On moduli spaces of symplectic forms

TL;DR

This paper demonstrates that certain four-manifolds can have multiple distinct symplectic forms, leading to a moduli space with many disconnected components, highlighting the complexity of symplectic structures.

Contribution

It constructs examples of simply-connected four-manifolds with multiple symplectic forms distinguished by their Chern class divisibilities, revealing new topological diversity.

Findings

Existence of four-manifolds with n distinct symplectic forms

Moduli space of symplectic forms can have at least n components

Different divisibilities of Chern classes distinguish symplectic forms

Abstract

We prove that, for any n, there are simply-connected four-manifolds which admit n-tuples of symplectic forms whose first Chern classes have pairwise different divisibilities in integral cohomology. It follows that the moduli space of symplectic forms modulo diffeomorphisms on such a manifold has at least n connected components.