On maximally elliptic singularities

Abstract:

Let p be the unique singularity of a normal two-dimensional Stein space V. Let m be the maximal ideal in $_V{\mathcal {O}_p}$, the local ring of germs of holomorphic functions at p. We first define the maximal ideal cycle which serves to identify the maximal ideal. We give an “upper” estimate for maximal ideal cycle in terms of the canonical divisor which is computable via the topological information, i.e., the weighted dual graph of the singularity. Let $M \to V$ be a resolution of V. It is known that $h = \dim {H^1}(M, \mathcal {O})$ is independent of resolution. Rational singularities in the sense of M. Artin are equivalent to $h = 0$. Minimally elliptic singularity in the sense of Laufer is equivalent to saying that $h = 1$ and $_V{\mathcal {O}_p}$ is Gorenstein. In this paper we develop a theory for a general class of weakly elliptic singularities which satisfy a maximality condition. Maximally elliptic singularities may have h arbitrarily large. Also minimally elliptic singlarities are maximally elliptic singularities. We prove that maximally elliptic singularities are Gorenstein singularities. We are able to identify the maximal ideal. Therefore, the important invariants of the singularities (such as multiplicity) are extracted from the topological information. For weakly elliptic singularities we introduce a new concept called “elliptic sequence". This elliptic sequence is defined purely topologically, i.e., it can be computed explicitly via the intersection matrix. We prove that —K, where K is the canonical divisor, is equal to the summation of the elliptic sequence. Moreover, the analytic data $\dim {H^1}(M, \mathcal {O})$ is bounded by the topological data, the length of elliptic sequence. We also prove that $h = 2$ and $_V{\mathcal {O}_p}$ Gorenstein implies that the singularity is weakly elliptic.