A Yau–Tian–Donaldson correspondence on a class of toric fibrations
Annales de l'Institut Fourier, Volume 73 (2023) no. 6, pp. 2567-2604.

We establish a Yau–Tian–Donaldson type correspondence, expressed in terms of a single Delzant polytope, concerning the existence of extremal Kähler metrics on a large class of toric fibrations, introduced by Apostolov–Calderbank–Gauduchon–Tonnesen-Friedman and called semi-simple principal toric fibrations. We use that an extremal metric on the total space corresponds to a weighted constant scalar curvature Kähler metric (in the sense of Lahdili) on the corresponding toric fiber in order to obtain an equivalence between the existence of extremal Kähler metrics on the total space and a suitable notion of weighted uniform K-stability of the corresponding Delzant polytope. As an application, we show that the projective plane bundle ( 0 1 2 ), where i are holomorphic line bundles over an elliptic curve, admits an extremal metric in every Kähler class.

Nous établissons une correspondance du type Yau–Tian–Donaldson, exprimée en terme d’un polytope de Delzant, concernant l’existence de métriques Kähler extrémales sur une large classe de fibrations toriques définie par Apostolov–Calderbank–Gauduchon–Tonnesen-Friedman et appelée semi-simple principal toric fibrations. Nous utilisons qu’une extrémale sur l’espace total correspond à une métrique à courbure scalaire constante pondérée (dans le sens de Lahdili) sur la fibre torique correspondante pour obtenir une équivalence entre l’existence des métriques extrémales sur l’espace total et une notion appropriée de K-stabilité uniforme pondéree du polytope de Delzant correspondant. En tant qu’application, nous montrons que le fibré en plan projectif ( 0 1 2 ), où les i sont des fibrés holomorphes au dessus d’une courbe elliptique, admet une métrique extrémale dans chaque classe de Kähler.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3580
Classification: 00X99
Keywords: Differential geometry, complex geometry, complex analysis, toric geometry.
Mot clés : Géométrie différentielle, géométrie complexe, fibration toric principal semisimple, métrique Kähler extrémale, métrique à csc pondérée, K-stabilité uniforme.

Jubert, Simon 1, 2

1 Departement de Mathématiques UQAM, C.P. 8888 Succursale Centre-ville, Montréal (Quebec) H3C 3P8 (Canada)
2 Institut de Mathématiques de Toulouse Université Paul Sabatier 118 route de Narbonne 31062, Toulouse (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2023__73_6_2567_0,
     author = {Jubert, Simon},
     title = {A {Yau{\textendash}Tian{\textendash}Donaldson} correspondence on a class of toric fibrations},
     journal = {Annales de l'Institut Fourier},
     pages = {2567--2604},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {73},
     number = {6},
     year = {2023},
     doi = {10.5802/aif.3580},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3580/}
}
TY  - JOUR
AU  - Jubert, Simon
TI  - A Yau–Tian–Donaldson correspondence on a class of toric fibrations
JO  - Annales de l'Institut Fourier
PY  - 2023
SP  - 2567
EP  - 2604
VL  - 73
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3580/
DO  - 10.5802/aif.3580
LA  - en
ID  - AIF_2023__73_6_2567_0
ER  - 
%0 Journal Article
%A Jubert, Simon
%T A Yau–Tian–Donaldson correspondence on a class of toric fibrations
%J Annales de l'Institut Fourier
%D 2023
%P 2567-2604
%V 73
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3580/
%R 10.5802/aif.3580
%G en
%F AIF_2023__73_6_2567_0
Jubert, Simon. A Yau–Tian–Donaldson correspondence on a class of toric fibrations. Annales de l'Institut Fourier, Volume 73 (2023) no. 6, pp. 2567-2604. doi : 10.5802/aif.3580. https://aif.centre-mersenne.org/articles/10.5802/aif.3580/

[1] Abreu, Miguel Kähler geometry of toric varieties and extremal metrics, Internat. J. Math., Volume 9 (1998) no. 6, pp. 641-651 | DOI | MR | Zbl

[2] Abreu, Miguel Kähler geometry of toric manifolds in symplectic coordinates, Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001) (Fields Inst. Commun.), Volume 35, Amer. Math. Soc., Providence, RI, 2003, pp. 1-24 | MR | Zbl

[3] Apostolov, Vestislav The Kähler geometry of toric manifolds (Lecture notes availabe at http://profmath.uqam.ca//~apostolo/notes.html)

[4] Apostolov, Vestislav; Calderbank, David M. J.; Gauduchon, Paul Hamiltonian 2-forms in Kähler geometry. I. General theory, J. Differential Geom., Volume 73 (2006) no. 3, pp. 359-412 | DOI | MR | Zbl

[5] Apostolov, Vestislav; Calderbank, David M. J.; Gauduchon, Paul; Tønnesen-Friedman, Christina W. Hamiltonian 2-forms in Kähler geometry. II. Global classification, J. Differential Geom., Volume 68 (2004) no. 2, pp. 277-345 | DOI | MR | Zbl

[6] 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

[7] Apostolov, Vestislav; Calderbank, David M. J.; Gauduchon, Paul; Tønnesen-Friedman, Christina W. Extremal Kähler metrics on projective bundles over a curve, Adv. Math., Volume 227 (2011) no. 6, pp. 2385-2424 | DOI | MR | Zbl

[8] Apostolov, Vestislav; Calderbank, David M. J.; Legendre, Eveline Weighted K-stability of polarized varieties and extremality of Sasaki manifolds (to appear in Adv. Math., https://arxiv.org/abs/2012.08628)

[9] Apostolov, Vestislav; Jubert, Simon; Lahdili, Abdellah Weighted K-stability and coercivity with applications to extremal Kähler and Sasaki metrics (to appear in Geometry and Topology, https://arxiv.org/abs/2104.09709)

[10] Apostolov, Vestislav; Maschler, Gideon Conformally Kähler, Einstein–Maxwell geometry, J. Eur. Math. Soc. (JEMS), Volume 21 (2019) no. 5, pp. 1319-1360 | DOI | MR | Zbl

[11] Atiyah, M. F. Convexity and commuting Hamiltonians, Bull. London Math. Soc., Volume 14 (1982) no. 1, pp. 1-15 | DOI | MR | Zbl

[12] Berman, Robert J.; Berndtsson, Bo Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics, J. Amer. Math. Soc., Volume 30 (2017) no. 4, pp. 1165-1196 | DOI | MR | Zbl

[13] Berman, Robert J.; Boucksom, Sébastien; Jonsson, Mattias A variational approach to the Yau–Tian–Donaldson conjecture, J. Amer. Math. Soc., Volume 34 (2021) no. 3, pp. 605-652 | DOI | MR | Zbl

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

[15] Calabi, Eugenio Extremal Kähler metrics, Seminar on Differential Geometry (Ann. of Math. Stud.), Volume 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 259-290 | MR | Zbl

[16] Calabi, Eugenio Extremal Kähler metrics. II, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 95-114 | MR | Zbl

[17] Chen, Bohui; Li, An-Min; Sheng, Li Uniform K-stability for extremal metrics on toric varieties, J. Differential Equations, Volume 257 (2014) no. 5, pp. 1487-1500 | DOI | MR | Zbl

[18] Chen, Xiuxiong The space of Kähler metrics, J. Differential Geom., Volume 56 (2000) no. 2, pp. 189-234 | DOI | MR | Zbl

[19] Chen, Xiuxiong On the existence of constant scalar curvature Kähler metric: a new perspective, Ann. Math. Qué., Volume 42 (2018) no. 2, pp. 169-189 | DOI | MR | Zbl

[20] Chen, Xiuxiong; Cheng, Jingrui On the constant scalar curvature Kähler metrics (I) – A priori estimates, J. Amer. Math. Soc., Volume 34 (2021) no. 4, pp. 909-936 | DOI | MR | Zbl

[21] Chen, Xiuxiong; Cheng, Jingrui On the constant scalar curvature Kähler metrics (II) – Existence results, J. Amer. Math. Soc., Volume 34 (2021) no. 4, pp. 937-1009 | DOI | MR | Zbl

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

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

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

[25] Chen, Xiuxiong; Paun, Mihai; Zeng, Yu On deformation of extremal metrics (https://arxiv.org/abs/1506.01290v2)

[26] Darvas, Tamás The Mabuchi completion of the space of Kähler potentials, Amer. J. Math., Volume 139 (2017) no. 5, pp. 1275-1313 | DOI | MR | Zbl

[27] Darvas, Tamás; Rubinstein, Yanir A. Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics, J. Amer. Math. Soc., Volume 30 (2017) no. 2, pp. 347-387 | DOI | MR | Zbl

[28] Delcroix, Thibaut The Yau–Tian–Donaldson conjecture for cohomogeneity one manifolds (https://arxiv.org/abs/2011.07135)

[29] Delzant, Thomas Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France, Volume 116 (1988) no. 3, pp. 315-339 | DOI | MR | Zbl

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

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

[32] Donaldson, S. K. Scalar curvature and stability of toric varieties, J. Differential Geom., Volume 62 (2002) no. 2, pp. 289-349 | DOI | MR | Zbl

[33] Donaldson, S. K. Extremal metrics on toric surfaces: a continuity method, J. Differential Geom., Volume 79 (2008) no. 3, pp. 389-432 | DOI | MR | Zbl

[34] Fujiki, Akira Remarks on extremal Kähler metrics on ruled manifolds, Nagoya Math. J., Volume 126 (1992), pp. 89-101 | DOI | MR | Zbl

[35] Futaki, Akito; Mabuchi, Toshiki Bilinear forms and extremal Kähler vector fields associated with Kähler classes, Math. Ann., Volume 301 (1995) no. 2, pp. 199-210 | DOI | MR | Zbl

[36] Gauduchon, Paul Calabi’s extremal Kähler metrics: An elementary introduction (Lecture Notes)

[37] Guan, Daniel Existence of extremal metrics on compact almost homogeneous Kähler manifolds with two ends, Trans. Amer. Math. Soc., Volume 347 (1995) no. 6, pp. 2255-2262 | DOI | MR | Zbl

[38] Guan, Daniel On modified Mabuchi functional and Mabuchi moduli space of Kähler metrics on toric bundles, Math. Res. Lett., Volume 6 (1999) no. 5-6, pp. 547-555 | DOI | MR | Zbl

[39] Guillemin, V.; Sternberg, S. Convexity properties of the moment mapping, Invent. Math., Volume 67 (1982) no. 3, pp. 491-513 | DOI | MR | Zbl

[40] Guillemin, Victor Kaehler structures on toric varieties, J. Differential Geom., Volume 40 (1994) no. 2, pp. 285-309 | DOI | MR | Zbl

[41] Hashimoto, Yoshinori Existence of twisted constant scalar curvature Kähler metrics with a large twist, Math. Z., Volume 292 (2019) no. 3-4, pp. 791-803 | DOI | MR | Zbl

[42] He, Weiyong On Calabi’s extremal metric and properness, Trans. Amer. Math. Soc., Volume 372 (2019) no. 8, pp. 5595-5619 | DOI | MR | Zbl

[43] Hisamoto, Tomoyuki Stability and coercivity for toric polarizations (https://arxiv.org/abs/1610.07998)

[44] Hwang, Andrew D. On existence of Kähler metrics with constant scalar curvature, Osaka J. Math., Volume 31 (1994) no. 3, pp. 561-595 | MR | Zbl

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

[46] Inoue, Eiji Constant μ-scalar curvature Kähler metric – formulation and foundational results (to appear in J. Geom. Anal., https://arxiv.org/abs/1902.00664)

[47] Koiso, Norihito; Sakane, Yusuke Nonhomogeneous Kähler–Einstein metrics on compact complex manifolds, Curvature and topology of Riemannian manifolds (Katata, 1985) (Lecture Notes in Math.), Volume 1201, Springer, Berlin, 1986, pp. 165-179 | DOI | MR | Zbl

[48] Lahdili, Abdellah Convexity of the weighted Mabuchi functional and the uniqueness of weighted extremal metrics (https://arxiv.org/abs/2007.01345)

[49] Lahdili, Abdellah Kähler metrics with constant weighted scalar curvature and weighted K-stability, Proc. Lond. Math. Soc. (3), Volume 119 (2019) no. 4, pp. 1065-1114 | DOI | MR | Zbl

[50] Legendre, Eveline A note on extremal toric almost Kähler metrics (https://arxiv.org/abs/1811.05693)

[51] Lejmi, Mehdi Extremal almost-Kähler metrics, Internat. J. Math., Volume 21 (2010) no. 12, pp. 1639-1662 | DOI | MR | Zbl

[52] Li, An-Min; Lian, Zhao; Sheng, Li Some estimates for the generalized Abreu equation, Differential Geom. Appl., Volume 48 (2016), pp. 87-103 | DOI | MR | Zbl

[53] Li, An-Min; Lian, Zhao; Sheng, Li Interior regularity for the generalized Abreu equation, Internat. J. Math., Volume 28 (2017) no. 7, 1750053, 18 pages | MR | Zbl

[54] Li, An-Min; Sheng, Li; Zhao, Guosong Differential inequalities on homogeneous toric bundles, J. Geom., Volume 108 (2017) no. 2, pp. 775-790 | DOI | MR | Zbl

[55] Li, Chi Geodesic rays and stability in the cscK problem (accepted by Ann. Sci. Éc. Norm. Supér., https://arxiv.org/abs/2001.01366v3)

[56] Li, Chi; Tian, Gang; Wang, Feng The uniform version of Yau-Tian-Donaldson conjecture for singular Fano varieties (https://arxiv.org/abs/1903.01215)

[57] Mabuchi, Toshiki K-energy maps integrating Futaki invariants, Tohoku Math. J. (2), Volume 38 (1986) no. 4, pp. 575-593 | DOI | MR | Zbl

[58] Mabuchi, Toshiki Some symplectic geometry on compact Kähler manifolds. I, Osaka J. Math., Volume 24 (1987) no. 2, pp. 227-252 | MR | Zbl

[59] Sakane, Yusuke Examples of compact Einstein Kähler manifolds with positive Ricci tensor, Osaka J. Math., Volume 23 (1986) no. 3, pp. 585-616 | MR | Zbl

[60] Schwarz, Gerald W. Smooth functions invariant under the action of a compact Lie group, Topology, Volume 14 (1975), pp. 63-68 | DOI | MR | Zbl

[61] Sjöström Dyrefelt, Zakarias K-semistability of cscK manifolds with transcendental cohomology class, J. Geom. Anal., Volume 28 (2018) no. 4, pp. 2927-2960 | DOI | MR | Zbl

[62] Székelyhidi, Gábor Extremal metric and K-stability (Phd Thesis) (https://arxiv.org/abs/math/0611002)

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

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

[65] Tønnesen-Friedman, Christina W. Kähler Yamabe minimizers on minimal ruled surfaces, Math. Scand., Volume 90 (2002) no. 2, pp. 180-190 | DOI | MR | Zbl

[66] Zhou, Bin; Zhu, Xiaohua Relative K-stability and modified K-energy on toric manifolds, Adv. Math., Volume 219 (2008) no. 4, pp. 1327-1362 | DOI | MR | Zbl

Cited by Sources: