Bounds on genus and configurations of embedded surfaces in 4-manifolds

TL;DR

This paper establishes genus bounds for embedded surfaces with zero self-intersection in 4-manifolds, using Seiberg-Witten equations to derive adjunction inequalities in specific complex projective space configurations.

Contribution

It introduces new genus bounds for surfaces in 4-manifolds with zero self-intersection, expanding the understanding of surface configurations via Seiberg-Witten theory.

Findings

Adjunction inequalities for surfaces in $m\mathbb{CP}^2\# n(-\mathbb{CP}^2)$

Genus bounds for at least one surface under certain conditions

Application of Seiberg-Witten equations to derive inequalities

Abstract

For several embedded surfaces with zero self-intersection number in 4-manifolds, we show that an adjunction-type genus bound holds for at least one of the surfaces under certain conditions. For example, we derive certain adjunction inequalities for surfaces embedded in $m\mathbb{CP}^2\# n(-\mathbb{CP}^2)$ ($m, n \geq 2$). The proofs of these results are given by studying a family of Seiberg-Witten equations.