2009/03/25 by Bennequin, Daniel, Le, Thanh-Tam
#53D12 (Primary) 53C15 #53D35 #57R17 #57R57 #57R95 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.0903.4490
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.