Geometric Hodge structures with prescribed Hodge numbers

Abstract

Call a pure Hodge structure geometric if it is contained in the cohomology of a smooth complex projective variety. The main goal is to show that for any set of Hodge numbers (subject to the obvious constraints), there exists a geometric Hodge structure with precisely these Hodge numbers. This is related to recent work of Schreieder, but the construction here is simpler. This also contains some speculations about 2 dimensional geometric Hodge structures.