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.
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.
Keywords: Local volumes, Hilbert-Samuel multiplicity, plurigenera, asymptotic invariants, Okounkov body
Mot 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}, pages = {1793--1847}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {63}, number = {5}, year = {2013}, doi = {10.5802/aif.2815}, zbl = {1297.14015}, mrnumber = {3186509}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2815/} }
TY - JOUR AU - Fulger, Mihai TI - Local volumes of Cartier divisors over normal algebraic varieties JO - Annales de l'Institut Fourier PY - 2013 SP - 1793 EP - 1847 VL - 63 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2815/ DO - 10.5802/aif.2815 LA - en ID - AIF_2013__63_5_1793_0 ER -
%0 Journal Article %A Fulger, Mihai %T Local volumes of Cartier divisors over normal algebraic varieties %J Annales de l'Institut Fourier %D 2013 %P 1793-1847 %V 63 %N 5 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2815/ %R 10.5802/aif.2815 %G en %F AIF_2013__63_5_1793_0
Fulger, Mihai. Local volumes of Cartier divisors over normal algebraic varieties. Annales de l'Institut Fourier, Volume 63 (2013) no. 5, pp. 1793-1847. doi : 10.5802/aif.2815. http://www.numdam.org/articles/10.5802/aif.2815/
[1] The volume of an isolated singularity, 2011 (arXiv: 1011.2847v3 [math.AG])
[2] Asymptotic growth of saturated powers and epsilon multiplicity, Math. Res. Lett., Volume 18 (2011) no. 1, pp. 93-106 | DOI | MR | Zbl
[3] Asymptotic behavior of the length of local cohomology, Canad. J. Math., Volume 57 (2005), pp. 1178-1192 | DOI | MR | Zbl
[4] Singularities on normal varieties, Compos. Math., Volume 2 (2009), pp. 393-414 | DOI | MR | Zbl
[5] Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. (1996) no. 83, pp. 51-93 | DOI | Numdam | MR | Zbl
[6] Complex tori and abelian varieties, 11, SMF/AMS texts and monographs, 2005 | MR | Zbl
[7] Introduction to toric varieties, Annals of Mathematics Studies, 1997 | Zbl
[8] Properties of for Gorenstein surface singularities, Math. Z., Volume 223 (1996) no. 3, pp. 411-419 | DOI | MR | Zbl
[9] Cohomologie locale des faisceaux cohérents et Théorèmes de Lefschetz locaux et globaux(SGA 2) (1962) | MR | Zbl
[10] Boundedness of pluricanonical maps of varieties of general type, Invent. Math., Volume 166 (2006), pp. 1-25 | DOI | MR | Zbl
[11] On the birational automorphisms of varieties of general type, 2010 (arXiv:1011.1464v1 [math.AG])
[12] Algebraic Geometry, Graduate texts in Mathematics, Springer-Verlag, New York, 1977 | MR | Zbl
[13] Algebraic Geometry: An introduction to Birational Geometry of algebraic varieties, Iwanami Shoten, Tokyo, 1977 | Zbl
[14] The asymptotic behavior of plurigenera for a normal isolated singularity, Math. Ann., Volume 286 (1990), pp. 803-812 | DOI | MR | Zbl
[15] A measure of integrity for local analytic algebras, Publ. RIMS, Kyoto Univ., Volume 21 (1985), pp. 719-735 | DOI | MR | Zbl
[16] Introduction to the minimal model problem, Algebraic Geometry, Sendai (1985) (Adv. Stud. Pure Math.), Volume 10, North-Holland, Amsterdam, 1987, pp. 283-360 | MR | Zbl
[17] 2-dimensionale singularitäten und differentialformen, Math. Ann., Volume 206 (1973), pp. 205-213 | DOI | MR | Zbl
[18] Asymptotic cohomological functions on projective varieties, Amer. J. Math., Volume 128 (2006), pp. 1475-1519 | DOI | MR | Zbl
[19] Positivity in Algebraic Geometry I, II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 49, Springer-Verlag, Berlin, 2004 | MR | Zbl
[20] Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4), Volume 42 (2009) no. 5, pp. 783-835 | Numdam | MR | Zbl
[21] Resolution of quasihomogeneous singularities and plurigenera, Compos. Math., Volume 64 (1987), pp. 311-327 | Numdam | MR | Zbl
[22] The pluri–genera of surface singularities, Tôhoku Math. J., Volume 50 (1998), pp. 119-132 | DOI | MR | Zbl
[23] Plurigenera of surface singularities, Nova Science Publishers, Inc., 2000
[24] Izumi’s Theorem, Commutative Algebra, Springer-Verlag, 1989, pp. 407-416 | MR | Zbl
[25] Kodaira dimensions of complements of divisors, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 239-257 | MR | Zbl
[26] Pluricanonical systems on algebraic varieties of general type, Invent. Math., Volume 165 (2006) no. 3, pp. 551-587 | DOI | MR | Zbl
[27] On –plurigenera of not-log–canonical Gorenstein isolated singularities, Proceedings of the AMS, Volume 109 (1990) no. 4, pp. 931-935 | MR | Zbl
[28] Higher-dimensional analogues of periodic continued fractions and cusp singularities, Tohoku Math. J. (2), Volume 35 (1983) no. 4, pp. 607-639 | DOI | MR | Zbl
[29] Pluricanonical systems of projective varieties of general type, v1-v10, 1999–2004 (arXiv: math.AG/9909021)
[30] Discrepancies of non-Gorenstein varieties, 2010 (arXiv:1001.2930 [math.AG])
[31] The behavior of the second pluri–genus of normal surface singularities of type ,, , , and , Math. J. Okayama Univ., Volume 45 (2003), pp. 45-58 | MR | Zbl
[32] A characteristic number for links of surface singularities, Journal of The AMS, Volume 3 (1990) no. 3, pp. 625-637 | MR | Zbl
[33] On plurigenera of normal isolated singularities. I, Math. Ann., Volume 250 (1980), pp. 65-94 | DOI | MR | Zbl
[34] Two theorems in higher dimensional singularities, Math. Ann., Volume 231 (1977), pp. 44-59 | DOI | MR | Zbl
Cited by Sources: