Geometric Hodge structures with prescribed Hodge numbers

摘要

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