Browse issues
no. 3
Restricted volumes on Kähler manifolds
Tristan C. Collins1; Valentino Tosatti2
1 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139
2 Department of Mathematics and Statistics, McGill University, Montréal, Québec H3A 0B9, Canada
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 3, pp. 907-947.

We study numerical restricted volumes of (1,1) 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 (1,1) 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.

Published online:
DOI: 10.5802/afst.1708

Tristan C. Collins 1; Valentino Tosatti 2

1 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139
2 Department of Mathematics and Statistics, McGill University, Montréal, Québec H3A 0B9, Canada
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{AFST_2022_6_31_3_907_0,
     author = {Tristan C. Collins and Valentino Tosatti},
     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 = {https://afst.centre-mersenne.org/articles/10.5802/afst.1708/}
}
TY  - JOUR
AU  - Tristan C. Collins
AU  - Valentino Tosatti
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  - https://afst.centre-mersenne.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 Tristan C. Collins
%A Valentino Tosatti
%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 https://afst.centre-mersenne.org/articles/10.5802/afst.1708/
%R 10.5802/afst.1708
%G en
%F AFST_2022_6_31_3_907_0
Tristan C. Collins; Valentino Tosatti. 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. https://afst.centre-mersenne.org/articles/10.5802/afst.1708/

[1] Thomas Bauer; Alex Küronya; Tomasz Szemberg Zariski chambers, volumes, and stable base loci, J. Reine Angew. Math., Volume 576 (2004), pp. 209-233 | MR | Zbl

[2] Sébastien Boucksom Cônes positifs des variétés complexes compactes, Ph. D. Thesis, Institut Fourier Grenoble (France) (2002)

[3] Sébastien Boucksom On the volume of a line bundle, Int. J. Math., Volume 13 (2002) no. 10, pp. 1043-1063 | DOI | MR | Zbl

[4] Sébastien Boucksom 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] Sébastien Boucksom; Salvatore Cacciola; Angelo Felice Lopez Augmented base loci and restricted volumes on normal varieties, Math. Z., Volume 278 (2014) no. 3, pp. 3-4 | MR | Zbl

[6] Sébastien Boucksom; Jean-Pierre Demailly; Mihai Păun; Thomas Peternell 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] Sébastien Boucksom; Philippe Eyssidieux; Vincent Guedj; Ahmed Zeriahi Monge–Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010) no. 2, pp. 199-262 | DOI | Zbl

[8] Sébastien Boucksom; Charles Favre; Mattias Jonsson 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] Salvatore Cacciola; Angelo Felice Lopez Nakamaye’s theorem on log canonical pairs, Ann. Inst. Fourier, Volume 64 (2014) no. 6, pp. 2283-2298 | DOI | Numdam | MR | Zbl

[10] Paolo Cascini; Michael Nakamaye Seshadri constants on smooth threefolds, Adv. Geom., Volume 14 (2014) no. 1, pp. 59-79 | DOI | MR | Zbl

[11] Sung Rak Choi; Yoonsuk Hyun; Jinhyung Park; Joonyeong Won Okounkov bodies associated to pseudoeffective divisors, J. Lond. Math. Soc., Volume 97 (2018) no. 2, pp. 170-195 | DOI | MR | Zbl

[12] Tristan C. Collins; Valentino Tosatti 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] Tristan C. Collins; Valentino Tosatti Kähler currents and null loci, Invent. Math., Volume 202 (2015) no. 3, pp. 1167-1198 | DOI | Zbl

[14] Tristan C. Collins; Valentino Tosatti A singular Demailly–Păun theorem, C. R. Math. Acad. Sci. Paris, Volume 354 (2016) no. 1, pp. 91-95 | DOI | Zbl

[15] Jean-Pierre Demailly Regularization of closed positive currents and intersection theory, J. Algebr. Geom., Volume 1 (1992) no. 3, pp. 361-409 | MR | Zbl

[16] Jean-Pierre Demailly 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] Jean-Pierre Demailly; Mihai Paun Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. Math., Volume 159 (2004) no. 3, pp. 1247-1274 | DOI | Zbl

[18] Ya Deng Transcendental Morse inequality and generalized Okounkov bodies, Algebr. Geom., Volume 4 (2017) no. 2, pp. 177-202 | DOI | MR | Zbl

[19] Lorenzo Di Biagio; Gianluca Pacienza Restricted volumes of effective divisors, Bull. Soc. Math. Fr., Volume 144 (2016) no. 2, pp. 299-337 | DOI | MR | Zbl

[20] Lawrence Ein; Robert Lazarsfeld; Mircea Mustaţă; Michael Nakamaye; Mihnea Popa Asymptotic invariants of base loci, Ann. Inst. Fourier, Volume 56 (2006) no. 6, pp. 1701-1734 | Numdam | MR | Zbl

[21] Lawrence Ein; Robert Lazarsfeld; Mircea Mustaţă; Michael Nakamaye; Mihnea Popa Restricted volumes and base loci of linear series, Am. J. Math., Volume 131 (2009) no. 3, pp. 607-651 | DOI | MR | Zbl

[22] Charles Favre 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] Christopher D. Hacon; James McKernan Boundedness of pluricanonical maps of varieties of general type, Invent. Math., Volume 166 (2006) no. 1, pp. 1-25 | DOI | MR | Zbl

[24] Tomoyuki Hisamoto Restricted Bergman kernel asymptotics, Trans. Am. Math. Soc., Volume 364 (2012) no. 7, pp. 3585-3607 | DOI | MR | Zbl

[25] Christer O. Kiselman 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] Robert Lazarsfeld Positivity in algebraic geometry I: Classical setting: Line bundles and linear series. II: Positivity for vector bundles, and multiplier ideals, Springer, 2004

[27] Robert Lazarsfeld; Mircea Mustaţă Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér., Volume 42 (2009) no. 5, pp. 783-835 | DOI | Numdam | MR | Zbl

[28] Brian Lehmann Comparing numerical dimensions, Algebra Number Theory, Volume 7 (2013) no. 5, pp. 1065-1100 | DOI | MR | Zbl

[29] John Lesieutre The diminished base locus is not always closed, Compos. Math., Volume 150 (2014) no. 10, pp. 1729-1741 | DOI | MR | Zbl

[30] Chi Li; Xiaowei Wang; Chenyang Xu 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] Angelo Felice Lopez 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] Shin-Ichi Matsumura Restricted volumes and divisorial Zariski decompositions, Am. J. Math., Volume 135 (2013) no. 3, pp. 637-662 | DOI | MR | Zbl

[33] Shin-Ichi Matsumura 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] Michael Nakamaye Stable base loci of linear series, Math. Ann., Volume 318 (2000) no. 4, pp. 837-847 | DOI | MR | Zbl

[35] Michael Nakamaye Base loci of linear series are numerically determined, Trans. Am. Math. Soc., Volume 355 (2003) no. 2, pp. 551-566 | DOI | MR | Zbl

[36] Noboru Nakayama Zariski-decomposition and abundance, MSJ Memoirs, 14, Mathematical Society of Japan, 2004

[37] Gianluca Pacienza; Shigeharu Takayama 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] Duong H. Phong; Jacob Sturm 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] Georg Schumacher; Hajime Tsuji Quasi-projectivity of moduli spaces of polarized varieties, Ann. Math., Volume 159 (2004) no. 2, pp. 597-639 | DOI | MR | Zbl

[40] Yum-Tong Siu 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] Shigeharu Takayama Pluricanonical systems on algebraic varieties of general type, Invent. Math., Volume 165 (2006) no. 3, pp. 551-587 | DOI | MR | Zbl

[42] Shigeharu Takayama A local ampleness criterion of torsion free sheaves, Bull. Sci. Math., Volume 137 (2013) no. 5, pp. 659-670 | DOI | MR | Zbl

[43] Valentino Tosatti The Calabi–Yau Theorem and Kähler currents, Adv. Theor. Math. Phys., Volume 20 (2016) no. 2, pp. 381-404 | DOI | MR | Zbl

[44] Valentino Tosatti 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] Valentino Tosatti Orthogonality of divisorial Zariski decompositions for classes with volume zero, Tôhoku Math. J., Volume 71 (2019) no. 1, pp. 1-8 | MR | Zbl

[46] David Witt-Nyström 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] Oscar Zariski 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

Cited by Sources: