“Weakly” elliptic Gorenstein singularities of surfaces. (English) Zbl 0934.32018

Némethi, András

Invent. Math. 137, No. 1, 145-167 (1999).

Let \(p\) be a singularity of a normal two-dimensional analytic surface \(X\) and \(\chi (D)\) be the Euler-characteristic of any cycle \(D\) and \(p_g=h^1 (M, {\Theta}_M)\).
M. Artin proved for rational singularities that \((X, p)\) is rational if and only if \(\chi = (Z_{\text{num}})=1\) and if and only if \(\min_{D\geq 0} \chi (D) \geq 1\). P. Wagreich proved that \((X, p)\) is elliptic if and only if \(\chi (Z_{\text{num}})=0\) and if and only if \(\min_{D\geq 0} \chi (D) = 0\). Later H. Laufer proved that the Gorenstein singularities with \(p_g =1\) can be characterized topologically, and for these singularities all the above analytical invariants are topological.
The author of this paper shows that for Gorenstein singularities with \(H^1 (A, Z) = 0\) the Artin- Laufer program can be continued and gives the complete answer in the case of elliptic singularities.

Reviewer: Jiancheng Zou (Beijing)

MSC:

32S15	Equisingularity (topological and analytic)
32S50	Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
14E15	Global theory and resolution of singularities (algebro-geometric aspects)

Keywords:

elliptic singularity; topological invariant; line bundle