We study numerical restricted volumes of classes on compact Kähler manifolds, as introduced by Boucksom. Inspired by work of Ein–Lazarsfeld–Mustaţă–Nakamaye–Popa on restricted volumes of line bundles on projective manifolds, we pose a natural conjecture to the effect that irreducible components of the non-Kähler locus of a big class should have vanishing numerical restricted volume. We prove this conjecture when the class has a Zariski decomposition, and give several applications.
Nous étudions les volumes restreints numériques de classes sur des variétés kähleriennes compactes, introduits par Boucksom. Inspirés par les travaux de Ein–Lazarsfeld–Mustaţă–Nakamaye–Popa sur les volumes restreints de fibrés en droites sur des variétés projectives, nous proposons la conjecture naturelle que les composantes irréductibles du lieu non-kählerien d’une classe grosse ont un volume restreint numérique identiquement nul. Nous établissons cette conjecture sous l’hypothèse que la classe admet une décomposition de Zariski, puis donnons plusieurs applications.
@article{AFST_2022_6_31_3_907_0, author = {Collins, Tristan C. and Tosatti, Valentino}, title = {Restricted volumes on {K\"ahler} manifolds}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {907--947}, publisher = {Universit\'e Paul Sabatier, Toulouse}, volume = {Ser. 6, 31}, number = {3}, year = {2022}, doi = {10.5802/afst.1708}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/afst.1708/} }
TY - JOUR AU - Collins, Tristan C. AU - Tosatti, Valentino TI - Restricted volumes on Kähler manifolds JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2022 SP - 907 EP - 947 VL - 31 IS - 3 PB - Université Paul Sabatier, Toulouse UR - http://archive.numdam.org/articles/10.5802/afst.1708/ DO - 10.5802/afst.1708 LA - en ID - AFST_2022_6_31_3_907_0 ER -
%0 Journal Article %A Collins, Tristan C. %A Tosatti, Valentino %T Restricted volumes on Kähler manifolds %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2022 %P 907-947 %V 31 %N 3 %I Université Paul Sabatier, Toulouse %U http://archive.numdam.org/articles/10.5802/afst.1708/ %R 10.5802/afst.1708 %G en %F AFST_2022_6_31_3_907_0
Collins, Tristan C.; Tosatti, Valentino. Restricted volumes on Kähler manifolds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 3, pp. 907-947. doi : 10.5802/afst.1708. http://archive.numdam.org/articles/10.5802/afst.1708/
[1] Zariski chambers, volumes, and stable base loci, J. Reine Angew. Math., Volume 576 (2004), pp. 209-233 | MR | Zbl
[2] Cônes positifs des variétés complexes compactes, Ph. D. Thesis, Institut Fourier Grenoble (France) (2002)
[3] On the volume of a line bundle, Int. J. Math., Volume 13 (2002) no. 10, pp. 1043-1063 | DOI | MR | Zbl
[4] Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 1, pp. 45-76 | DOI | Numdam | MR | Zbl
[5] Augmented base loci and restricted volumes on normal varieties, Math. Z., Volume 278 (2014) no. 3, pp. 3-4 | MR | Zbl
[6] The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebr. Geom., Volume 22 (2013) no. 2, pp. 201-248 | DOI | Zbl
[7] Monge–Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010) no. 2, pp. 199-262 | DOI | Zbl
[8] Differentiability of volumes of divisors and a problem of Teissier, J. Algebr. Geom., Volume 18 (2009) no. 2, pp. 279-308 | DOI | MR | Zbl
[9] Nakamaye’s theorem on log canonical pairs, Ann. Inst. Fourier, Volume 64 (2014) no. 6, pp. 2283-2298 | DOI | Numdam | MR | Zbl
[10] Seshadri constants on smooth threefolds, Adv. Geom., Volume 14 (2014) no. 1, pp. 59-79 | DOI | MR | Zbl
[11] Okounkov bodies associated to pseudoeffective divisors, J. Lond. Math. Soc., Volume 97 (2018) no. 2, pp. 170-195 | DOI | MR | Zbl
[12] An extension theorem for Kähler currents with analytic singularities, Ann. Fac. Sci. Toulouse, Math., Volume 23 (2014) no. 4, pp. 893-905 | DOI | Numdam | Zbl
[13] Kähler currents and null loci, Invent. Math., Volume 202 (2015) no. 3, pp. 1167-1198 | DOI | Zbl
[14] A singular Demailly–Păun theorem, C. R. Math. Acad. Sci. Paris, Volume 354 (2016) no. 1, pp. 91-95 | DOI | Zbl
[15] Regularization of closed positive currents and intersection theory, J. Algebr. Geom., Volume 1 (1992) no. 3, pp. 361-409 | MR | Zbl
[16] Singular Hermitian metrics on positive line bundles, Complex algebraic varieties (Bayreuth, 1990) (Lecture Notes in Mathematics), Volume 1507, Springer, 1992, pp. 87-104 | MR | Zbl
[17] Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. Math., Volume 159 (2004) no. 3, pp. 1247-1274 | DOI | Zbl
[18] Transcendental Morse inequality and generalized Okounkov bodies, Algebr. Geom., Volume 4 (2017) no. 2, pp. 177-202 | DOI | MR | Zbl
[19] Restricted volumes of effective divisors, Bull. Soc. Math. Fr., Volume 144 (2016) no. 2, pp. 299-337 | DOI | MR | Zbl
[20] Asymptotic invariants of base loci, Ann. Inst. Fourier, Volume 56 (2006) no. 6, pp. 1701-1734 | Numdam | MR | Zbl
[21] Restricted volumes and base loci of linear series, Am. J. Math., Volume 131 (2009) no. 3, pp. 607-651 | DOI | MR | Zbl
[22] Note on pull-back and Lelong number of currents, Bull. Soc. Math. Fr., Volume 127 (1999) no. 3, pp. 445-458 | DOI | Numdam | MR | Zbl
[23] Boundedness of pluricanonical maps of varieties of general type, Invent. Math., Volume 166 (2006) no. 1, pp. 1-25 | DOI | MR | Zbl
[24] Restricted Bergman kernel asymptotics, Trans. Am. Math. Soc., Volume 364 (2012) no. 7, pp. 3585-3607 | DOI | MR | Zbl
[25] Ensembles de sous-niveau et images inverses des fonctions plurisousharmoniques, Bull. Sci. Math., Volume 124 (2000) no. 1, pp. 75-92 | DOI | MR | Zbl
[26] Positivity in algebraic geometry I: Classical setting: Line bundles and linear series. II: Positivity for vector bundles, and multiplier ideals, Springer, 2004
[27] Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér., Volume 42 (2009) no. 5, pp. 783-835 | DOI | Numdam | MR | Zbl
[28] Comparing numerical dimensions, Algebra Number Theory, Volume 7 (2013) no. 5, pp. 1065-1100 | DOI | MR | Zbl
[29] The diminished base locus is not always closed, Compos. Math., Volume 150 (2014) no. 10, pp. 1729-1741 | DOI | MR | Zbl
[30] Quasi-projectivity of the moduli space of smooth Kähler–Einstein Fano manifolds, Ann. Sci. Éc. Norm. Supér., Volume 51 (2018) no. 3, pp. 739-772 | DOI | Zbl
[31] Augmented base loci and restricted volumes on normal varieties, II: The case of real divisors, Math. Proc. Camb. Philos. Soc., Volume 159 (2015) no. 3, pp. 517-527 | DOI | MR | Zbl
[32] Restricted volumes and divisorial Zariski decompositions, Am. J. Math., Volume 135 (2013) no. 3, pp. 637-662 | DOI | MR | Zbl
[33] A Nadel vanishing theorem for metrics with minimal singularities on big line bundles, Adv. Math., Volume 280 (2015), pp. 188-207 | DOI | MR | Zbl
[34] Stable base loci of linear series, Math. Ann., Volume 318 (2000) no. 4, pp. 837-847 | DOI | MR | Zbl
[35] Base loci of linear series are numerically determined, Trans. Am. Math. Soc., Volume 355 (2003) no. 2, pp. 551-566 | DOI | MR | Zbl
[36] Zariski-decomposition and abundance, MSJ Memoirs, 14, Mathematical Society of Japan, 2004
[37] On volumes along subvarieties of line bundles with nonnegative Kodaira–Iitaka dimension, Mich. Math. J., Volume 60 (2011) no. 1, pp. 35-49 | MR | Zbl
[38] On the singularities of the pluricomplex Green’s function, Advances in analysis. The legacy of Elias M. Stein (Princeton Mathematical Series), Volume 50, Princeton University Press, 2014, pp. 419-435 | DOI | MR | Zbl
[39] Quasi-projectivity of moduli spaces of polarized varieties, Ann. Math., Volume 159 (2004) no. 2, pp. 597-639 | DOI | MR | Zbl
[40] Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math., Volume 27 (1974), pp. 53-156 | MR | Zbl
[41] Pluricanonical systems on algebraic varieties of general type, Invent. Math., Volume 165 (2006) no. 3, pp. 551-587 | DOI | MR | Zbl
[42] A local ampleness criterion of torsion free sheaves, Bull. Sci. Math., Volume 137 (2013) no. 5, pp. 659-670 | DOI | MR | Zbl
[43] The Calabi–Yau Theorem and Kähler currents, Adv. Theor. Math. Phys., Volume 20 (2016) no. 2, pp. 381-404 | DOI | MR | Zbl
[44] Nakamaye’s Theorem on complex manifolds, Algebraic geometry: Salt Lake City 2015 (Proceedings of Symposia in Pure Mathematics), Volume 97.1, American Mathematical Society, 2018, pp. 633-655 | Zbl
[45] Orthogonality of divisorial Zariski decompositions for classes with volume zero, Tôhoku Math. J., Volume 71 (2019) no. 1, pp. 1-8 | MR | Zbl
[46] Duality between the pseudoeffective and the movable cone on a projective manifold. With an appendix by Sébastien Boucksom, J. Am. Math. Soc., Volume 32 (2019) no. 3, pp. 675-689 | MR | Zbl
[47] The theorem of Riemann–Roch for high multiples of an effective divisor on an algebraic surface, Ann. Math., Volume 76 (1962), pp. 560-615 | DOI | MR | Zbl
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