Degenerating complex variations of Hodge structure in dimension one

TL;DR

This paper extends the analysis of polarized complex variations of Hodge structure on punctured disks by removing the roots of unity assumption on monodromy eigenvalues, providing new proofs for key theorems.

Contribution

It generalizes Schmid's results to non-integral cases and offers more conceptual proofs for fundamental theorems in Hodge theory.

Findings

Established growth estimates for Hodge norms

Proved existence of limiting mixed Hodge structures

Simplified the SL(2)-orbit theorem

Abstract

We analyze the behavior of polarized complex variations of Hodge structure on the punctured unit disk. For integral variations of Hodge structure, this analysis was first carried out by Wilfried Schmid. We get rid of the assumption that the eigenvalues of the monodromy transformation are roots of unity. In this generality, we give new (and, we think, more conceptual) proofs for all the major results in Schmid's paper, such as the estimates for the rate of growth of the Hodge norm; the existence of a limiting mixed Hodge structure; the nilpotent orbit theorem; and a simplified (but still sufficiently powerful) version of the SL(2)-orbit theorem.