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- W2083830154 abstract "Let <italic>p</italic> be the unique singularity of a normal two-dimensional Stein space <italic>V</italic>. Let <italic>m</italic> be the maximal ideal in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Subscript upper V Baseline script upper O Subscript p> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi /> <mml:mi>V</mml:mi> </mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>_V{mathcal {O}_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the local ring of germs of holomorphic functions at <italic>p</italic>. 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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M right-arrow upper V> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi>V</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>M to V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a resolution of <italic>V</italic>. It is known that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=h equals dimension upper H Superscript 1 Baseline left-parenthesis upper M comma script upper O right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mspace width=thinmathspace /> <mml:mo>=</mml:mo> <mml:mspace width=thinmathspace /> <mml:mi>dim</mml:mi> <mml:mspace width=thinmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>h, = ,dim ,{H^1}(M,,mathcal {O})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is independent of resolution. Rational singularities in the sense of M. Artin are equivalent to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=h equals 0> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mspace width=thinmathspace /> <mml:mo>=</mml:mo> <mml:mspace width=thinmathspace /> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>h, = ,0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Minimally elliptic singularity in the sense of Laufer is equivalent to saying that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=h equals 1> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mspace width=thinmathspace /> <mml:mo>=</mml:mo> <mml:mspace width=thinmathspace /> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>h, = ,1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Subscript upper V Baseline script upper O Subscript p> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi /> <mml:mi>V</mml:mi> </mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>_V{mathcal {O}_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <italic>h</italic> 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 —<italic>K</italic>, where <italic>K</italic> is the canonical divisor, is equal to the summation of the elliptic sequence. Moreover, the analytic data <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=dimension upper H Superscript 1 Baseline left-parenthesis upper M comma script upper O right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>dim</mml:mi> <mml:mspace width=thinmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>dim ,{H^1}(M,,mathcal {O})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is bounded by the topological data, the length of elliptic sequence. We also prove that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=h equals 2> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mspace width=thinmathspace /> <mml:mo>=</mml:mo> <mml:mspace width=thinmathspace /> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>h, = ,2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Subscript upper V Baseline script upper O Subscript p> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi /> <mml:mi>V</mml:mi> </mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>_V{mathcal {O}_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Gorenstein implies that the singularity is weakly elliptic." @default.
- W2083830154 created "2016-06-24" @default.
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- W2083830154 date "1980-01-01" @default.
- W2083830154 modified "2023-10-03" @default.
- W2083830154 title "On maximally elliptic singularities" @default.
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