  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.
