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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On maximally elliptic singularities

Author: Stephen Shing Toung Yau
Journal: Trans. Amer. Math. Soc. 257 (1980), 269-329
MSC: Primary 32C45; Secondary 14J17, 32B30
MathSciNet review: 552260
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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.

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Keywords: Weakly elliptic singularity, elliptic sequence, maximally elliptic singularity, fundamental cycle, minimally elliptic cycle, Gorenstein singularity, canonical divisor, maximal ideal cycle, elliptic double points, Hilbert function, weighted dual graphs
Article copyright: © Copyright 1980 American Mathematical Society