Zeta functions of alternate mirror Calabi-Yau families
type: Item
instance of: scholarly article
title: Zeta functions of alternate mirror Calabi-Yau families
zbMATH Open document ID: 1403.14055
author: C. F. Doran
author: Tyler L. Kelly
author: Adriana Salerno
author: John Voight
author: Ursula Whitcher
author: Steven Sperber
published in: Israel Journal of Mathematics
publication date: 2018-11-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1612.09249
review text: An invertible polynomial is a polynomial of the form \[ F_A=\sum_{i=0}^n\prod_{j=0}^n x_j^{a_{ij}}\in{\mathbb{Z}}[x_0,\dots,x_n] \] where the matrix of exponents \(A=(a_{ij})\) is a square matrix of size \(n+1\) with nonnegative integer entries satisfying (1) \(\det(A)\neq 0\), (2) \(F_A\) is quasi-homogeneous, and (3) \(F_A:{\mathbb{C}}^{n+1}\to{\mathbb{C}}\) is nonsingular at the origin. When \(F_A\) is invertible and homogeneous of degree \(d=n+1\), \(F_A=0\) defines a Calabi-Yau variety \(X_A\) in \({\mathbb{P}}^n\). For an invertible polynomial \(F_A\), there is the (dual) polynomial \[ F_{A^T}:=\sum_{i=0}^n\prod_{j=0}^n x_j^{a_{ji}}. \] Then \(F_{A^T}\) is again an invertible polynomial, quasihomogeneous with weights \(q_0,\dots, q_n\) (\(\gcd(q_0,\dots, q_n)=1\)), so that \(F_{A^T}=0\) defines a hypersurface \(X_{A^T}\) in \({\mathbf {WP}}(q_0,\dots, q_n)\) with the dual weights \((q_0,\dots, q_n)\). Let \(d^T=\sum_{i=0}^nq_i\). Define a one-parameter deformation of \(F_A\) by \[ F_{A,\psi}:=\sum_{i=0}^n\prod_{j=0}^n x_j^{q_{ij}}-d^T\psi\prod_{i=0}^n x_j \in{\mathbb{Z}}[\psi][x_0,\dots, x_n]. \] Then \(X_{A,\psi}: F_{A,\psi}=0\) is a family of hypersurfaces in \({\mathbb{P}}^n\) in the parameter \(\psi\), which is called an invertible pencil. \textit{S. Gährs} [in: Arithmetic and geometry of \(K3\) surfaces and Calabi-Yau threefolds. Proceedings of the workshop, Toronto, Canada, August 16--25, 2011. New York, NY: Springer. 285--310 (2013; Zbl 1302.14034)] showed the Picard-Fuchs differential equation for \(X_{A,\psi}\) is determined by the \((n+1)\)-tuple of dual weights \((q_0,\dots, q_n)\), and that its order \(D(q_0,\dots, q_n)\) depends only on the dual weights. This paper shows that invertible pencils whose Berglund-Hübsch-Krawits (BHK) mirrors have common properties share arithmetic properties as well. In particular, invertible pencils whose BHK mirrors are hypersurfaces in quotients of the same weighted projective space have the same Picard-Fuchs equat…
reviewed by: Noriko Yui
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11G42
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 32Q25
zbMATH DE Number: 6976747
zbMATH Keywords: zeta function
zbMATH Keywords: BHK mirror
zbMATH Keywords: Calabi-Yau hypersurfaces
zbMATH Keywords: one-parameter deformation
zbMATH Keywords: Picard-Fuchs equation
zbMATH Keywords: \(K3\) surface
zbMATH Keywords: arithmetic mirror symmetry
describes a project that uses: Magma
MaRDI profile type: Publication
OpenAlex ID: W2962751730
arXiv ID: 1612.09249
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Wikidata QID: Q129244957
DOI: 10.1007/S11856-018-1783-0