On the infinitesimal Torelli problem of elliptic surfaces
type: Item
instance of: scholarly article
title: On the infinitesimal Torelli problem of elliptic surfaces
zbMATH Open document ID: 0532.14019
author: Masa-Hiko Saito
published in: Journal of Mathematics of Kyoto University
publication date: 1983-01-01T00:00:00Z
review text: The author studies the injectivity of the map \(\delta:H^ 1(X,\Theta)\to Hom_{{\mathbb{C}}}(H^{2,0}(X),H^{1,1}(X))\) given by the cup-product \(H^ 0(X,\Omega^ 2)\otimes H^ 1(X,\Theta)\to H^ 1(X,\Omega^ 1)\) deduced from the contraction \(\Theta \otimes \Omega^ 2\to \Omega^ 1\) for an elliptic surface \(\phi\) :\(X\to C\) with base curve C. He proves that: \(\delta\) is injective if \(h^{2,0}(X)>O\) and one of the following three conditions holds: (i) the functional invariant J(X) is not constant; (ii) J(X) is constant, distinct from O and 1, and either C is \({\mathbb{P}}_ 1({\mathbb{C}})\) or \((iii)\quad \chi(X,{\mathcal O}_ X))\geq 3.\) If \(\phi\) :\(X\to C\) is an elliptic bundle, then: (i) \(\delta\) is injective if \(g(C)=h^{1,0}(C)=1\), (ii) if \(b_ 1(X)\) is even and g(C)\(\geq 2\), \(\delta\) is not injective iff \(g(C)>2\) and C is hyperelliptic; (iii) if \(b_ 1(X)\) is odd and g(C)\(\geq 2\), \(\delta\) is never injective. - For Kodaira surfaces, the global Torelli does not hold, although the map \(\delta\) is injective. Editorial remark: \textit{A. Ikeda} constructed in [Adv. Math. 349, 125--161 (2019; Zbl 1414.14004)] an elliptic surface $Y$ and showed that the period map for the Hodge structure on $H^2(Y,\mathbb{Z})$ has one dimensional fibers and the general fiber has two irreducible components, which contradicts the main theorem of this work.
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 32J25
Mathematics Subject Classification ID: 32G05
zbMATH DE Number: 3843907
zbMATH Keywords: compact complex surface
zbMATH Keywords: second cohomology
zbMATH Keywords: Hodge structure
zbMATH Keywords: infinitesimal Torelli problem
zbMATH Keywords: elliptic surfaces
zbMATH Keywords: elliptic bundle
zbMATH Keywords: Kodaira surfaces
reviewed by: Frédéric Campana
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250521474
OpenAlex ID: W1591563964
DOI: 10.1215/KJM/1250521474
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description: scientific article; zbMATH DE number 3843907