From Smooth to Almost Complex

Abstract

This article mainly aims to overview the recent efforts on developing algebraic geometry for an arbitrary compact almost complex manifold. We review the results obtained by the guiding philosophy that a statement for smooth maps between smooth manifolds in terms of Ren\'e Thom's transversality should also have its counterpart in pseudoholomorphic setting without requiring genericity of the almost complex structures. These include intersection of compact almost complex submanifolds, structure of pseudoholomorphic maps, zero locus of certain harmonic forms, and eigenvalues of Laplacian. In addition to reviewing the compact manifolds situation, we also extend these results to orbifolds and non-compact manifolds. Motivations, methodologies, applications, and further directions are discussed. The structural results on the pseudoholomorphic maps lead to a notion of birational morphism between almost complex manifolds. This motivates the study of various birational invariants, including Kodaira dimensions and plurigenera, in this setting. Some other aspects of almost complex algebraic geometry in dimension $4$ are also reviewed. These include cones of (co)homology classes and subvarieties in spherical classes.