no. 2
Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor
[La courbure scalaire et l’invariant de Futaki des métriques kählériennes avec des singularités coniques le long d’un diviseur]
Hashimoto, Yoshinori1
1 University College London Department of Mathematics Gower Street, London WC1E 6BT (UK)
Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 591-652.

Nous étudions la courbure scalaire de certaines classes de métriques kählériennes qui ont des singularités coniques le long d’un diviseur. Notre résultat principal concerne le complété projectif du fibré pluricanonique sur un produit de variétés Fano Kähler–Einstein avec second nombre de Betti égal à 1. Sur cette variété, en utilisant une construction due à Hwang, nous démontrons qu’il existe une métrique kählérienne à courbure scalaire constante et singularités coniques si et seulement si l’invariant logarithmique de Futaki est nul pour l’action de ℂ * sur le fibré. Ce résultat conforte la version logarithmique de la conjecture de Yau–Tian–Donaldson pour des polarisations quelconques.

Nous démontrons aussi que la courbure scalaire peut être définie sur le lieu singulier comme un courant pour certaines classes de métriques à singularités coniques, ce qui nous permet de calculer l’invariant logarithmique de Futaki par rapport à des métriques singulières.

We study the scalar curvature of Kähler metrics that have cone singularities along a divisor, with a particular focus on certain specific classes of such metrics that enjoy some curvature estimates. Our main result is that, on the projective completion of a pluricanonical bundle over a product of Kähler–Einstein Fano manifolds with the second Betti number 1, momentum-constructed constant scalar curvature Kähler metrics with cone singularities along the ∞-section exist if and only if the log Futaki invariant vanishes on the fibrewise ℂ * -action, giving a supporting evidence to the log version of the Yau–Tian–Donaldson conjecture for general polarisations.

We also show that, for these classes of conically singular metrics, the scalar curvature can be defined on the whole manifold as a current, so that we can compute the log Futaki invariant with respect to them. Finally, we prove some partial invariance results for them.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3252
Classification : 53C55, 53C25
Keywords: constant scalar curvature Kähler metrics, Futaki invariant, Kähler metrics with cone singularities
Mot clés : métriques kählériennes de courbure scalaire constante, l’invariant de Futaki, métriques kählériennes avec les singularités coniques

Hashimoto, Yoshinori 1

1 University College London Department of Mathematics Gower Street, London WC1E 6BT (UK)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2019__69_2_591_0,
     author = {Hashimoto, Yoshinori},
     title = {Scalar curvature and {Futaki} invariant of {K\"ahler} metrics with cone singularities along a divisor},
     journal = {Annales de l'Institut Fourier},
     pages = {591--652},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {2},
     year = {2019},
     doi = {10.5802/aif.3252},
     zbl = {07067413},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3252/}
}
TY  - JOUR
AU  - Hashimoto, Yoshinori
TI  - Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor
JO  - Annales de l'Institut Fourier
PY  - 2019
SP  - 591
EP  - 652
VL  - 69
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3252/
DO  - 10.5802/aif.3252
LA  - en
ID  - AIF_2019__69_2_591_0
ER  - 
%0 Journal Article
%A Hashimoto, Yoshinori
%T Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor
%J Annales de l'Institut Fourier
%D 2019
%P 591-652
%V 69
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3252/
%R 10.5802/aif.3252
%G en
%F AIF_2019__69_2_591_0
Hashimoto, Yoshinori. Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor. Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 591-652. doi : 10.5802/aif.3252. https://aif.centre-mersenne.org/articles/10.5802/aif.3252/

[1] Apostolov, Vestislav; Calderbank, David M. J.; Gauduchon, Paul Hamiltonian 2-forms in Kähler geometry. I. General theory, J. Differ. Geom., Volume 73 (2006) no. 3, pp. 359-412 http://projecteuclid.org/euclid.jdg/1146169934 | DOI | MR | Zbl

[2] Apostolov, Vestislav; Calderbank, David M. J.; Gauduchon, Paul; Tønnesen-Friedman, Christina W. Hamiltonian 2-forms in Kähler geometry. II. Global classification, J. Differ. Geom., Volume 68 (2004) no. 2, pp. 277-345 http://projecteuclid.org/euclid.jdg/1115669513 | DOI | MR | Zbl

[3] Apostolov, Vestislav; Calderbank, David M. J.; Gauduchon, Paul; Tønnesen-Friedman, Christina W. Hamiltonian 2-forms in Kähler geometry. III. Extremal metrics and stability, Invent. Math., Volume 173 (2008) no. 3, pp. 547-601 | DOI | MR | Zbl

[4] Auvray, Hugues Note on Poincaré type Futaki characters, Ann. Inst. Fourier, Volume 68 (2018) no. 1, pp. 319-344 | DOI | MR | Zbl

[5] Berman, Robert J. K-polystability of ℚ-Fano varieties admitting Kähler-Einstein metrics, Invent. Math., Volume 203 (2016) no. 3, pp. 973-1025 | DOI | MR | Zbl

[6] Boucksom, Sébastien; Hisamoto, Tomoyuki; Jonsson, Mattias Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs, Ann. Inst. Fourier., Volume 67 (2017) no. 2, pp. 743-841 | DOI | MR | Zbl

[7] Brendle, Simon Ricci flat Kähler metrics with edge singularities, Int. Math. Res. Not. (2013) no. 24, pp. 5727-5766 | DOI | MR | Zbl

[8] Calamai, Simone; Zheng, Kai Geodesics in the space of Kähler cone metrics, I, Am. J. Math., Volume 137 (2015) no. 5, pp. 1149-1208 | DOI | MR | Zbl

[9] Chen, Xiuxiong; Donaldson, Simon K.; Sun, Song Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 183-197 | DOI | MR | Zbl

[10] Chen, Xiuxiong; Donaldson, Simon K.; Sun, Song Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π, J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 199-234 | DOI | MR | Zbl

[11] Chen, Xiuxiong; Donaldson, Simon K.; Sun, Song Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof, J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 235-278 | DOI | MR | Zbl

[12] Debarre, Olivier Higher-dimensional algebraic geometry, Universitext, Springer, 2001, xiv+233 pages | DOI | MR | Zbl

[13] Demailly, Jean-Pierre Complex Analytic and Differential Geometry (Textbook available at https://www-fourier.ujf-grenoble.fr/ demailly/manuscripts/agbook.pdf) | Numdam | Zbl

[14] Dervan, Ruadhaí Alpha invariants and K-stability for general polarizations of Fano varieties, Int. Math. Res. Not. (2015) no. 16, pp. 7162-7189 | DOI | MR | Zbl

[15] Dervan, Ruadhaí Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not. (2016) no. 15, pp. 4728-4783 | DOI | MR | Zbl

[16] Dervan, Ruadhaí; Ross, Julius K-stability for Kähler manifolds, Math. Res. Lett., Volume 24 (2017) no. 3, pp. 689-739 | DOI | Zbl

[17] Donaldson, Simon K. Remarks on gauge theory, complex geometry and 4-manifold topology, Fields Medallists’ lectures (World Scientific Series in 20th Century Mathematics), Volume 5, World Scientific, 1997, pp. 384-403 | DOI | MR

[18] Donaldson, Simon K. Scalar curvature and projective embeddings. I, J. Differ. Geom., Volume 59 (2001) no. 3, pp. 479-522 http://projecteuclid.org/getrecord?id=euclid.jdg/1090349449 | DOI | MR | Zbl

[19] Donaldson, Simon K. Scalar curvature and stability of toric varieties, J. Differ. Geom., Volume 62 (2002) no. 2, pp. 289-349 http://projecteuclid.org/getrecord?id=euclid.jdg/1090950195 | DOI | MR | Zbl

[20] Donaldson, Simon K. Lower bounds on the Calabi functional, J. Differ. Geom., Volume 70 (2005) no. 3, pp. 453-472 http://projecteuclid.org/getrecord?id=euclid.jdg/1143642909 | DOI | MR | Zbl

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

[22] Futaki, Akito An obstruction to the existence of Einstein Kähler metrics, Invent. Math., Volume 73 (1983) no. 3, pp. 437-443 | DOI | MR | Zbl

[23] Guenancia, Henri Kähler-Einstein metrics: from cones to cusps (2015) (https://arxiv.org/abs/1504.01947, to appear in J. Reine Angew. Math.)

[24] Hwang, Andrew D. On existence of Kähler metrics with constant scalar curvature, Osaka J. Math., Volume 31 (1994) no. 3, pp. 561-595 http://projecteuclid.org/getrecord?id=euclid.ojm/1200785464 | MR | Zbl

[25] Hwang, Andrew D.; Singer, Michael A. A momentum construction for circle-invariant Kähler metrics, Trans. Am. Math. Soc., Volume 354 (2002) no. 6, pp. 2285-2325 | DOI | MR | Zbl

[26] Jeffres, Thalia Uniqueness of Kähler-Einstein cone metrics, Publ. Mat., Volume 44 (2000) no. 2, pp. 437-448 | DOI | MR | Zbl

[27] Jeffres, Thalia; Mazzeo, Rafe; Rubinstein, Yanir A. Kähler-Einstein metrics with edge singularities, Ann. Math., Volume 183 (2016) no. 1, pp. 95-176 | DOI | MR | Zbl

[28] Keller, Julien About canonical Kähler metrics on Mumford semistable projective bundles over a curve, J. Lond. Math. Soc., Volume 93 (2016) no. 1, pp. 159-174 | DOI | MR | Zbl

[29] Keller, Julien; Zheng, Kai Construction of constant scalar curvature Kähler cone metrics, Proc. Lond. Math. Soc., Volume 117 (2018) no. 3, pp. 527-573 | DOI | Zbl

[30] Kobayashi, Shoshichi Transformation groups in differential geometry, Classics in Mathematics, Springer, 1995, viii+182 pages (Reprint of the 1972 edition) | MR | Zbl

[31] Kobayashi, Shoshichi; Nomizu, Katsumi Foundations of differential geometry. Vol. II, Wiley Classics Library, John Wiley & Sons, 1996, xvi+468 pages (Reprint of the 1969 original, A Wiley-Interscience Publication) | MR | Zbl

[32] LeBrun, Claude; Simanca, Santiago R. Extremal Kähler metrics and complex deformation theory, Geom. Funct. Anal., Volume 4 (1994) no. 3, pp. 298-336 | DOI | MR | Zbl

[33] Li, Chi; Xu, Chenyang Special test configuration and K-stability of Fano varieties, Ann. Math., Volume 180 (2014), pp. 1-36 | MR | Zbl

[34] Li, Haozhao Extremal Kähler metrics and energy functionals on projective bundles, Ann. Global Anal. Geom., Volume 41 (2012) no. 4, pp. 423-445 | DOI | MR | Zbl

[35] Li, Long Subharmonicity of conic Mabuchi’s functional, I, Ann. Inst. Fourier, Volume 68 (2018) no. 2, pp. 805-845 | DOI | MR | Zbl

[36] Li, Long; Zheng, Kai Uniqueness of constant scalar curvature Kähler metrics with cone singularities, I: Reductivity, Math. Ann. (2017) (published online) | DOI | Zbl

[37] McOwen, Robert C. Point singularities and conformal metrics on Riemann surfaces, Proc. Am. Math. Soc., Volume 103 (1988) no. 1, pp. 222-224 | DOI | MR | Zbl

[38] Mondello, Gabriele; Panov, Dmitri Spherical metrics with conical singularities on a 2-sphere: angle constraints, Int. Math. Res. Not. (2016) no. 16, pp. 4937-4995 | DOI | MR | Zbl

[39] Odaka, Yuji; Sun, Song Testing log K-stability by blowing up formalism, Ann. Fac. Sci. Toulouse Math. (6), Volume 24 (2015) no. 3, pp. 505-522 | DOI | MR | Zbl

[40] Ross, Julius; Thomas, Richard An obstruction to the existence of constant scalar curvature Kähler metrics, J. Differ. Geom., Volume 72 (2006) no. 3, pp. 429-466 http://projecteuclid.org/getrecord?id=euclid.jdg/1143593746 | DOI | MR | Zbl

[41] Singer, Michael A. private communication

[42] Sjöström Dyrefelt, Zakarias K-semistability of cscK manifolds with transcendental cohomology class (2016) (https://arxiv.org/abs/1601.07659, to appear in J. Geom. Anal.) | Zbl

[43] Song, Jian; Wang, Xiaowei The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality, Geom. Topol., Volume 20 (2016) no. 1, pp. 49-102 | DOI | MR | Zbl

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

[45] Székelyhidi, Gábor Introduction to Extremal Kähler metrics, Graduate Studies in Mathematics, 152, American Mathematical Society, 2014, xvi+192 pages | Zbl

[46] Székelyhidi, Gábor Filtrations and test-configurations, Math. Ann., Volume 362 (2015) no. 1-2, pp. 451-484 | DOI | MR | Zbl

[47] Tian, Gang Kähler-Einstein metrics on algebraic manifolds, Transcendental methods in algebraic geometry (Cetraro, 1994) (Lecture Notes in Mathematics), Volume 1646, Springer, 1996, pp. 143-185 | DOI | MR | Zbl

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

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

[50] Troyanov, Marc Prescribing curvature on compact surfaces with conical singularities, Trans. Am. Math. Soc., Volume 324 (1991) no. 2, pp. 793-821 | DOI | MR | Zbl

[51] Yau, Shing Tung Problem section, Seminar on Differential Geometry (Annals of Mathematics Studies), Volume 102, Princeton University Press, 1982, pp. 669-706 | MR | Zbl

[52] Zheng, Kai Kähler metrics with cone singularities and uniqueness problem, Current trends in analysis and its applications (Trends Math.), Birkhäuser/Springer, Cham, 2015, pp. 395-408 | DOI | MR | Zbl

Cité par Sources :