The symplectic Thom conjecture

TL;DR

This paper proves the symplectic Thom conjecture by establishing relations among Seiberg-Witten invariants for embedded surfaces with negative self-intersection, demonstrating genus-minimization of symplectic surfaces in four-manifolds.

Contribution

It introduces new relations among Seiberg-Witten invariants that, combined with existing theorems, prove the symplectic Thom conjecture and derive a general adjunction inequality.

Findings

Proof of the symplectic Thom conjecture.

New relations among Seiberg-Witten invariants.

A general adjunction inequality for embedded surfaces.

Abstract

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes' basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the symplectic Thom conjecture: a symplectic surface in a symplectic four-manifold is genus-minimizing in its homology class. Another corollary of the relations is a general adjunction inequality for embedded surfaces of negative self-intersection in four-manifolds.