  A theorem of O. Haupt, rediscovered by M. Kapovich and celebrated by his
proof invoking Ratner theory, describes the set of de Rham cohomology classes
on a topological orientable surface, which can be realized by an abelian
differential in some respective complex structure, in purely topological terms.
We make an attempt to describe similarly pairs and triples of cohomology
classes, which can be realized by abelian differentials in some complex
structure. This leads us to some interesting problems in algebraic geometry of
curves, and gives an unexpected local description of the Teichm\"uller space.
