Representants lagrangiens de l'homologie des surfaces projectives complexes

TL;DR

This paper demonstrates the existence of embedded Lagrangian surfaces representing certain homology classes in complex projective surfaces with specific geometric properties, using advanced techniques in symplectic geometry and complex analysis.

Contribution

It establishes conditions under which smooth Lagrangian surfaces exist in complex projective surfaces with particular canonical divisors, extending previous results in symplectic topology.

Findings

Existence of Lagrangian surfaces in minimal surfaces of general type with specific properties.

Identification of a convex cone in homology classes representable by Lagrangian surfaces.

Corollary linking the surface's properties to being of simple type in gauge theory.

Abstract

Using results by Donaldson and Auroux on pseudo-holomorphic curves as well as Duval's rational convexity construction, the paper investigates the existence of smooth Lagrangian surfaces representing 2-dimensional homology classes in complex projective surfaces. We prove that if the projective surface X is minimal, of general type, with uneven geometric genus, and has an effective, smooth and connected canonical divisor K, then there exists a non-empty convex open cone in the real 2-dimensional homology group H2(X) such that a multiple of every integral homology class in this cone can be represented by an embedded Lagrangian surface in X\K. A corollary asserts that such a surface X is of simple type in the sense of Kronheimer and Mrowka.