$p$-symplectic and $p$-pluriclosed structures on solvmanifolds

TL;DR

This paper investigates the existence of $p$-symplectic and $p$-pluriclosed structures on compact complex manifolds, providing obstructions and examples of manifolds with these structures and special Hermitian metrics.

Contribution

It introduces new obstructions to the existence of $p$-symplectic and $p$-pluriclosed structures on compact complex manifolds and constructs examples of manifolds with these structures.

Findings

Identified obstructions to $p$-symplectic and $p$-pluriclosed structures.

Constructed families of manifolds with both $(n-1)$-symplectic structures and special Hermitian metrics.

Demonstrated the coexistence of these structures on certain compact complex manifolds.

Abstract

Let $(M,J)$ be a $n$-dimensional complex manifold: a $p$-K\"ahler structure (resp. $p$-symplectic structure) on $M$ is a real, closed $(p,p)$-transverse form $\Omega$ (resp. real, closed $2p$-form whose $(p,p)$-component is transverse). We give obstructions to the existence of such structures on compact complex manifolds. We provide several families of compact complex manifolds which admit both $(n-1)$-symplectic structures and special Hermitian metrics.