Intersection Forms and the Adjunction Formula for Four-manifolds via CR Geometry

Abstract

This is primarily an expository note showing that earlier work of Lai on CR geometry provides a clean interpretation, in terms of a Gauss map, for an adjunction formula for embedded surfaces in an almost complex four manifold. We will see that if F is a surface with genus g in an almost complex four-manifold M, then 2 - 2 g + F \cdot F - i^{*} c_{1}(M) - 2 F\cdot C = 0, where C is a two-cycle on M pulled back from the cycle of two planes with complex structure in a Grassmannian Gr (2, C^N) via a Gauss map and where i^{*} c_{1}(M) is the restriction of the first Chern class of M to F. The key new term of interest is F \cdot C, which will capture the points of F whose tangent planes inherit a complex structure from the almost complex structure of the ambient manifold M. These complex jump points then determine the genus of smooth representative of a homology class in H_{2}(M, Z). Further, via polarization, we can use this formula to determine the intersection form on M from knowing the nature of the complex jump points of M's surfaces.