Variation of Mixed Hodge Structures associated to an equisingular one-dimensional family of Calabi-Yau threefolds

TL;DR

This paper investigates how mixed Hodge structures vary in families of Calabi-Yau threefolds with specific singularities, introducing new concepts like homologically good position to better understand the limiting behaviors.

Contribution

It introduces the notion of homologically good position for singular points and analyzes its impact on the variation of Hodge structures in Calabi-Yau families.

Findings

Determined the LMHS for families with singular points in agp and hg positions.

Showed that agp alone does not fully describe the Hodge structure variations.

Analyzed a specific quintic hypersurface family with 100 nodes in detail.

Abstract

We study the Variations of mixed Hodge structures (VMHS) associated to a pencil ${\cal X}$ (parametrised by an open set $B \subset {\Bbb P}^1$) of equisingular hypersurfaces of degree $d$ in ${\Bbb P}^{4}$ with exactly $m$ ordinary double points as singularities as well as the variations of Hodge structures (VHS) associated to the desingularization of this family $ \widetilde{\cal X}$. The case where exactly $l \le m $ of those double points are in algebraic general position (short:agp) is studied in detail and determine the possible limiting mixed Hodge structures (LMHS) associated to each of the points in ${\Bbb P}^1\backslash B$. We find that the position of the singular points being in agp is not sufficient to describe the space of first one-adjoint conditions and naturally the notion of a set of singular points being in homologically good position (short: hg) is introduced. By requiring that the set of nodes in agp is also in hg, the $F^2$-term of the Hodge filtration of the desingularization is completely determined. The particular pencil $ {\cal X}$ of quintic hypersurfaces with $100$ singular double points with $86$ of them in agp which served as the starting point for this paper is treated with particular attention.