A note on positive definite, symplectic four-manifolds

TL;DR

This paper proves that certain positive definite four-manifolds with specific handle constraints cannot support symplectic structures, contributing to the understanding of the topology of symplectic four-manifolds.

Contribution

It establishes new restrictions on the topology of positive definite four-manifolds that can admit symplectic structures, focusing on handle decompositions.

Findings

Positive definite four-manifolds with no 1-handles cannot be symplectic.

Positive definite four-manifolds with no 3-handles cannot be symplectic.

The result applies when $b_2^+ \,\geq\, 2$.

Abstract

We prove that a positive definite smooth four-manifold with $b_2^+ \geq 2$ and having either no 1-handles or no 3-handles cannot admit a symplectic structure.