# Seiberg-Witten equation on a manifold with rank $2$-foliation

## TL;DR

This paper proves that on certain 4-manifolds with a rank-2 foliation and positive leafwise scalar curvature, the Seiberg-Witten invariants vanish for all ext{spin}^c structures when the Betti number $b^+>1$.

## Contribution

It establishes a vanishing theorem for Seiberg-Witten invariants on 4-manifolds with specific foliations and curvature conditions, extending previous results in the field.

## Key findings

- Seiberg-Witten invariants vanish under the given conditions
- The result applies to manifolds with rank-2 foliations and positive leafwise scalar curvature
- The theorem holds for all ext{spin}^c structures when $b^+>1$

## Abstract

Let $M$ be a closed oriented $4$-manifold admitting a rank-$2$ oriented foliation with a metric of leafwise positive scalar curvature. If $b^+>1$, then we will show that the Seiberg-Witten invariant vanishes for all \spinc structures.