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Variation of Mixed Hodge Structures Associated to an Equisingular One-dimensional Family of Calabi-Yau Threefolds

Part of: Surfaces and higher-dimensional varieties Families, fibrations Cycles and subschemes Computational aspects in algebraic geometry

Published online by Cambridge University Press:  16 January 2020

Isidro Nieto-Baños  and
Pedro Luis del Angel-Rodriguez
Isidro Nieto-Baños
Affiliation:
C.P. 36023, Guanajuato, Gto., México Email: [email protected]
Pedro Luis del Angel-Rodriguez
Affiliation:
CIMAT, A.C., Jalisco, S/N, Guanajuato, Gto., México Email: [email protected]
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Abstract

We study the variations of mixed Hodge structures (VMHS) associated with a pencil ${\mathcal{X}}$ of equisingular hypersurfaces of degree $d$ in $\mathbb{P}^{4}$ with only ordinary double points as singularities, as well as the variations of Hodge structures (VHS) associated with the desingularization of this family $\widetilde{{\mathcal{X}}}$. The notion of a set of singular points being in homologically good position is introduced, and, by requiring that the subset of nodes in (algebraic) general position is also in homologically good position, we can extend Griffiths’ description of the $F^{2}$-term of the Hodge filtration of the desingularization to this case, where we can also determine the possible limiting mixed Hodge structures (LMHS). The particular pencil ${\mathcal{X}}$ of quintic hypersurfaces with 100 singular double points with 86 of them in (algebraic) general position that served as the starting point for this paper is treated with particular attention.

Keywords

variations of hodge structurescalabi-yau manifoldscomputational aspects in higher dimensional varietiestranscendental methodshodge theory

MSC classification

Primary: 14D07: Variation of Hodge structures
Secondary: 14J32: Calabi-Yau manifolds 14C30: Transcendental methods, Hodge theory , Hodge conjecture 14Q15: Higher-dimensional varieties
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 2 , April 2021 , pp. 441 - 464
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The first author acknowledges partial support from CIMAT as well as from the CMO congress “Primer Congreso Nacional de Geometría Algebraica” in the year 2016 in Oaxaca where part of these results were presented. Both authors had fruitful discussions at various stages of this work with D. Van Straten, Ch. Peters, J. Carlson, X. Gomez-Mont, and H. Kanarek and thank them for it. We acknowledge partial support from CONACyT Grant 0181730.

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