Differentiability of volumes of divisors and a problem of Teissier
Authors:
Sébastien Boucksom, Charles Favre and Mattias Jonsson
Journal:
J. Algebraic Geom. 18 (2009), 279-308
DOI:
https://doi.org/10.1090/S1056-3911-08-00490-6
Published electronically:
April 23, 2008
MathSciNet review:
2475816
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We give an algebraic construction of the positive intersection products of pseudo-effective classes and use them to prove that the volume function on the Néron–Severi space of a projective variety is $\mathcal {C}^1$-differentiable, expressing its differential as a positive intersection product. We also relate the differential to the restricted volumes. We then apply our differentiability result to prove an algebro-geometric version of the Diskant inequality in convex geometry, allowing us to characterize the equality case of the Khovanskii–Teissier inequalities for nef and big classes.
References
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References
- Aluffi, P., Modification systems and integration in their Chow groups. Selecta Math. (N.S.) 11 (2005), no. 2, 155–202. MR 2183846 (2006k:14007)
- Bonnesen, T., Les problèmes des isopérimètres et des isépiphanes. 175 p. Paris, Gauthier-Villars (Collection de monographies sur la théorie des fonctions). (1929)
- Boucksom, S., On the volume of a line bundle. Internat. J. Math. 13 (2002), no. 10, 1043–1063. MR 1945706 (2003j:32025)
- Boucksom, S., Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 1, 45–76. MR 2050205 (2005i:32018)
- Boucksom, S., Demailly, J.-P., Paun, M., Peternell, T., The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. Preprint, 2004, arxiv.org/abs/math/0405285.
- Boucksom, S., Favre, C. and Jonsson M., Degree growth of meromorphic surface maps. To appear in Duke Math. J.
- Boucksom, S., Favre, C. and Jonsson M., Valuations and plurisubharmonic singularities. Preprint, 2007, arxiv.org/abs/math/0702487.
- Cantat, S., Sur les groupes de transformations birationnelles des surfaces. Preprint.
- Demailly, J.-P., Ein, L., Lazarsfeld, R., A subadditivity property of multiplier ideals. Michigan Math. J., 48 (2000), 137–156. MR 1786484 (2002a:14016)
- Diskant, V., A generalization of Bonnesen’s inequalities. Soviet Math. Dokl., 14 (1973), 1728–1731.
- Ein, L., Lazarsfeld, R., Mustaţǎ, M, Nakamaye, M., Popa, M., Asymptotic invariants of line bundles. Pure Appl. Math. Q., 1 (2005), no. 2, 379–403. MR 2194730 (2007e:14007)
- Ein, L., Lazarsfeld, R., Mustaţǎ, M, Nakamaye, M., Popa, M., Asymptotic invariants of base loci. Ann. Inst. Fourier, 56 (2006), 1701–1734. MR 2282673
- Ein, L., Lazarsfeld, R., Mustaţǎ, M, Nakamaye, M., Popa, M., Restricted volumes and base loci of linear series. Preprint, 2006, arxiv.org/abs/math/0607221.
- Favre, C. and Jonsson M., The valuative tree. Lecture Notes in Mathematics, 1853. Springer-Verlag, Berlin, 2004. MR 2097722 (2006a:13008)
- Fujita, T., Approximating Zariski decomposition of big line bundles. Kodai, Math. J., 17 (1994), no. 1, 1–3. MR 1262949 (95c:14053)
- Fulton, W., Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)
- Hacon, C., McKernan, J., Boundedness of pluricanonical maps of varieties of general type. Invent. Math. 166 (2006), no. 1, 1–25. MR 2242631 (2007e:14022)
- Iskovskikh, V. A., $b$-divisors and Shokurov functional algebras. (Russian) Proc. Steklov Inst. Math. 2003, no. 1 (240), 4–15. MR 1993746 (2004g:14019)
- Lazarsfeld, R., Positivity in algebraic geometry. I and II. Ergebnisse der Mathematik und ihrer Grenzgebiete, 48 and 49. Springer-Verlag, Berlin, 2004. MR 2095471 (2005k:14001a)
- Manin, Y., Cubic forms. Algebra, geometry, arithmetic. North-Holland Mathematical Library, 4. North-Holland Publishing Co., Amsterdam, 1986. MR 833513 (87d:11037)
- Nakayama, N., Zariski-decomposition and abundance. MSJ Memoirs, 14. Mathematical Society of Japan, Tokyo, 2004. MR 2104208 (2005h:14015)
- Oda, T., Convex bodies and algebraic geometry. An introduction to the theory of toric varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, 15. Springer-Verlag, Berlin, 1988. MR 922894 (88m:14038)
- Oda, T., Park, H.S., Linear Gale transforms and Gel′fand-Kapranov-Zelevinskij decompositions. Tohoku Math. J. (2) 43 (1991), no. 3, 375–399. MR 1117211 (92d:14042)
- Prokhorov, Yu. G., On the Zariski decomposition problem. Proc. Steklov Inst. Math. 2003, no. 1 (240), 37–65. MR 1993748 (2004m:14021)
- Schneider, R., Convex bodies: the Brunn-Minkowski theory. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, 1993. MR 1216521 (94d:52007)
- Shokurov, V.V., Prelimiting flips. Proc. Steklov Inst. Math. 2003, no. 1 (240), 75–213. MR 1993750 (2004k:14024)
- Takayama, S., Pluricanonical systems on algebraic varieties of general type. Invent. Math. 165 (2006), no. 3, 551–587. MR 2242627
- Teissier, B., Du théorème de l’index de Hodge aux inégalités isopérimétriques. C. R. Acad. Sci. Paris Sér. A-B, 288 (1979), no. 4, A287–A289. MR 524795 (80k:14014)
- Teissier, B., Bonnesen-type inequalities in algebraic geometry. I. Introduction to the problem. In Seminar on Differential Geometry, pp. 85–105. Ann. Math. Stud., 102. Princeton University Press, 1982. MR 645731 (83d:52010)
- Teissier, B., Monômes, volumes et multiplicités. In Introduction à la théorie des singularités, II, pp. 127-141. Travaux en Cours, 37. Hermann, Paris, 1988 MR 1074593 (92e:14003)
- Vaquie, M., Valuations. Resolution of singularities (Obergurgl, 1997), 539–590, Progr. Math., 181, Birkhäuser, Basel, 2000. MR 1748635 (2001i:13005)
- Voisin, C., Hodge theory and complex algebraic geometry. I. Cambridge Studies in Advanced Mathematics, 76. Cambridge University Press, Cambridge, 2002. MR 1967689 (2004d:32020)
- Zariski, O., Samuel, P., Commutative algebra. Vol. 2. Graduate Texts in Mathematics, No. 29. Springer-Verlag, New York-Heidelberg-Berlin (1975)
Additional Information
Sébastien Boucksom
Affiliation:
CNRS-Université Paris 7 Institut de Mathématiques, F-75251 Paris Cedex 05, France
MR Author ID:
688226
Email:
boucksom@math.jussieu.fr
Charles Favre
Affiliation:
CNRS-Université Paris 7, Institut de Mathématiques, F-75251 Paris Cedex 05, France
MR Author ID:
641179
Email:
favre@math.jussieu.fr
Mattias Jonsson
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043 USA and Department of Mathematics, KTH, SE-100 44 Stockholm, Sweden
MR Author ID:
631360
Email:
mattiasj@umich.edu; mattiasj@kth.se
Received by editor(s):
December 21, 2006
Received by editor(s) in revised form:
April 16, 2007
Published electronically:
April 23, 2008
Additional Notes:
The second author was supported by the Japanese Society for the Promotion of Science. The third author was supported by NSF Grant No. DMS-0449465, the Swedish Science Council, and the Gustafsson Foundation.
