Infinitesimal deformations of Tsuchihashi's cusp singularities
type: Item
instance of: scholarly article
title: Infinitesimal deformations of Tsuchihashi's cusp singularities
zbMATH Open document ID: 0601.32024
author: Shoetsu Ogata
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1986-01-01T00:00:00Z
review text: In this paper, we can see some interesting deformations of normal isolated singularities called Tsuchihashi's cusp types. There are three main theorems. The third one shows that Tsuchihashi's cusp singularities of dimensions three are not rigid in general. Our cusp singularity \((X,x_ 0)\) is a singular point of the analytic space \[ X:=((N\otimes_{{\mathbb{Z}}}{\mathbb{R}}+\sqrt{-1}C)/N\cdot \Gamma)\cup \{x_ 0\} \] where N is a free \({\mathbb{Z}}\)-module of rank \(>1\), C is a nondegenerate open \(\Gamma\)-invariant convex cone in \(N_{{\mathbb{R}}}\), \(\Gamma\) is a subgroup in \(Aut_{{\mathbb{Z}}}(N)\) whose action on \(D:=C/{\mathbb{R}}_+\) is purely discontinuous and fixed point free and D/\(\Gamma\) is compact. Theorem 1. Let \(U:=X-\{x_ 0\}\). When rank (N)\(\geq 3\), we have canonical isomorphisms \[ T^ 1_ X\cong H^ 1(U,\Theta_ X)\cong H^ 1(\Gamma,N_ C), \] where \(\Theta_ X\) is the holomorphic tangent sheaf of X. Theorem 2. C is decomposable \(\Rightarrow H^ 1(\Gamma,N_ C)=0\) and, rank (N)\(\geq 3\) and C is decomposable \(\Rightarrow T^ 1_ X=0.\) Theorem 3. When rank (N)\(=3\), \[ 3(1-\chi (D/\Gamma)) \geq \dim_{{\mathbb{C}}} T^ 1_ X \geq -3\chi (D/\Gamma), \] where \(\chi\) (D/\(\Gamma)\) is the Euler number of the compact real manifold D/\(\Gamma\).
Mathematics Subject Classification ID: 32S30
Mathematics Subject Classification ID: 32Sxx
Mathematics Subject Classification ID: 32C20
zbMATH DE Number: 3969062
zbMATH Keywords: infinitesimal deformation
zbMATH Keywords: rigid singularity
zbMATH Keywords: deformations of normal isolated singularities
zbMATH Keywords: Tsuchihashi's cusp singularities
reviewed by: S. Ohyanagi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228493
OpenAlex ID: W2136208351
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DOI: 10.2748/TMJ/1178228493
description: scientific article; zbMATH DE number 3969062