zbMATH — the first resource for mathematics

Differentiability of volumes of divisors and a problem of Teissier. (English) Zbl 1162.14003
Let $$L$$ be a line bundle on an $$n$$-dimensional projective variety $$X$$. The volume of $$L$$ measures the rate of growth of the dimensions of the spaces of global sections $$H^0(X,kL)$$. It is defined by $\text{vol}(L)=\text{lim sup}_{k\to \infty} \frac {n!}{k^n}h^0(X,kL).$ It is known that $$\text{vol}(L)>0$$ if and only if $$H^0(X,kL)$$ defines a birational map for some $$k>0$$. In this case we say that $$L$$ is big. It is also the case that the volume of $$L$$ only depends on its numerical class $$[L]\in N^1(X)$$ and that it defines a continuous function on $$N^1(X)$$ such that $$\text{vol}^{1/n}$$ is concave and homogeneous of degree $$1$$. In the article under review, it is shown that the volume function is $$\mathcal C^1$$-differentiable on the big cone of $$N^1(X)$$ and for any $$\alpha, \gamma \in N^1(X)$$ such that $$\alpha$$ is big, we have $\frac d{dt}\Big| _{t=0} \text{vol}(\alpha +t \gamma )=n<\alpha ^{n-1}>\cdot \gamma .$ Here $$<\alpha ^{n-1}>\in N^1(X)^*$$ denotes the positive intersection product of the big class $$\alpha$$. When $$D$$ is a prime divisor and $$\gamma$$ is the numerical class of $$D$$, the authors also show that $$<\alpha ^{n-1}>\cdot \gamma$$ is given by the restricted volume $$\text{vol }_{X|D}(L)=\text{lim sup}_{k\to \infty} \frac {(n-1)!}{k^{n-1}}h^0(X|V,kL)$$ where $$h^0(X|V,kL)$$ denotes the dimension of the image of the restriction homomorphism $$H^0(X,kL)\to H^0(V,kL|_V)$$. The authors then show that the function $$\alpha \to (\alpha ^n)^{1/n}$$ is strictly concave on the big and nef cone of $$N^1(X)$$, they give a result characterizing the equality case of the Khovanskii-Teissier inequalities and they prove an algebro-geometric version of the Diskant inequality in convex geometry.

MSC:
 14C20 Divisors, linear systems, invertible sheaves
Keywords:
volumes of divisors
Full Text:
References:
 [1] Paolo Aluffi, Modification systems and integration in their Chow groups, Selecta Math. (N.S.) 11 (2005), no. 2, 155 – 202. · Zbl 1090.14001 [2] Bonnesen, T., Les problèmes des isopérimètres et des isépiphanes. 175 p. Paris, Gauthier-Villars (Collection de monographies sur la théorie des fonctions). (1929) · JFM 55.0431.08 [3] Sébastien Boucksom, On the volume of a line bundle, Internat. J. Math. 13 (2002), no. 10, 1043 – 1063. · Zbl 1101.14008 [4] Sébastien Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 1, 45 – 76 (English, with English and French summaries). · Zbl 1054.32010 [5] 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. [6] Boucksom, S., Favre, C. and Jonsson M., Degree growth of meromorphic surface maps. To appear in Duke Math. J. · Zbl 1185.32009 [7] Boucksom, S., Favre, C. and Jonsson M., Valuations and plurisubharmonic singularities. Preprint, 2007, arxiv.org/abs/math/0702487. · Zbl 1146.32017 [8] Cantat, S., Sur les groupes de transformations birationnelles des surfaces. Preprint. · Zbl 1233.14011 [9] Jean-Pierre Demailly, Lawrence Ein, and Robert Lazarsfeld, A subadditivity property of multiplier ideals, Michigan Math. J. 48 (2000), 137 – 156. Dedicated to William Fulton on the occasion of his 60th birthday. · Zbl 1077.14516 [10] Diskant, V., A generalization of Bonnesen’s inequalities. Soviet Math. Dokl., 14 (1973), 1728-1731. · Zbl 0298.52009 [11] L. Ein, R. Lazarsfeld, M. Mustaţǎ, M. Nakamaye, and M. Popa, Asymptotic invariants of line bundles, Pure Appl. Math. Q. 1 (2005), no. 2, Special Issue: In memory of Armand Borel., 379 – 403. · Zbl 1139.14008 [12] Lawrence Ein, Robert Lazarsfeld, Mircea Mustaţă, Michael Nakamaye, and Mihnea Popa, Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 6, 1701 – 1734 (English, with English and French summaries). · Zbl 1127.14010 [13] Ein, L., Lazarsfeld, R., Mustaţa, M, Nakamaye, M., Popa, M., Restricted volumes and base loci of linear series. Preprint, 2006, arxiv.org/abs/math/0607221. · Zbl 1179.14006 [14] Charles Favre and Mattias Jonsson, The valuative tree, Lecture Notes in Mathematics, vol. 1853, Springer-Verlag, Berlin, 2004. · Zbl 1064.14024 [15] Takao Fujita, Approximating Zariski decomposition of big line bundles, Kodai Math. J. 17 (1994), no. 1, 1 – 3. · Zbl 0814.14006 [16] William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. · Zbl 0885.14002 [17] Christopher D. Hacon and James McKernan, Boundedness of pluricanonical maps of varieties of general type, Invent. Math. 166 (2006), no. 1, 1 – 25. · Zbl 1121.14011 [18] V. A. Iskovskikh, \?-divisors and Shokurov functional algebras, Tr. Mat. Inst. Steklova 240 (2003), no. Biratsion. Geom. Lineĭn. Sist. Konechno Porozhdennye Algebry, 8 – 20 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 1(240) (2003), 4 – 15. · Zbl 1081.14022 [19] Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. [20] Yu. I. Manin, Cubic forms, 2nd ed., North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986. Algebra, geometry, arithmetic; Translated from the Russian by M. Hazewinkel. · Zbl 0582.14010 [21] Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. · Zbl 1061.14018 [22] Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. · Zbl 0628.52002 [23] Tadao Oda and Hye Sook Park, Linear Gale transforms and Gel$$^{\prime}$$fand-Kapranov-Zelevinskij decompositions, Tohoku Math. J. (2) 43 (1991), no. 3, 375 – 399. · Zbl 0782.52006 [24] Yu. G. Prokhorov, On the Zariski decomposition problem, Tr. Mat. Inst. Steklova 240 (2003), no. Biratsion. Geom. Lineĭn. Sist. Konechno Porozhdennye Algebry, 43 – 72 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 1(240) (2003), 37 – 65. · Zbl 1092.14024 [25] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. · Zbl 0798.52001 [26] V. V. Shokurov, Prelimiting flips, Tr. Mat. Inst. Steklova 240 (2003), no. Biratsion. Geom. Lineĭn. Sist. Konechno Porozhdennye Algebry, 82 – 219; English transl., Proc. Steklov Inst. Math. 1(240) (2003), 75 – 213. · Zbl 1082.14019 [27] Shigeharu Takayama, Pluricanonical systems on algebraic varieties of general type, Invent. Math. 165 (2006), no. 3, 551 – 587. · Zbl 1108.14031 [28] Bernard Teissier, Du théorème de l’index de Hodge aux inégalités isopérimétriques, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 4, A287 – A289 (French, with English summary). · Zbl 0406.14011 [29] B. Teissier, Bonnesen-type inequalities in algebraic geometry. I. Introduction to the problem, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 85 – 105. · Zbl 0494.52009 [30] Bernard Teissier, Monômes, volumes et multiplicités, Introduction à la théorie des singularités, II, Travaux en Cours, vol. 37, Hermann, Paris, 1988, pp. 127 – 141 (French). · Zbl 0698.14001 [31] Michel Vaquié, Valuations, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 539 – 590 (French). · Zbl 1003.13001 [32] Claire Voisin, Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2002. Translated from the French original by Leila Schneps. · Zbl 1032.14001 [33] Zariski, O., Samuel, P., Commutative algebra. Vol. 2. Graduate Texts in Mathematics, No. 29. Springer-Verlag, New York-Heidelberg-Berlin (1975) · Zbl 0313.13001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.