Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs

Boucksom, Sébastien; Hisamoto, Tomoyuki; Jonsson, Mattias

Annales de l'Institut Fourier, Tome 67 (2017) no. 2, pp. 743-841.

Le but de cet article est de mettre en place un formalisme inspiré par la géométrie non-archimédienne pour étudier la K-stabilité. Nous fournissons d’abord une analyse détaillée des mesures de Duistermaat–Heckman dans le contexte des configurations test de schémas polarisés arbitraires, caractérisant en particulier les configurations test presque triviales. Nous introduisons et étudions ensuite, pour toute variété normale polarisée (ou, plus généralement, toute paire polarisée au sens du Programme du Modèle Minimal), les analogues non-archimédiens de certaines fonctionnelles classiques de la géométrie kählérienne. Ces fonctionnelles sont définies sur l’espace des configurations test, et l’invariant de Donaldson–Futaki est en particulier interprété comme la version non-archimédienne de la fonctionnelle de Mabuchi, à un terme d’erreur explicite près. Enfin, nous étudions en détail les liens entre K-stabilité uniforme et singularités des paires, redémontrant et améliorant des résultats de Y. Odaka dans notre formalisme. Ceci fournit divers exemples de variétés uniformément K-stables.

The purpose of this paper is to set up a formalism inspired by non-Archimedean geometry to study K-stability. We first provide a detailed analysis of Duistermaat–Heckman measures in the context of test configurations for arbitrary polarized schemes, characterizing in particular almost trivial test configurations. Second, for any normal polarized variety (or, more generally, polarized pair in the sense of the Minimal Model Program), we introduce and study non-Archimedean analogues of certain classical functionals in Kähler geometry. These functionals are defined on the space of test configurations, and the Donaldson–Futaki invariant is in particular interpreted as the non-Archimedean version of the Mabuchi functional, up to an explicit error term. Finally, we study in detail the relation between uniform K-stability and singularities of pairs, reproving and strengthening Y. Odaka’s results in our formalism. This provides various examples of uniformly K-stable varieties.

Classification : 14L30, 14E30
Keywords: K-stability, Duistermaat–Heckman measures, singularities of pairs.
Mot clés : K-stabilité, mesures de Duistermaat–Heckman, singularités de paires.

[1] Arezzo, Claudio; Della Vedova, Alberto; La Nave, Gabriele Singularities and $K$-semistability, Int. Math. Res. Not. IMRN (2012) no. 4, pp. 849-869 | DOI

[2] Arezzo, Claudio; Pacard, Frank Blowing up and desingularizing constant scalar curvature Kähler manifolds, Acta Math., Volume 196 (2006) no. 2, pp. 179-228 | DOI

[3] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, Providence, RI, 1990, x+169 pages

[4] Berman, Robert J. K-polystability of $\mathbb{Q}$-Fano varieties admitting Kähler-Einstein metrics, Invent. Math., Volume 203 (2016) no. 3, pp. 973-1025 | DOI

[5] Berman, Robert J.; Berndtsson, Bo Convexity of the K-energy on the space of Kahler metrics and uniqueness of extremal metrics (2014) (https://arxiv.org/abs/1405.0401)

[6] Berman, Robert J.; Boucksom, Sébastien; Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties (2011) (https://arxiv.org/abs/1111.7158)

[7] Berman, Robert J.; Boucksom, Sébastien; Jonsson, Mattias A variational approach to the Yau-Tian-Donaldson conjecture (2015) (https://arxiv.org/abs/1509.04561)

[8] Berman, Robert J.; Nystrom, David Witt Complex optimal transport and the pluripotential theory of Kähler-Ricci solitons (2014) (https://arxiv.org/abs/1401.8264)

[9] Berndtsson, Bo Probability measures related to geodesics in the space of Kähler metrics (2009) (https://arxiv.org/abs/0907.1806)

[10] Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., Volume 23 (2010) no. 2, pp. 405-468 | DOI

[11] Boucksom, Sébastien; Chen, Huayi Okounkov bodies of filtered linear series, Compos. Math., Volume 147 (2011) no. 4, pp. 1205-1229 | DOI

[12] Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias The non-Archimedean Monge-Ampere equation, Nonarchimedean and Tropical Geometry (Simons Symposia), Springer, 2015, pp. 31-49

[13] Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias Solution to a non-Archimedean Monge-Ampère equation, J. Amer. Math. Soc., Volume 28 (2015) no. 3, pp. 617-667 | DOI

[14] Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias Singular semipositive metrics in non-Archimedean geometry, J. Algebraic Geom., Volume 25 (2016) no. 1, pp. 77-139 | DOI

[15] Boucksom, Sébastien; de Fernex, Tommaso; Favre, Charles The volume of an isolated singularity, Duke Math. J., Volume 161 (2012) no. 8, pp. 1455-1520 | DOI

[16] Boucksom, Sébastien; Hisamoto, Tomoyuki; Jonsson, Mattias Uniform K-stability and asymptotics of energy functionals in Kähler geometry (2016) (https://arxiv.org/abs/1603.01026)

[17] Boucksom, Sébastien; Jonsson, Mattias Singular semipositive metrics on line bundles on varieties over trivially valued fields (In preparation)

[18] Boucksom, Sébastien; Küronya, Alex; Maclean, Catriona; Szemberg, Tomasz Vanishing sequences and Okounkov bodies, Math. Ann., Volume 361 (2015) no. 3-4, pp. 811-834 | DOI

[19] Brion, Michel; Procesi, Claudio Action d’un tore dans une variété projective, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) (Progr. Math.), Volume 92, Birkhäuser Boston, Boston, MA, 1990, pp. 509-539 | DOI

[20] Chen, Xiuxiong On the lower bound of the Mabuchi energy and its application, Internat. Math. Res. Notices (2000) no. 12, pp. 607-623 | DOI

[21] Chen, Xiuxiong; Donaldson, Simon; Sun, Song Kähler-Einstein metrics on Fano manifolds, J. Amer. Math. Soc., Volume 28 (2015) no. 1, p. 183-197, 199–234, 235–278 | DOI

[22] Darvas, Tamás The Mabuchi geometry of finite energy classes, Adv. Math., Volume 285 (2015), pp. 182-219 | DOI

[23] Darvas, Tamás; Rubinstein, Yanir A. Tian’s properness conjectures and Finsler geometry of the space of Kahler metrics (2015) (https://arxiv.org/abs/1506.07129)

[24] Dervan, Ruadhaí Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds, Ann. Fac. Sci. Toulouse, Volume 25 (2016) no. 4, pp. 919-934 | DOI

[25] Dervan, Ruadhaí Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not. IMRN, Volume 15 (2016), pp. 4728-4783 | DOI

[26] Ding, Wei Yue Remarks on the existence problem of positive Kähler-Einstein metrics, Math. Ann., Volume 282 (1988) no. 3, pp. 463-471 | DOI

[27] Donaldson, S. K. Scalar curvature and projective embeddings. I, J. Differ. Geom., Volume 59 (2001) no. 3, pp. 479-522 http://projecteuclid.org/euclid.jdg/1090349449 | DOI

[28] Donaldson, S. K. Scalar curvature and stability of toric varieties, J. Differ. Geom., Volume 62 (2002) no. 2, pp. 289-349 http://projecteuclid.org/euclid.jdg/1090950195 | DOI

[29] Donaldson, S. K. Lower bounds on the Calabi functional, J. Differ. Geom., Volume 70 (2005) no. 3, pp. 453-472 http://projecteuclid.org/euclid.jdg/1143642909 | DOI

[30] Donaldson, S. K. Kähler metrics with cone singularities along a divisor, Essays in mathematics and its applications, Springer, Heidelberg, 2012, pp. 49-79 | DOI

[31] Duistermaat, Johannes J.; Heckman, Gerrit J. On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math., Volume 69 (1982) no. 2, pp. 259-268 | DOI

[32] Edidin, Dan; Graham, William Characteristic classes in the Chow ring, J. Algebraic Geom., Volume 6 (1997) no. 3, pp. 431-443

[33] Edidin, Dan; Graham, William Equivariant intersection theory, Invent. Math., Volume 131 (1998) no. 3, pp. 595-634 | DOI

[34] Edidin, Dan; Graham, William Riemann-Roch for equivariant Chow groups, Duke Math. J., Volume 102 (2000) no. 3, pp. 567-594 | DOI

[35] Ein, Lawrence; Lazarsfeld, Robert; Mustaţă, Mircea; Nakamaye, Michael; Popa, Mihnea Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble), Volume 56 (2006) no. 6, pp. 1701-1734 http://aif.cedram.org/item?id=AIF_2006__56_6_1701_0 | DOI

[36] Ein, Lawrence; Lazarsfeld, Robert; Smith, Karen E. Uniform approximation of Abhyankar valuation ideals in smooth function fields, Amer. J. Math., Volume 125 (2003) no. 2, pp. 409-440 http://muse.jhu.edu/journals/american_journal_of_mathematics/v125/125.2ein.pdf | DOI

[37] Flenner, Hubert Die Sätze von Bertini für lokale Ringe, Math. Ann., Volume 229 (1977) no. 2, pp. 97-111 | DOI

[38] Fujita, Kento On K-stability and the volume functions of $\mathbb{Q}$-Fano varieties (2015) (https://arxiv.org/abs/1508.04052)

[39] Fujita, Kento Optimal bounds for the volumes of Kähler-Einstein Fano manifolds (2015) (https://arxiv.org/abs/1508.04578)

[40] Fujita, Kento A valuative criterion for uniform K-stability of $\mathbb{Q}$-Fano varieties (2016) (https://arxiv.org/abs/1602.00901)

[41] Fujita, Kento; Odaka, Yuji On the K-stability of Fano varieties and anticanonical divisors (2016) (https://arxiv.org/abs/1602.01305)

[42] Fulton, William Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2, Springer-Verlag, Berlin, 1998, xiv+470 pages | DOI

[43] Hartshorne, Robin Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, 156, Springer-Verlag, Berlin-New York, 1970, xiv+256 pages (Notes written in collaboration with C. Musili)

[44] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York-Heidelberg, 1977, xvi+496 pages

[45] Hisamoto, Tomoyuki On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold, J. Reine Angew. Math., Volume 713 (2016), pp. 129-148 | DOI

[46] Hübl, Reinhold; Swanson, Irena Discrete valuations centered on local domains, J. Pure Appl. Algebra, Volume 161 (2001) no. 1-2, pp. 145-166 | DOI

[47] Huneke, Craig; Swanson, Irena Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, 336, Cambridge University Press, Cambridge, 2006, xiv+431 pages

[48] Jonsson, Mattias; Mustaţă, Mircea Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble), Volume 62 (2012) no. 6, p. 2145-2209 (2013) | DOI

[49] Kaveh, Kiumars; Khovanskii, Askold G. Convex bodies associated to actions of reductive groups, Mosc. Math. J., Volume 12 (2012) no. 2, p. 369-396, 461

[50] Kleiman, Steven L. Toward a numerical theory of ampleness, Ann. Math., Volume 84 (1966), pp. 293-344 | DOI

[51] Knaf, Hagen; Kuhlmann, Franz-Viktor Abhyankar places admit local uniformization in any characteristic, Ann. Sci. Éc. Norm. Sup., Volume 38 (2005) no. 6, pp. 833-846 | DOI

[52] Kollár, János Singularities of the minimal model program, Cambridge Tracts in Mathematics, 200, Cambridge University Press, Cambridge, 2013, x+370 pages (With a collaboration of Sándor Kovács) | DOI

[53] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, 1998, viii+254 pages (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original) | DOI

[54] Lazarsfeld, Robert Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 49, Springer-Verlag, Berlin, 2004, xviii+385 pages

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

[56] Li, Chi K-semistability is equivariant volume minimization (2015) (https://arxiv.org/abs/1512.07205)

[57] Li, Chi Minimizing normalized volumes of valuations (2015) (https://arxiv.org/abs/1511.08164)

[58] Li, Chi; Sun, Song Conical Kähler-Einstein metrics revisited, Comm. Math. Phys., Volume 331 (2014) no. 3, pp. 927-973 | DOI

[59] Li, Chi; Xu, Chenyang Special test configuration and K-stability of Fano varieties, Ann. Math., Volume 180 (2014) no. 1, pp. 197-232 | DOI

[60] Liu, Yuchen The volume of singular Kähler-Einstein Fano varieties (2016) (https://arxiv.org/abs/1605.01034)

[61] Odaka, Yuji The Calabi conjecture and K-stability, Int. Math. Res. Not. IMRN (2012) no. 10, pp. 2272-2288 | DOI

[62] Odaka, Yuji A generalization of the Ross-Thomas slope theory, Osaka J. Math., Volume 50 (2013) no. 1, pp. 171-185 http://projecteuclid.org/euclid.ojm/1364390425

[63] Odaka, Yuji The GIT stability of polarized varieties via discrepancy, Ann. Math., Volume 177 (2013) no. 2, pp. 645-661 | DOI

[64] Odaka, Yuji On parametrization, optimization and triviality of test configurations, Proc. Amer. Math. Soc., Volume 143 (2015) no. 1, pp. 25-33 | DOI

[65] Odaka, Yuji; Sano, Yuji Alpha invariant and K-stability of $\mathbb{Q}$-Fano varieties, Adv. Math., Volume 229 (2012) no. 5, pp. 2818-2834 | DOI

[66] Odaka, Yuji; Sun, Song Testing log K-stability by blowing up formalism, Ann. Fac. Sci. Toulouse, Volume 24 (2015) no. 3, pp. 505-522 | DOI

[67] Odaka, Yuji; Xu, Chenyang Log-canonical models of singular pairs and its applications, Math. Res. Lett., Volume 19 (2012) no. 2, pp. 325-334 | DOI

[68] Okounkov, Andrei Brunn-Minkowski inequality for multiplicities, Invent. Math., Volume 125 (1996) no. 3, pp. 405-411 | DOI

[69] Paul, Sean T.; Tian, Gang CM Stability and the Generalized Futaki Invariant I (2006) (https://arxiv.org/abs/math/0605278)

[70] Paul, Sean Timothy; Tian, Gang CM stability and the generalized Futaki invariant II, Astérisque (2009) no. 328, p. 339-354 (2010)

[71] Phong, Duong H.; Ross, Julius; Sturm, Jacob Deligne pairings and the Knudsen-Mumford expansion, J. Differ. Geom., Volume 78 (2008) no. 3, pp. 475-496 http://projecteuclid.org/euclid.jdg/1207834553 | DOI

[72] Phong, Duong H.; Sturm, Jacob Test configurations for K-stability and geodesic rays, J. Symplectic Geom., Volume 5 (2007) no. 2, pp. 221-247 http://projecteuclid.org/euclid.jsg/1202004456 | DOI

[73] Ross, Julius; Thomas, Richard A study of the Hilbert-Mumford criterion for the stability of projective varieties, J. Algebraic Geom., Volume 16 (2007) no. 2, pp. 201-255 | DOI

[74] Stoppa, Jacopo K-stability of constant scalar curvature Kähler manifolds, Adv. Math., Volume 221 (2009) no. 4, pp. 1397-1408 | DOI

[75] Stoppa, Jacopo A note on the definition of K-stability (2011) (https://arxiv.org/abs/1111.5826)

[76] Sun, Song Note on K-stability of pairs, Math. Ann., Volume 355 (2013) no. 1, pp. 259-272 | DOI

[77] Székelyhidi, Gábor Extremal metrics and K-stability (PhD thesis) (2006) (https://arxiv.org/abs/math/0611002)

[78] Székelyhidi, Gábor Filtrations and test-configurations, Math. Ann., Volume 362 (2015) no. 1-2, pp. 451-484 (With an appendix by Sébastien Boucksom) | DOI

[79] The Stacks Project Authors Stacks Project (2016) (http://stacks.math.columbia.edu)

[80] Tian, Gang Kähler-Einstein metrics with positive scalar curvature, Invent. Math., Volume 130 (1997) no. 1, pp. 1-37 | DOI

[81] Tian, Gang K-stability and Kähler-Einstein metrics, Comm. Pure Appl. Math., Volume 68 (2015) no. 7, pp. 1085-1156 | DOI

[82] Wang, Xiaowei Height and GIT weight, Math. Res. Lett., Volume 19 (2012) no. 4, pp. 909-926 | DOI

[83] Witt Nyström, David Test configurations and Okounkov bodies, Compos. Math., Volume 148 (2012) no. 6, pp. 1736-1756 | DOI

[84] Yuan, Xinyi; Zhang, Shou-Wu The arithmetic Hodge index theorem for adelic line bundles, Mathematische Annalen (2016), pp. 1-49

[85] Zariski, Oscar; Samuel, Pierre Commutative algebra. Vol. II, Springer-Verlag, New York-Heidelberg, 1975, x+414 pages (Reprint of the 1960 edition, Graduate Texts in Mathematics, Vol. 29)