no. 2
Restricted volumes of effective divisors
[Volumes restreints de diviseurs effectifs]
Bulletin de la Société Mathématique de France, Tome 144 (2016) no. 2, pp. 299-337

We study the restricted volume of effective divisors, its properties and the relationship with the related notion of reduced volume, defined via multiplier ideals, and with the asymptotic intersection number. We build upon the fundamental work of Lazarsfeld and Mustăţa relating the restricted volume of big divisors to the volume of the associated Okounkov body. We extend their constructions and results to the case of effective divisors, recovering some results of Kaveh and Khovanskii, proving a Fujita-type approximation in this larger setting and studying the restricted volume function. In order to relate the reduced volume and the asymptotic intersection number we investigate a boundedness property of asymptotic multiplier ideals and prove it holds, for instance, for finitely generated divisors. In this way we obtain also a complete picture for the canonical divisor of an arbitrary smooth projective variety and for nef divisors on varieties of dimension at most 3.

Nous étudions le volume restreint de diviseurs effectifs, ses propriétés et la relation avec le volume réduit, défini en termes d'idéaux multiplicateurs, ainsi qu'avec le nombre d'intersection asymptotique. Nous nous basons sur le travail fondamental de Lazarsfeld et Mustăţa qui met en relation le volume restreint d'un diviseur gros avec le volume du corps d'Okounkov associé. Nous étendons leurs constructions et résultats au cas des diviseurs effectifs. Nous retrouvons en particulier certains résultats de Kaveh et Khovanskii, démontrons une approximation de Fujita dans ce cadre plus large et étudions la fonction volume restreint. Afin de relier le volume réduit et le nombre d'intersection asymptotique nous étudions une propriété d'encadrement des idéaux multiplicateurs asymptotiques et montrons qu'elle est valable, par exemple, dans le cas des diviseurs de type fini. De cette manière nous obtenons une description complète pour le diviseur canonique d'une variété lisse et projective quelconque et pour les diviseurs nef sur les variétés de dimension au plus 3.

Publié le :
DOI : 10.24033/bsmf.2715
Classification : 14C20, 14F18
Keywords: Asymptotic intersection number, canonical divisor, Fujita approximation, (multi)graded series, multiplier ideal, Okounkov body, restricted volumes. 
@article{BSMF_2016__144_2_299_0,
     author = {Biagio, Lorenzo Di and Pacienza, Gianluca},
     title = {Restricted volumes of effective divisors},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {299--337},
     year = {2016},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {144},
     number = {2},
     doi = {10.24033/bsmf.2715},
     mrnumber = {3499083},
     zbl = {1401.14038},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2715/}
}
TY  - JOUR
AU  - Biagio, Lorenzo Di
AU  - Pacienza, Gianluca
TI  - Restricted volumes of effective divisors
JO  - Bulletin de la Société Mathématique de France
PY  - 2016
SP  - 299
EP  - 337
VL  - 144
IS  - 2
PB  - Société mathématique de France
UR  - https://www.numdam.org/articles/10.24033/bsmf.2715/
DO  - 10.24033/bsmf.2715
LA  - en
ID  - BSMF_2016__144_2_299_0
ER  - 
%0 Journal Article
%A Biagio, Lorenzo Di
%A Pacienza, Gianluca
%T Restricted volumes of effective divisors
%J Bulletin de la Société Mathématique de France
%D 2016
%P 299-337
%V 144
%N 2
%I Société mathématique de France
%U https://www.numdam.org/articles/10.24033/bsmf.2715/
%R 10.24033/bsmf.2715
%G en
%F BSMF_2016__144_2_299_0
Biagio, Lorenzo Di; Pacienza, Gianluca. Restricted volumes of effective divisors. Bulletin de la Société Mathématique de France, Tome 144 (2016) no. 2, pp. 299-337. doi: 10.24033/bsmf.2715

Bădescu, Lucian Algebraic surfaces, Universitext, Springer, New York, 2001, 258 pages (ISBN: 0-387-98668-5) | MR | Zbl | DOI

Boucksom, Sébastien; Demailly, Jean-Pierre; Păun, Mihai; Peternell, Thomas The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom., Volume 22 (2013), pp. 201-248 (ISSN: 1056-3911) | MR | Zbl | DOI

Cacciola, Salvatore Asymptotic invariants of line bundles, semiampleness and finite generation, Ph. D. Thesis , Università degli studi Roma Tre (2008) ( http://ricerca.mat.uniroma3.it/users/lopez/Tesi-Cacciola.pdf )

Cacciola, Salvatore; Di Biagio, Lorenzo Asymptotic base loci on singular varieties, Math. Z., Volume 275 (2013), pp. 151-166 (ISSN: 0025-5874) | MR | Zbl | DOI

Debarre, Olivier Higher-dimensional algebraic geometry, Universitext, Springer, New York, 2001, 233 pages (ISBN: 0-387-95227-6) | MR | Zbl | DOI

Demailly, Jean-Pierre; Ein, Lawrence; Lazarsfeld, Robert A subadditivity property of multiplier ideals, Michigan Math. J., Volume 48 (2000), pp. 137-156 (ISSN: 0026-2285) | MR | Zbl | DOI

Ein, Lawrence; Lazarsfeld, Robert; Mustaţă, Mircea; Nakamaye, Michael; Popa, Mihnea Asymptotic invariants of line bundles, Pure Appl. Math. Q., Volume 1 (2005), pp. 379-403 (ISSN: 1558-8599) | MR | Zbl | DOI

Ein, Lawrence; Lazarsfeld, Robert; Mustaţă, Mircea; Nakamaye, Michael; Popa, Mihnea Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble), Volume 56 (2006), pp. 1701-1734 (ISSN: 0373-0956) | Numdam | MR | Zbl | DOI

Ein, Lawrence; Lazarsfeld, Robert; Mustaţă, Mircea; Nakamaye, Michael; Popa, Mihnea Restricted volumes and base loci of linear series, Amer. J. Math., Volume 131 (2009), pp. 607-651 (ISSN: 0002-9327) | MR | Zbl | DOI

de Fernex, Tommaso; Hacon, Christopher D. Singularities on normal varieties, Compos. Math., Volume 145 (2009), pp. 393-414 (ISSN: 0010-437X) | MR | Zbl | DOI

Hacon, Christopher D.; McKernan, James Boundedness of pluricanonical maps of varieties of general type, Invent. math., Volume 166 (2006), pp. 1-25 (ISSN: 0020-9910) | MR | Zbl | DOI

Hartshorne, Robin Algebraic geometry, Graduate Texts in Math., 52, Springer, New York-Heidelberg, 1977, 496 pages (ISBN: 0-387-90244-9) | MR | Zbl | DOI

Jow, Shin-Yao Multigraded Fujita approximation, Pacific J. Math., Volume 251 (2011), pp. 331-336 (ISSN: 0030-8730) | MR | Zbl | DOI

Kaveh, Kiumars; Khovanskii, A. G. Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math., Volume 176 (2012), pp. 925-978 (ISSN: 0003-486X) | MR | Zbl | DOI

Kawamata, Yujiro Pluricanonical systems on minimal algebraic varieties, Invent. math., Volume 79 (1985), pp. 567-588 (ISSN: 0020-9910) | MR | Zbl | DOI

Kawamata, Yujiro; Matsuda, Katsumi; Matsuki, Kenji, Algebraic geometry, Sendai, 1985 (Adv. Stud. Pure Math.), Volume 10, North-Holland, Amsterdam, 1987, pp. 283-360 | MR | Zbl | DOI

Lazarsfeld, Robert Positivity in algebraic geometry. I, Ergebn. Math. Grenzg., 48, Springer, Berlin, 2004, 387 pages (ISBN: 3-540-22533-1) | MR | Zbl | DOI

Lazarsfeld, Robert Positivity in algebraic geometry. II, Ergebn. Math. Grenzg., 49, Springer, Berlin, 2004, 385 pages (ISBN: 3-540-22534-X) | MR | Zbl | DOI

Lazarsfeld, Robert; Mustaţă, Mircea Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér., Volume 42 (2009), pp. 783-835 (ISSN: 0012-9593) | MR | Zbl | Numdam | DOI

Mourougane, Christophe; Russo, Francesco Some remarks on nef and good divisors on an algebraic variety, C. R. Acad. Sci. Paris Sér. I Math., Volume 325 (1997), pp. 499-504 (ISSN: 0764-4442) | MR | Zbl | DOI

Nakayama, Noboru Zariski-decomposition and abundance, MSJ Memoirs, 14, Mathematical Society of Japan, Tokyo, 2004, 277 pages (ISBN: 4-931469-31-0) | MR | Zbl

Okounkov, Andrei Brunn-Minkowski inequality for multiplicities, Invent. math., Volume 125 (1996), pp. 405-411 (ISSN: 0020-9910) | MR | Zbl | DOI

Okounkov, Andrei, The orbit method in geometry and physics (Marseille, 2000) (Progr. Math.), Volume 213, Birkhäuser, 2003, pp. 329-347 | MR | Zbl | DOI

Pacienza, Gianluca; Takayama, Shigeharu On volumes along subvarieties of line bundles with nonnegative Kodaira-Iitaka dimension, Michigan Math. J., Volume 60 (2011), pp. 35-49 (ISSN: 0026-2285) | MR | Zbl | DOI

Russo, Francesco A characterization of nef and good divisors by asymptotic multiplier ideals, Bull. Belg. Math. Soc. Simon Stevin, Volume 16 (2009), pp. 943-951 http://projecteuclid.org/euclid.bbms/1260369408 (ISSN: 1370-1444) | MR | Zbl

Takayama, Shigeharu Pluricanonical systems on algebraic varieties of general type, Invent. math., Volume 165 (2006), pp. 551-587 (ISSN: 0020-9910) | MR | Zbl | DOI

Cité par Sources :