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

• Published in: arXiv.org
• Main Authors: Nieto-Baños, Isidro, Del Angel-Rodriguez, Pedro Luis
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 23.03.2020
• Subjects: Homology; Hyperspaces; Mathematics - Algebraic Geometry; Singularities
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1501.00339

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.