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Existence of valuations with smallest normalized volume

Part of: Local theory Topological fields

Published online by Cambridge University Press:  08 March 2018

Harold Blum
Harold Blum*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA email [email protected]://www-personal.umich.edu/∼blum/
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Abstract

Li introduced the normalized volume of a valuation due to its relation to K-semistability. He conjectured that over a Kawamata log terminal (klt) singularity there exists a valuation with smallest normalized volume. We prove this conjecture and give an explicit example to show that such a valuation need not be divisorial.

Keywords

singularitiesvaluationsnormalized volume

MSC classification

Primary: 14B05: Singularities
Secondary: 12J20: General valuation theory
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 4 , April 2018 , pp. 820 - 849
Copyright
© The Author 2018 

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References

Bennett, B., On the characteristic function of a local ring , Ann. of Math. (2) 91 (1970), 25–87.CrossRefGoogle Scholar
Blum, H., On divisors computing mlds and lcts, Preprint (2016), arXiv:1605.09662v2.Google Scholar
Boucksom, S., Favre, C. and Jonsson, M., Valuations and plurisubharmonic singularities , Publ. Res. Inst. Math. Sci. 44 (2008), 449–494.Google Scholar
Boucksom, S., de Fernex, T., Favre, C. and Urbinati, S., Valuation spaces and multiplier ideals on singular varieties , in Recent advances in algebraic geometry. Volume in honor of Rob Lazarsfeld’s 60th birthday, London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, 2015), 29–51.CrossRefGoogle Scholar
Boucksom, S., Hisamoto, T. and Jonsson, M., Uniform K-stability and asymptotics of energy functionals in Kähler geometry, Ann. Inst Fourier, to appear. Preprint (2016),arXiv:1603.01026.Google Scholar
Cutkosky, D., Multiplicities associated to graded families of ideals , Algebra Number Theory 7 (2013), 2059–2083.Google Scholar
Demailly, J.-P., Ein, L. and Lazarsfeld, R., A subadditivity property of multiplier ideals , Michigan Math. J. 48 (2000), 137–156.Google Scholar
Ein, L., Lazarsfeld, R. and Smith, K. E., Uniform approximation of Abhyankar valuation ideals in smooth function fields , Amer. J. Math. 125 (2003), 409–440.Google Scholar
Eisenbud, D., Commutative algebra, Graduate Texts in Mathematics, vol. 150 (Springer, New York, NY, 1995).Google Scholar
Eisenstein, E., Inversion of adjunction in high codimension, PhD thesis, University of Michigan (2011).Google Scholar
Favre, C. and Jonsson, M., The valuative tree, Lecture Notes in Mathematics, vol. 1853 (Springer, Berlin, 2004).Google Scholar
de Fernex, T., Birationally rigid hypersurfaces , Invent. Math. 192 (2013), 533–566.CrossRefGoogle Scholar
de Fernex, T., Ein, L. and Mustaţă, M., Bounds for log canonical thresholds with applications to birational rigidity , Math. Res. Lett. 10 (2003), 219–236.Google Scholar
de Fernex, T., Ein, L. and Mustaţă, M., Multiplicities and log canonical threshold , J. Algebraic Geom. 13 (2004), 603–615.Google Scholar
de Fernex, T., Ein, L. and Mustaţă, M., Shokurovs ACC conjecture for log canonical thresholds on smooth varieties , Duke Math. J. 152 (2010), 93–114.Google Scholar
de Fernex, T., Ein, L. and Mustaţă, M., Log canonical thresholds on varieties with bounded singularities , in Classification of algebraic varieties, EMS Series of Congress Reports (European Mathematical Society, Zürich, 2011), 221–257.Google Scholar
de Fernex, T. and Mustaţă, M., Limits of log canonical thresholds , Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 491–515.Google Scholar
Fulton, W., Introduction to toric varieties, Annals of Mathematics Studies, vol. 131 (Princeton University Press, Princeton, NJ, 1993).Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes , Inst. Hautes Études Sci. Publ. Math. 8 (1961), 5–222.Google Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, Heidelberg, 1977).Google Scholar
Jonsson, M. and Mustaţǎ, M., Valuations and asymptotic invariants for sequences of ideals , Ann. Inst. Fourier (Grenoble) 62 (2012), 2145–2209.Google Scholar
Kaveh, K. and Khovanskii, A. G., Convex bodies and multiplicities of ideals , Proc. Steklov Inst. Math. 286 (2014), 268–284.CrossRefGoogle Scholar
Kollár, J., Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 32 (Springer, Berlin, 1996).Google Scholar
Kollár, J., Singularities of pairs , in Algebraic geometry – Santa Cruz 1995, Proceedings of Symposia in Pure Mathematics, vol. 62, Part 1 (American Mathematical Society, Providence, RI, 1997), 221–287.Google Scholar
Kollár, J., Which powers of holomorphic functions are integrable? Preprint (2008), arXiv:0805.0756.Google Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
Lazarsfeld, R., Positivity in algebraic geometry, I and II , Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 48–49 (Springer, Berlin, 2004).Google Scholar
Lazarsfeld, R. and Mustaţǎ, M., Convex bodies associated to linear series , Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 783–835.CrossRefGoogle Scholar
Li, C., Minimizing normalized volumes of valuations, Math. Z., to appear. Preprint (2015),arXiv:1511.08164v3.Google Scholar
Li, C., K-semistability is equivariant volume minimization , Duke Math. J. 166 (2017), 3147–3218.CrossRefGoogle Scholar
Li, C. and Liu, Y., Kähler–Einstein metrics and volume minimization, Adv. Math., to appear. Preprint (2016), arXiv:1602.05094.Google Scholar
Li, C. and Xu, C., Stability of valuations and Kollár components, Preprint (2016), arXiv:1604.05398.Google Scholar
Lipman, J., Equimultiplicity, reduction, and blowing up , in Commutative algebra (Fairfax, Va. 1979), Lecture Notes in Pure and Applied Mathematics, vol. 68 (Dekker, New York, NY, 1982).Google Scholar
Liu, Y., The volume of singular Kähler–Einstein Fano varieties, Compositio Math., to appear. Preprint (2016), arXiv:1605.01034v2.Google Scholar
Martelli, D. and Sparks, J., Toric geometry, Sasaki–Einstein manifolds and a new infinite class of AdS/CFT duals , Comm. Math. Phys. 262 (2006), 51–89.CrossRefGoogle Scholar
Mustaţǎ, M., On multiplicities of graded sequences of ideals , J. Algebra 256 (2002), 229–249.Google Scholar
Ramanujam, C., On a geometric interpretation of multiplicity , Invent. Math. 22 (1973), 63–67.Google Scholar
Takagi, S., Formulas for multiplier ideals on singular varieties , Amer. J. Math. 128 (2006), 1345–1362.Google Scholar
Takagi, S., A subadditivity formula for multiplier ideals associated to log pairs , Proc. Amer. Math. Soc. 141 (2013), 93–102.Google Scholar
Teissier, B., Sur une inégalité à la Minkowski pour les multiplicités, appendix to D. Eisenbud and H. I. Levine, The degree of a C ∞ map germ , Ann. of Math. (2) 106 (1977), 19–44.Google Scholar