Infinitesimal deformations of Tsuchihashi's cusp singularities (Q1081001)

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/\(\Gamma\).