Frobenius and the Hodge Spectral Sequence

Frobenius and the Hodge Spectral Sequence

Let X k be a smooth proper scheme over a perfect field of characteristic p and let n be a natural number. A fundamental theorem of Barry Mazur relates the Hodge numbers of X k to the action of Frobenius on the crystalline cohomology H cris (X W) of X over the Witt ring W of k, which can be viewed as a linear map 8: F*W H n cris (X W) H n cris (X W). If H n cris (X W) is torsion free, then since...

Frobenius modules and Hodge asymptotics

We exhibit a direct correspondence between the potential defining the H small quantum module structure on the cohomology of a Calabi-Yau manifold and the asymptotic data of the A-model variation of Hodge structure. This is done in the abstract context of polarized variations of Hodge structure and Frobenius modules.

Exotic Torsion, Frobenius Splitting and the Slope Spectral Sequence

Eigenvalues of Frobenius and Hodge Numbers

We study the connection between Hodge purity of the cohomology of algebraic varieties over fields of different characteristics. Specifically, we study varieties over number fields, whose cohomology in some fixed degree 2i consists entirely of Hodge classes, that is, whose Hodge cohomology in degree 2i is entirely of type (i, i). Among other things, we give sufficient conditions for varieties ov...

Opposite Filtrations, Variations of Hodge Structure, and Frobenius Modules

In this article we describe three constructions of complex variations of Hodge structure, proving the existence of interesting opposite filtrations that generalize a construction of Deligne. We also analyze the relation between deformations of Frobenius modules and certain maximally degenerate variations of Hodge structures. Finally, under a certain generation hypothesis, we show how to constru...