Singularities on normal varieties

By Tommaso de Fernex, Christopher D. Hacon. Compositio Mathematica, 2008

89 citations recorded by the source corpus.

Abstract In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being ℚ-Gorenstein or the pair being log ℚ-Gorenstein. The main features of the theory extend to this setting in a natural way.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7499b3ac08174e19abd6d863d15d14ccb4b77620

How researchers see this paper

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In Algebra and Number Theory

1. Karl Schwede
2. Chi Li
3. Manuel Blickle
4. S. Boucksom
5. Wenliang Zhang
6. Yuchen Liu
7. Tommaso de Fernex
8. S. Takagi

In Geometry and Topology

1. Chi Li
2. Yuchen Liu
3. S. Boucksom
4. Mattias Jonsson
5. Karl Schwede
6. H. Blum
7. M. Mustaţă
8. Manuel Blickle

Citations in the corpus

• Discreteness and rationality of F-jumping numbers on singular varieties (2009, background): “3 In [9], de Fernex and Hacon define a multiplier ideal for normal nonQ-Gorenstein varieties.”
• The volume of an isolated singularity (2010, methodology): “ition is given by Env X(D)π where D is the Cartier b-divisor induced by D. In higher dimensions D →EnvX(D) is non-linear in general, and EnvX(D)π coincides up to sign with the pull-back π∗D defined in [dFH09]. It is however this approach via b-divisors and nef envelopes that brings to light the cruci”
• Kähler–Einstein metrics and volume minimization (2016, background): “Notice that Gis Q-Cartier. Because W is smooth and C(W) ∼= C(Y), we can define, for any v∈ ValY ∼= ValW, A(Y,E)(v) := AW(v) +v(G), where v(G) := 1 kv(IkG) if kGis Cartier with the ideal sheaf IkG (see [dFH09]). With the above definition, we define for any v∈ ValY,o: volc (Y,E)(v) = ˆ A(Y,E)(v) nvol(v),”
• Valuation spaces and multiplier ideals on singular varieties (2013, methodology): “′ = OX′(−D) with D an effective Cartier divisor. For each exponent c > 0 we then have J(ac) = π∗OX′ ⌈Knum X′/X −cD⌉ . In dimension two, this result says that the multiplier ideals introduced in [dFH09] agree with the numerical multiplier ideals defined using Mumford’s numerical pull-back. Since t”
• Minimizing normalized volumes of valuations (2015, methodology): “aper we will only work with the class of Q-Gorenstein singularities. Notice that the log discrepancy and klt condition can be defined for all normal singularities following the work of de-Fernex-Hacon [dFH09] (see also [BFF12, BFFU13]) and most discussions in this paper can indeed be generalized corr”