From symplectic deformation to isotopy

TL;DR

This paper proves that on certain 4-manifolds, any two symplectic forms that are cohomologous and deformation equivalent are isotopic, establishing uniqueness results for symplectic structures and extending previous work.

Contribution

It demonstrates the isotopy of cohomologous symplectic forms on specific 4-manifolds and proves the uniqueness of their symplectic structures, extending Biran's results.

Findings

Cohomologous, deformation equivalent symplectic forms are isotopic on certain 4-manifolds.

Blow-ups of these manifolds have unique symplectic structures.

Established uniqueness of symplectic structures for specific fibered 4-manifolds.

Abstract

Let $X$ be an oriented 4-manifold which does not have simple SW-type, for example a blow-up of a rational or ruled surface. We show that any two cohomologous and deformation equivalent symplectic forms on $X$ are isotopic. This implies that blow-ups of these manifolds are unique, thus extending work of Biran. We also establish uniqueness of structure for certain fibered 4-manifolds.