Upper bounds for virtual dimensions of Seiberg-Witten moduli spaces

TL;DR

This paper establishes upper bounds on the virtual dimensions of Seiberg-Witten moduli spaces for certain four-manifolds, leading to new inequalities for embedded surfaces with negative self-intersection.

Contribution

It provides the first bounds on virtual dimensions mod prime powers and derives adjunction inequalities for specific embedded surfaces.

Findings

Bounded the virtual dimension by 2r(p-1)-2 for mod p^r basic classes.

Derived adjunction inequalities for surfaces with negative self-intersection.

Applied bounds to specific four-manifold cases.

Abstract

Given a closed four-manifold with $b_1=0$ and a prime number $p$, we prove that for any mod $p^r$ basic class, the virtual dimension of the Seiberg-Witten moduli space is bounded above by $2r(p-1)-2$ under some conditions on $r$ and $b_2^+$. As an application, we obtain adjunction inequalities for embedded surfaces with negative self-intersection number.