Bridge trisections in $\mathbb{CP}^2$ and the Thom conjecture (with Corrigendum)

Abstract

In this paper, we develop new techniques for understanding surfaces in $\mathbb{CP}^2$ via bridge trisections. Trisections are a novel approach to smooth 4-manifold topology, introduced by Gay and Kirby, that provide an avenue to apply 3-dimensional tools to 4-dimensional problems. Meier and Zupan subsequently developed the theory of bridge trisections for smoothly embedded surfaces in 4-manifolds. The main application of these techniques is a new proof of the Thom conjecture, which posits that algebraic curves in $\mathbb{CP}^2$ have minimal genus among all smoothly embedded, oriented surfaces in their homology class. This new proof is notable as it completely avoids any gauge theory or pseudoholomorphic curve techniques. Corrigendum: This paper contains a fatal error in the proof of Theorem 1.1, which is the headline result of the paper. The error is localized to Section 6 and is described in a Corrigendum at the end of this updated version. The remaining results in Sections 1 through 5 remain valid.