Zeta-Functions for Families of Calabi--Yau n-folds with Singularities

TL;DR

This paper explores the relationship between singularities in families of Calabi-Yau n-folds and their local Weil zeta-functions, offering new insights into their combinatorial structure and supporting conjectures in higher dimensions.

Contribution

It introduces a novel analysis of singularities' impact on zeta-function decomposition in Calabi-Yau families, extending previous approaches to higher dimensions.

Findings

Singularity structure influences zeta-function decomposition.

Predictions match observed changes in zeta-function degree.

Supports Lauder's conjectured analogue of Clemens-Schmid sequence.

Abstract

We consider families of Calabi-Yau n-folds containing singular fibres and study relations between the occurring singularity structure and the decomposition of the local Weil zeta-function. For 1-parameter families, this provides new insights into the combinatorial structure of the strong equivalence classes arising in the Candelas - de la Ossa - Rodrigues-Villegas approach for computing the zeta-function. This can also be extended to families with more parameters as is explored in several examples, where the singularity analysis provides correct predictions for the changes of degree in the decomposition of the zeta-function when passing to singular fibres. These observations provide first evidence in higher dimensions for Lauder's conjectured analogue of the Clemens-Schmid exact sequence.