## Duke Mathematical Journal

### The volume of an isolated singularity

#### Abstract

We introduce a notion of volume of a normal isolated singularity that generalizes Wahl’s characteristic number of surface singularities to arbitrary dimensions. We prove a basic monotonicity property of this volume under finite morphisms. We draw several consequences regarding the existence of noninvertible finite endomorphisms fixing an isolated singularity. Using a cone construction, we deduce that the anticanonical divisor of any smooth projective variety carrying a noninvertible polarized endomorphism is pseudoeffective.

Our techniques build on Shokurov’s $b$-divisors. We define the notions of nef Weil $b$-divisors and of nef envelopes of $b$-divisors. We relate the latter to the pullback of Weil divisors introduced by de Fernex and Hacon. Using the subadditivity theorem for multiplier ideals with respect to pairs recently obtained by Takagi, we carry over to the isolated singularity case the intersection theory of nef Weil $b$-divisors formerly developed by Boucksom, Favre, and Jonsson in the smooth case.

#### Article information

Source
Duke Math. J., Volume 161, Number 8 (2012), 1455-1520.

Dates
First available in Project Euclid: 22 May 2012

https://projecteuclid.org/euclid.dmj/1337690406

Digital Object Identifier
doi:10.1215/00127094-1593317

Mathematical Reviews number (MathSciNet)
MR2931273

Zentralblatt MATH identifier
1251.14026

#### Citation

Boucksom, Sebastien; de Fernex, Tommaso; Favre, Charles. The volume of an isolated singularity. Duke Math. J. 161 (2012), no. 8, 1455--1520. doi:10.1215/00127094-1593317. https://projecteuclid.org/euclid.dmj/1337690406

#### References

• [BCHM] C. Birkar, P. Cascini, C. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405–468.
• [BDPP] S. Boucksom, J.-P. Demailly, M. Paun, and T. Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, preprint, arXiv:math/0405285v1 [math.AG]
• [BFJ1] S. Boucksom, C. Favre, and M. Jonsson, Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci. 44 (2008), 449–494.
• [BFJ2] S. Boucksom, C. Favre and M. Jonsson. Non-archimedean pluripotential theory over a smooth point, in preparation.
• [Cor] A. Corti, ed., Flips for 3-Folds and 4-Folds, Oxford Lecture Ser. Math. Appl. 35, Oxford Univ. Press, Oxford, 2007.
• [CLS] D. Cox, J. Little, and H. Schenck, Toric Varieties, Grad. Stud. Math. 124, Amer. Math. Soc., Providence, 2011.
• [Cut] S. D. Cutkosky, Irrational asymptotic behaviour of Castelnuovo–Mumford regularity, J. Reine Angew. Math. 522 (2000), 93–103.
• [dFH] T. de Fernex and C. Hacon, Singularities on normal varieties, Compos. Math. 145 (2009), 393–414.
• [DEL] J.-P. Demailly, L. Ein, and R. Lazarsfeld, A subadditivity property of multiplier ideals, Michigan Math. J. 48 (2000), 137–156.
• [ELS] L. Ein, R. Lazarsfeld, and K. E. Smith, Uniform approximation of Abhyankar valuation ideals in smooth function fields, Amer. J. Math. 125 (2003), 409–440.
• [Eis] E. Eisenstein, Generalization of the restriction theorem for multiplier ideals, preprint, arXiv:1001.2841v1 [math.AG]
• [Fak] N. Fakhruddin, Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), 109–122.
• [Fav] C. Favre, Holomorphic self-maps of singular rational surfaces, Publ. Mat. 54 (2010), 389–432.
• [FN] Y. Fujimoto and N. Nakayama, Compact complex surfaces admitting non-trivial surjective endomorphisms, Tohoku Math. J. (2) 57 (2005), 395–426.
• [Fulg] M. Fulger, Local volumes on normal algebraic varieties, preprint, arXiv:1105.2981v1 [math.AG]
• [Fult] W. Fulton, Introduction to Toric Varieties, Ann. of Math. Stud. 131, Princeton Univ. Press, Princeton, 1993.
• [Gan] F. M. Ganter, Properties of −P ⋅ P for Gorenstein surface singularities, Math. Z. 223 (1996), 411–419.
• [Gra] H. Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368.
• [Gro] A. Grothendieck, Éléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, Inst. Hautes Études Sci. Publ. Math. 32, 1967.
• [Ish] S. Ishii, The asymptotic behaviour of plurigenera for a normal isolated singularity, Math. Ann. 286 (1990), 803–812.
• [Isk] V. A. Iskovskikh, b-divisors and Shokurov functional algebras (in Russian), Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 8–20; English translation in Proc. Steklov Inst. Math. 240 (2003), no. 1, 4–15.
• [Izu] S. Izumi, Linear complementary inequalities for orders of germs of analytic functions, Invent. Math. 65 (1981/82), 459–471.
• [Kaw] M. Kawakita, On a comparison of minimal log discrepancies in terms of motivic integration, J. Reine Angew. Math. 620 (2008), 55–65.
• [Knö] F. W. Knöller, 2-dimensionale Singularitäten und Differentialformen, Math. Ann. 206 (1973), 205–213.
• [Kol1] J. Kollár, “Singularities of pairs” in Algebraic Geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math. 62, Part 1, Amer. Math. Soc., Providence, 1997, 221–287.
• [Kol2] J. Kollár, “Exercises in the birational geometry of algebraic varieties” in Analytic and Algebraic Geometry, IAS/Park City Math. Ser. 17, Amer. Math. Soc., Providence, 2010, 495–524.
• [Kol3] J. Kollár, Book on moduli of surfaces, ongoing project, http://math.princeton.edu/~Kollar (accessed 30 April 2012)
• [KoMo] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998.
• [Kür] A. Küronya, A divisorial valuation with irrational volume, J. Algebra 262 (2003), 413–423.
• [KuMa] A. Küronya and C. MacLean, Zariski decomposition of b-divisors, to appear in Math. Z., preprint, arXiv:0807.2809v2 [math.AG]
• [Laz] R. Lazarsfeld, Positivity in Algebraic Geometry, I, II, Ergeb. Math. Grenzgeb. (3) 49, Springer, Berlin, 2004.
• [LM] R. Lazarsfeld and M. Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 783–835.
• [Loo] E. J. N. Looijenga, Isolated Singular Points on Complete Intersections, London Math. Soc. Lecture Note Ser. 77, Cambridge Univ. Press, Cambridge, 1984.
• [Mor] M. Morales, Resolution of quasi-homogeneous singularities and plurigenera, Compos. Math. 64 (1987), 311–327.
• [Nak] N. Nakayama, On complex normal projective surfaces admitting non-isomorphic surjective endomorphisms, preprint, 2008.
• [Oda] T. Oda, Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties, Ergeb. Math. Grenzgeb. (3) 15, Springer, Berlin, 1988.
• [Sak] F. Sakai, Anticanonical models of rational surfaces, Math. Ann. 269 (1984), 389–410.
• [Sch] K. Schwede, Test ideals in non-ℚ-Gorenstein rings, Trans. Amer. Math. Soc. 363 (2011), 5925–5941.
• [Tak1] S. Takagi, Formulas for multiplier ideals on singular varieties, Amer. J. Math. 128 (2006), 1345–1362.
• [Tak2] S. Tagaki, A subadditivity formula for multiplier ideals associated to log pairs, to appear in Proc. Amer. Math. Soc., preprint, arXiv:1103.1179v2 [math.AG]
• [Tsu] H. Tsuchihashi, Higher-dimensional analogues of periodic continued fractions and cusp singularities, Tohoku Math. J. (2) 35 (1983), 607–639.
• [Urb] S. Urbinati, Discrepancies of non-ℚ-Gorenstein varieties, to appear in Mich. Math J., preprint, arXiv:1001.2930v3 [math.AG]
• [Wa] J. Wahl, A characteristic number for links of surface singularities, J. Amer. Math. Soc. 3 (1990), 625–637.
• [Wat1] K. Watanabe, On plurigenera of normal isolated singularities, I, Math. Ann. 250 (1980), 65–94.
• [Wat2] K. Watanabe, “On plurigenera of normal isolated singularities, II” in Complex Analytic Singularities, Adv. Stud. Pure Math. 8, North-Holland, Amsterdam, 1987, 671–685.
• [ZS] O. Zariski and P. Samuel, Commutative Algebra, Vol. II, reprint of the 1960 ed., Grad. Texts in Math. 29, Springer, New York, 1975.
• [dqZ] D.-Q. Zhang, Polarized endomorphisms of uniruled varieties, Compos. Math. 146 (2010), 145–168.
• [swZ] S.-W. Zhang, “Distribution in algebraic dynamics” in Surveys in Differential Geometry, Vol. X, Surv. Differ. Geom. 10, Int. Press, Somerville, Mass., 2006, 381–430.