditions under which the Beauville-Bogomolov-Fujiki quadric is smooth or irreducible based on the $ar{ ext{d}}$-lemma and Hodge numbers for complex symplectic manifolds.

Findings

Quadric $Q_\sigma$ is smooth iff $h^{2,0}(X)=1$.

Quadric $Q_\sigma$ is irreducible iff $h^{1,1}(X)>0$.

Abstract

In this paper we study complex symplectic manifolds, i.e., compact complex manifolds $X$ which admit a holomorphic $(2, 0)$-form $\sigma$ which is $d$-closed and non-degenerate, and in particular the Beauville-Bogomolov-Fujiki quadric $Q_\sigma$ associated to them. We will show that if X satisfies the $\partial \bar{\partial}$-lemma, then $Q_\sigma$ is smooth if and only if $h^{2,0}(X) = 1$ and is irreducible if and only if $h^{1,1}(X) > 0$.