Families of \(K3\) surfaces over curves reaching the Arakelov-Yau type upper bounds and modularity
type: Item
instance of: scholarly article
title: Families of \(K3\) surfaces over curves reaching the Arakelov-Yau type upper bounds and modularity
zbMATH Open document ID: 1100.14004
author: Xiaotao Sun
author: Sheng-Li Tan
author: Kang Zuo
published in: Mathematical Research Letters
publication date: 2004-05-18T00:00:00Z
review text: This paper considers non-isotrivial algebraic families of semi-stable \(K3\) surfaces over curves when the Arakelov-Yau inequality reaches the upper bound and becomes equality. Let \(f : X\to C\) be a family of semi-stable \(K3\) surfaces over a projective curve \(C\). The Arakelov-Yau inequality implies that \(\text{deg }(f_*\omega_{X/C})\leq \text{deg }\Omega_C^1(\log S)\) where \(S\) is the discriminant locus. This paper studies non-isotrivial families which satisfy the equality \(\text{deg}(f_*\omega_{X/C})=\text{deg\,}\Omega_C^1(\log\,S)\). They prove that the general fiber of \(f\) has Picard number at least 19, and after finite étale base change, the restriction of the family over the open subset admits a Nikulin involution whose quotient is the family of abelian surfaces isogenous to the symmetric square of a family of elliptic curves. (Such a family is called a family coming from Nikulin-Kummer construction of the square product of a family of elliptic curves.) Also they prove that if the second iterated Kodaira-Spencer map of the family is zero and if the Arakelov-Yau upper bound is reached, then the general fiber of \(f\) has the Picard number at least 18. In both cases, the monodromy representations are determined as subgroups of \(\text{PSL}_2({\mathbb Z})\). As a corollary, the authors restrict to the special case that \(X\) is a Calabi-Yau threefold. Then \(\# S\geq 4\). Let \(f: X\to {\mathbb{P}}^1\) be a Calabi-Yau threefold fibered by non-constant families of semi-stable \(K3\) surfaces, reaching the Arakelov-Yau bound. Then the following hold true: (1) If the iterated Kodaira--Spencer map of \(f\) is non-zero, then \(f\) has at least 4 singular fibers. If \(f\) has exactly 4 singular fibers, then \(X\) is a rigid Calabi-Yau threefold, the general fibers have Picard number at least 19, and birational to the Nikulin-Kummer construction of a square product of a family of elliptic curves \(g: E\to{\mathbb{P}}^1\). After passing to (if necessary) a dou…
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 32G20
zbMATH DE Number: 2064162
zbMATH Keywords: non-isotrivial families
zbMATH Keywords: Kodaira-Spencer maps
zbMATH Keywords: \(K3\) fibered Calabi-Yau threefolds
reviewed by: Noriko Yui
MaRDI profile type: Publication
full work available at URL: mrl.2003.v10.n3.a4
OpenAlex ID: W2163117662
DOI: 10.4310/MRL.2003.V10.N3.A4
description: scientific article; zbMATH DE number 2064162