Local volumes of Cartier divisors over normal algebraic varieties  [ Volumes locaux de diviseurs de Cartier sur des variétés algébriques normales ]
Annales de l'Institut Fourier, Tome 63 (2013) no. 5, p. 1793-1847
Dans cet article, nous étudions une notion de volume local pour les diviseurs de Cartier sur des éclatements arbitraires de variétés algébriques complexes normales de dimension supérieure à un, avec un point distingué. Nous appliquons cela pour étudier un invariant de singularités isolées normales, en généralisant un volume défini par J. Wahl dans le cas des surfaces. Nous comparons également cet invariant à celui obtenu dans les travaux récents de T. de Fernex, S. Boucksom, et C. Favre.
In this paper we study a notion of local volume for Cartier divisors on arbitrary blow-ups of normal complex algebraic varieties of dimension greater than one, with a distinguished point. We apply this to study an invariant for normal isolated singularities, generalizing a volume defined by J. Wahl for surfaces. We also compare this generalization to a different one arising in recent work of T. de Fernex, S. Boucksom, and C. Favre.
DOI : https://doi.org/10.5802/aif.2815
Classification:  14E05,  14E15,  14B05,  14B15,  32S05
Mots clés: Volumes locaux, multiplicité de Hilbert-Samuel, plurigenres, invariants asymptotiques, corps de Okounkov
@article{AIF_2013__63_5_1793_0,
     author = {Fulger, Mihai},
     title = {Local volumes of Cartier divisors over normal algebraic varieties},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {5},
     year = {2013},
     pages = {1793-1847},
     doi = {10.5802/aif.2815},
     mrnumber = {3186509},
     zbl = {1297.14015},
     language = {en},
     url = {http://http://www.numdam.org/item/AIF_2013__63_5_1793_0}
}
Fulger, Mihai. Local volumes of Cartier divisors over normal algebraic varieties. Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 1793-1847. doi : 10.5802/aif.2815. http://www.numdam.org/item/AIF_2013__63_5_1793_0/

[1] Boucksom, S.; De Fernex, T.; Favre, C. The volume of an isolated singularity (2011) (arXiv: 1011.2847v3 [math.AG])

[2] Cutkosky, S. D. Asymptotic growth of saturated powers and epsilon multiplicity, Math. Res. Lett., Tome 18 (2011) no. 1, pp. 93-106 | Article | MR 2770584 | Zbl 1238.13012

[3] Cutkosky, S. D.; Hà, H. T.; Srinivasan, H.; Theodorescu, E. Asymptotic behavior of the length of local cohomology, Canad. J. Math., Tome 57 (2005), pp. 1178-1192 | Article | MR 2178557 | Zbl 1095.13015

[4] De Fernex, T.; Hacon, C. D. Singularities on normal varieties, Compos. Math., Tome 2 (2009), pp. 393-414 | Article | MR 2501423 | Zbl 1179.14003

[5] De Jong, A. J. Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. (1996) no. 83, pp. 51-93 | Article | Numdam | MR 1423020 | Zbl 0916.14005

[6] Debarre, O. Complex tori and abelian varieties, SMF/AMS texts and monographs Tome 11 (2005) | MR 2158864 | Zbl 1078.14061

[7] Fulton, W. Introduction to toric varieties, Annals of Mathematics Studies (1997) | Zbl 0813.14039

[8] Ganter, F. M. Properties of -P·P for Gorenstein surface singularities, Math. Z., Tome 223 (1996) no. 3, pp. 411-419 | Article | MR 1417852 | Zbl 0901.14006

[9] Grothendieck, A. Cohomologie locale des faisceaux cohérents et Théorèmes de Lefschetz locaux et globaux(SGA 2), Séminaire de Géométrie Algébrique du Bois Marie (1962) | MR 2171939 | Zbl 0197.47202

[10] Hacon, C.; Mckernan, J. Boundedness of pluricanonical maps of varieties of general type, Invent. Math., Tome 166 (2006), pp. 1-25 | Article | MR 2242631 | Zbl 1121.14011

[11] Hacon, C.; Mckernan, J.; Xu, C. On the birational automorphisms of varieties of general type (2010) (arXiv:1011.1464v1 [math.AG])

[12] Hartshorne, R. Algebraic Geometry, Springer-Verlag, New York, Graduate texts in Mathematics (1977) | MR 463157 | Zbl 0531.14001

[13] Iitaka, S. Algebraic Geometry: An introduction to Birational Geometry of algebraic varieties, Iwanami Shoten, Tokyo (1977) | Zbl 0656.14001

[14] Ishii, S. The asymptotic behavior of plurigenera for a normal isolated singularity, Math. Ann., Tome 286 (1990), pp. 803-812 | Article | MR 1045403 | Zbl 0668.14002

[15] Izumi, S. A measure of integrity for local analytic algebras, Publ. RIMS, Kyoto Univ., Tome 21 (1985), pp. 719-735 | Article | MR 817161 | Zbl 0587.32016

[16] Kawamata, Y.; Matsuda, K.; Matsuki, K. Introduction to the minimal model problem, Algebraic Geometry, Sendai (1985), North-Holland, Amsterdam (Adv. Stud. Pure Math.) Tome 10 (1987), pp. 283-360 | MR 946243 | Zbl 0672.14006

[17] Knöller, F. W. 2-dimensionale singularitäten und differentialformen, Math. Ann., Tome 206 (1973), pp. 205-213 | Article | MR 340260 | Zbl 0258.32002

[18] Küronya, A. Asymptotic cohomological functions on projective varieties, Amer. J. Math., Tome 128 (2006), pp. 1475-1519 | Article | MR 2275909 | Zbl 1114.14005

[19] Lazarsfeld, R. Positivity in Algebraic Geometry I, II, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Tome 49 (2004) | MR 2095471 | Zbl 0633.14016

[20] Lazarsfeld, R.; Mustaţă, M. Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4), Tome 42 (2009) no. 5, pp. 783-835 | Numdam | MR 2571958 | Zbl 1182.14004

[21] Morales, M. Resolution of quasihomogeneous singularities and plurigenera, Compos. Math., Tome 64 (1987), pp. 311-327 | Numdam | MR 918415 | Zbl 0648.14005

[22] Okuma, T. The pluri–genera of surface singularities, Tôhoku Math. J., Tome 50 (1998), pp. 119-132 | Article | MR 1604636 | Zbl 0928.14023

[23] Okuma, T. Plurigenera of surface singularities, Nova Science Publishers, Inc. (2000)

[24] Rees, D.; Hochster, M.; Huneke, C.; Sally, J. D. Izumi’s Theorem, Commutative Algebra, Springer-Verlag (1989), pp. 407-416 | MR 1015531 | Zbl 0741.13011

[25] Sakai, F. Kodaira dimensions of complements of divisors, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo (1977), pp. 239-257 | MR 590433 | Zbl 0375.14009

[26] Takayama, S. Pluricanonical systems on algebraic varieties of general type, Invent. Math., Tome 165 (2006) no. 3, pp. 551-587 | Article | MR 2242627 | Zbl 1108.14031

[27] Tomari, M.; Watanabe, K. On L 2 –plurigenera of not-log–canonical Gorenstein isolated singularities, Proceedings of the AMS, Tome 109 (1990) no. 4, pp. 931-935 | MR 1021213 | Zbl 0714.32010

[28] Tsuchihashi, H. Higher-dimensional analogues of periodic continued fractions and cusp singularities, Tohoku Math. J. (2), Tome 35 (1983) no. 4, pp. 607-639 | Article | MR 721966 | Zbl 0585.14004

[29] Tsuji, H. Pluricanonical systems of projective varieties of general type, v1-v10 (1999–2004) (arXiv: math.AG/9909021)

[30] Urbinati, S. Discrepancies of non--Gorenstein varieties (2010) (arXiv:1001.2930 [math.AG])

[31] Wada, K. The behavior of the second pluri–genus of normal surface singularities of type * A n , * D n , * E n , * A n ˜, * D n ˜ and * E n ˜, Math. J. Okayama Univ., Tome 45 (2003), pp. 45-58 | MR 2038838 | Zbl 1061.14030

[32] Wahl, J. A characteristic number for links of surface singularities, Journal of The AMS, Tome 3 (1990) no. 3, pp. 625-637 | MR 1044058 | Zbl 0743.14026

[33] Watanabe, K. On plurigenera of normal isolated singularities. I, Math. Ann., Tome 250 (1980), pp. 65-94 | Article | MR 581632 | Zbl 0414.32005

[34] Yau, S. S. T. Two theorems in higher dimensional singularities, Math. Ann., Tome 231 (1977), pp. 44-59 | Article | MR 492389 | Zbl 0343.32010