Deformations of a holomorphic map and its degeneracy locus
type: Item
instance of: scholarly article
title: Deformations of a holomorphic map and its degeneracy locus
zbMATH Open document ID: 1274.14007
author: Madoka Ebihara
published in: Tokyo Journal of Mathematics
publication date: 2013-03-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tjm/1358951317
review text: The author studies deformations of holomorphic maps. Given a deformation \(\Phi : \mathcal{X} \to Y \times M\) of a holomorphic map \(\Phi_0 : \mathcal{X}_0 \to Y\) between holomorphic manifolds, one can consider two Kodaira-Spencer maps. The first one has its image in the infinitesimal deformation space of \(\Phi_0 : \mathcal{X}_0 \to Y\) (here we allow both \(\mathcal{X}_0\) and the map to deform, however \(Y\) is kept fixed). The second one has its image in the infinitesimal deformation space of the degeneracy locus \(\Delta\). The author shows a compatibility between these two Kodaira-Spencer maps with the additional assumptions that \(f\) is surjective, \(\Delta\) is a smooth divisor and the ramification locus \(R\) maps isomorphically onto \(\Delta\). The method of the proof is by using the explicit descriptions of the above Kodaira-Spencer maps given by [\textit{E. Horikawa}, Math. Ann. 222, 275--282 (1976; Zbl 0334.32021)] and [\textit{K. Kodaira}, Ann. Math. (2) 75, 146--162 (1962; Zbl 0112.38404)]. The main application of the above deformation theoretic result is the rigidity of ordinary conic bundles \(f : X \to \mathbb{P}^m\) associated to vector bundles \(\mathcal{E}\) that are direct sums of line bundles. Here rigidity means that there are no deformations of \(f\) that fixes \(Y\) and the degeneracy locus \(\Delta\). This extends earlier results of the author on deformations of conic bundles.
reviewed by: Zsolt Patakfalvi
Mathematics Subject Classification ID: 14D15
zbMATH DE Number: 6146568
zbMATH Keywords: holomorphic map
zbMATH Keywords: deformation
zbMATH Keywords: conic bundle
MaRDI profile type: Publication
OpenAlex ID: W2007926114
cites work: Variétés de Prym et jacobiennes intermédiaires
cites work: Some remarks on infinitesimal deformations of a conic bundle. I
cites work: Some remarks on infinitesimal deformations of a conic bundle. II.
cites work: Q3533449
cites work: Q4898107
cites work: On deformations of holomorphic maps . I
cites work: A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds
DOI: 10.3836/TJM/1358951317
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description: scientific article; zbMATH DE number 6146568