Vanishing cycle sheaves of one-parameter smoothings and quasi-semistable degenerations

TL;DR

This paper investigates the structure of vanishing cycles in one-parameter smoothings of complex spaces, revealing their relation to monodromy and introducing quasi-semistable degenerations with applications to Hodge theory.

Contribution

It provides new insights into the weight filtration and monodromy filtration relationship, introduces quasi-semistable degenerations, and computes the limit mixed Hodge structure in this context.

Findings

The weight filtration closely approximates the monodromy filtration in the highest degree perverse cohomology.

1 is not an eigenvalue of monodromy iff the total space and singular fiber are rational homology manifolds.

Non-trivial vanishing cycles exist for Lefschetz pencils of tensor products of very ample line bundles, except in specific cases.

Abstract

We study the vanishing cycles of a one-parameter smoothing of a complex analytic space and show that the weight filtration on its perverse cohomology sheaf of the highest degree is quite close to the monodromy filtration so that its graded pieces have a modified Lefschetz decomposition. We describe its primitive part using the weight filtration on the perverse cohomology sheaves of the constant sheaves. As a corollary we show in the local complete intersection case that 1 is not an eigenvalue of the monodromy on the reduced Milnor cohomology at any points if and only if the total space and the singular fiber are both rational homology manifolds. Also we introduce quasi-semistable degenerations and calculate the limit mixed Hodge structure by constructing the weight spectral sequence. As a corollary we show non-triviality of the space of vanishing cycles of the Lefschetz pencil associated with a tensor product of any two very ample line bundles except for the case of even-dimensional projective space where two has to be replaced by three.