---
abstract: 'This survey paper discusses some uniqueness questions for symplectic forms on compact manifolds without boundary.'
author:
- |
    Dietmar Salamon[^1]\
    ETH Zürich
date: 12 December 2012
title: Uniqueness of Symplectic Structures
---

Introduction
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Let $M$ be an oriented $2n$-manifold. A symplectic form on $M$ is a closed $2$-form ${{\omega}}\in{{\Omega}}^2(M)$ whose top exterior power ${{\omega}}^n$ is a volume form. Fundamental questions in symplectic topology are which deRham cohomology classes $a\in H^2(M;{{\mathbb R}})$ are represented by symplectic forms (the existence problem) and whether any two symplectic forms on $M$ that represent the same cohomology class are related by a suitable equivalence relation (the uniqueness problem). Necessary conditions for the existence of symplectic forms are the existence of an almost complex structure and, in the closed case, the existence of a cohomology class $a\in H^2(M;{{\mathbb R}})$ with $a^n>0$. Whether or not these conditions are also sufficient, when $M$ is a closed manifold, is completely open in dimensions $2n\ge 6$. In dimension four additional necessary conditions for existence arise from Seiberg–Witten theory. In the present survey paper the focus is mainly on the uniqueness problem. Section \[sec:SYMP\] gives precise formulations of some relevant questions, Section \[sec:EX\] discusses what is known about 