Haupt-Kapovich theorem revisited

A theorem of O. Haupt, rediscovered by M. Kapovich using the Ratner theorem in ergodic theory, describes the set of de Rham cohomology classes on a topological surface which can be realised by an abelian differential in some complex structure, in purely topological terms. It turns out, almost all the positive classes can be realised. We prove that the pairs and triples of cohomology classes which can be simultaneously realised by abelian differentials in some complex structure also constitute open dense subsets in the Grassmannians of subspaces subject to the Hodge–Riemann relations, whereas for quadruples the exact opposite is true. We also prove similar results for hyperelliptic curves, Prym varieties, and cubic threefolds, and discuss the connection of these results to geometry of holomorphic Lagrangian fibrations.