×

Infinitesimal deformations of Tsuchihashi’s cusp singularities. (English) Zbl 0601.32024

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/\(\Gamma\).
Reviewer: S.Ohyanagi

MSC:

32S30 Deformations of complex singularities; vanishing cycles
32Sxx Complex singularities
32C20 Normal analytic spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] BANICA AND O. STANASILA, Algebraic Methods in the Global Theory of Complex Spaces, Editura, Academiei, Bucuresti and John Wiley & Sons, London, New York, Sydney and Toronto, 1976. · Zbl 0334.32001
[2] K. BEHNKE, Infinitesimal deformations of cusp sigularities, Math. Ann. 265 (1983) 407-422 · Zbl 0559.14001 · doi:10.1007/BF01455945
[3] K. BEHNKE, On the module of Zariski differentials and infinitesimal deformations of cus singularities, Math. Ann. 271 (1985), 133-142. · Zbl 0586.14002 · doi:10.1007/BF01455802
[4] E. FREITAG, Lokale und globale Invarianten der Hilbertschen Modulgruppen, Inventiones, Math. 17 (1972), 106-134. · Zbl 0272.32010 · doi:10.1007/BF01418935
[5] E. FREITAG AND R. KIEHL, Algebraische Eigenschaften der lokalen Ringe in den Spitze der Hilbertschen Modulgruppen, ibid. 24 (1974), 121-148. · Zbl 0304.32018 · doi:10.1007/BF01404302
[6] A. GROTHENDIECK, Sur quelques points d’algebre homologique, Thoku Math. J. 9 (1957), 119-221. · Zbl 0118.26104
[7] R. C. GUNNING AND H. Rossi, Analytic Functions of Several Complex Variables, Prentice Hall, Inc., Englewood Cliffs, N. J., 1965. · Zbl 0141.08601
[8] P. J. HILTON AND U. STAMMBACH, A Course in Homological Algebra, Springer-Verlag, New York, Heidelberg, Berlin, 1970. · Zbl 0863.18001
[9] I. NAKAMURA, Infinitesimal deformations of cusp singularities, Proc. Japan Acad. 60, Ser. A (1984), 35-38. · doi:10.3792/pjaa.60.35
[10] M. SCHLESSINGER, Rigidity of quotient singularities, Inventiones Math. 14 (1971), 17-26 · Zbl 0232.14005 · doi:10.1007/BF01418741
[11] J. -P. SERRE, Cohomologie des groupes discrete, Prospects in mathematics, Annals o Mathematical Studies 70, Princeton Univ. Press and Univ. of Tokyo Press, 1971, 77-169. · Zbl 0235.22020
[12] H. TSUCHIHASHI, Higher dimensional analogues of periodic continued fractions and cus singularities, Thoku Math. J. 35 (1983), 607-639. · Zbl 0585.14004 · doi:10.2748/tmj/1178228955
[13] E. B. VINBERG, Theory of homogeneous convex cones, Trans. Moscow Math. Soc. 12 (1967), 303-368. · Zbl 0138.43301
[14] K. YOSHIDA, Functional Analysis, Second edition, Springer-Verlag, Berlin, Heidelberg, New York, 1968.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.