1999/04/02 by Mikhail Chkhenkeli, Thomas Garrity, Chkhenkeli, Mikhail +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG
paper · pdf · doi:10.48550/arxiv.math/9904007
7 pages. The title has been changed, some references have been added and a formula has been corrected
arxiv created 2000/02/27 · arxiv updated 2009/11/30
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 ⋅ F - i* c1(M) - 2 F⋅ 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, CN) via a Gauss map and where i* c1(M) is the restriction of the first Chern class of M to F. The key new term of interest is F ⋅ 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 H2(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.