Homotopical obstructions to the existence of certain complex structures on smooth manifolds

Scott Wilson, City University of New York

A difficult open problem is to determine if there are topological obstructions to complex structures on smooth manifolds (of even dimension greater than or equal to six) beyond the fairly well-understood obstructions to almost complex structures.  In this talk, I will explain that certain types of complex structures are homotopically obstructed in these dimensions, where the “types” are organized by the structure of the underlying bicomplex of differential forms.  To establish this, I’ll describe some numerical inequalities for complex manifolds of the form “Topology is less than or equal to Complex-analysis”. The topology invariants roughly measure the failure of the algebra of differential forms to be equivalent to its cohomology, and the complex-analytic invariants measure the “wildness” of the bi-complex of differential forms. Explicit examples will be given. This is joint work with Jonas Stelzig.

MC 5417