Surgery, Yamabe invariant, and Seiberg-Witten theory

TL;DR

This paper computes the Yamabe invariant of certain 4-manifolds obtained via surgery on Kähler surfaces using Seiberg-Witten theory, revealing invariance properties under specific surgeries.

Contribution

It introduces a method to compute Yamabe invariants of surgically modified 4-manifolds using Seiberg-Witten invariants and provides explicit invariance results.

Findings

Yamabe invariant remains unchanged under certain surgeries on Kähler surfaces.

Seiberg-Witten invariants facilitate the computation of Yamabe invariants for complex 4-manifolds.

Explicit formulas for Yamabe invariants after surgeries are derived.

Abstract

By using the gluing formula of the Seiberg-Witten invariant, we compute the Yamabe invariant Y(X) of 4-manifolds X obtained by performing surgeries along points, circles or tori on compact Kaehler surfaces. For instance, if M is a compact Kaehler surface of nonnegative Kodaira dimension, and N is a smooth closed oriented 4-manifold with b_2^+(N)=0 and Y(N)>= 0, then we show that Y(M # N)=Y(M).