Monodromy at Infinity and the Weights of Cohomology

TL;DR

This paper investigates the monodromy at infinity for polynomial maps, establishing bounds on Jordan block sizes related to divisor multiplicities and linking these to global invariants and period integrals.

Contribution

It introduces bounds on Jordan block sizes of monodromy at infinity using divisor multiplicities and connects these to weight filtrations and global invariant cycles.

Findings

Bound on Jordan block size by divisor multiplicity

Relation between Jordan blocks and global invariant cycles

More precise monodromy description when no singularities at infinity

Abstract

We show that for a polynomial map, the size of the Jordan blocks for the eigenvalue 1 of the monodromy at infinity is bounded by the multiplicity of the reduced divisor at infinity of a good compactification of a general fiber. The existence of such Jordan blocks is related to global invariant cycles of the graded pieces of the weight filtration. These imply some applications to period integrals. We also show that such a Jordan block of size greater than 1 for the graded pieces of the weight filtration is the restriction of a strictly larger Jordan block for the total cohomology group. If there are no singularities at infinity, we have a more precise statement on the monodromy.