Convergence of cscK metrics on smooth minimal models of general type
[Convergence des métriques cscK sur des modèles lisses minimaux de type général]
Liu, Wanxing1
1 Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208 (USA)
Annales de l'Institut Fourier, Online first, 30 p.

On considère des métriques de Kähler à courbure scalaire constante sur une surface lissemodèle minimal de type général dans un voisinage de la classe canonique, qui estla perturbation de la classe canonique par une métrique de Kähler fixe. Nous montrons queles séquences de ces métriques convergent en douceur sur des sous-ensembles compacts loin d’un sous-ensemble. variété à la métrique singuliére de Kähler Einstein dans la classe canonique. Cela confirme partiellement une conjecture de Jian–Shi–Song sur le comportement de convergence detels séquences.

We consider constant scalar curvature Kähler metrics on a smooth minimal model of general type in a neighborhood of the canonical class, which is the perturbation of the canonical class by a fixed Kähler metric. We show that sequences of such metrics converge smoothly on compact subsets away from a subvariety to the singular Kähler–Einstein metric in the canonical class. This confirms partially a conjecture of Jian–Shi–Song about the convergence behavior of such sequences.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3637
Classification : 32W20
Keywords: Kähler geometry, Constant Scalar Curvature Kähler metric.
Mot clés : Géométrie de Kähler, Courbure scalaire constante Métrique de Kähler.
Liu, Wanxing 1

1 Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208 (USA) 
@unpublished{AIF_0__0_0_A77_0,
     author = {Liu, Wanxing},
     title = {Convergence of {cscK} metrics on smooth minimal models of general type},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3637},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Liu, Wanxing
TI  - Convergence of cscK metrics on smooth minimal models of general type
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3637
LA  - en
ID  - AIF_0__0_0_A77_0
ER  - 
%0 Unpublished Work
%A Liu, Wanxing
%T Convergence of cscK metrics on smooth minimal models of general type
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3637
%G en
%F AIF_0__0_0_A77_0
Liu, Wanxing. Convergence of cscK metrics on smooth minimal models of general type. Annales de l'Institut Fourier, Online first, 30 p.

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

[2] Aubin, Thierry Équations du type Monge–Ampère sur les variétés kählériennes compactes, Bull. Sci. Math., Volume 102 (1978) no. 1, pp. 63-95 | MR | Zbl

[3] 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

[4] Berman, Robert J.; Darvas, Tamás; Lu, Chinh H. Regularity of weak minimizers of the K-energy and applications to properness and K-stability, Ann. Sci. Éc. Norm. Supér., Volume 53 (2020) no. 2, pp. 267-289 | DOI | MR | Zbl

[5] Calabi, Eugenio Extremal Kähler metrics, Seminar on Differential Geometry (Annals of Mathematics Studies), Volume 102, Princeton University Press, 1982, pp. 259-290 | MR | Zbl

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

[7] Chen, Xiuxiong On the lower bound of the Mabuchi energy and its application, Int. Math. Res. Not. (2000) no. 12, pp. 607-623 | DOI | MR | Zbl

[8] 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

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

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

[11] 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

[12] 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

[13] 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

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

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

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

[17] Demailly, Jean-Pierre; Pali, Nefton Degenerate complex Monge–Ampère equations over compact Kähler manifolds, Int. J. Math., Volume 21 (2010) no. 3, pp. 357-405 | DOI | MR | Zbl

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

[19] Donaldson, Simon K. Scalar curvature and projective embeddings. I, J. Differ. Geom., Volume 59 (2001) no. 3, pp. 479-522 | MR | Zbl

[20] Donaldson, Simon K. Constant scalar curvature metrics on toric surfaces, Geom. Funct. Anal., Volume 19 (2009) no. 1, pp. 83-136 | DOI | MR | Zbl

[21] Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed A priori L ∞ -estimates for degenerate complex Monge–Ampère equations, Int. Math. Res. Not. (2008), rnn070, 8 pages | DOI | MR | Zbl

[22] Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Singular Kähler-Einstein metrics, J. Am. Math. Soc., Volume 22 (2009) no. 3, pp. 607-639 | DOI | MR | Zbl

[23] Fine, Joel Constant scalar curvature Kähler metrics on fibred complex surfaces, J. Differ. Geom., Volume 68 (2004) no. 3, pp. 397-432 | DOI | MR | Zbl

[24] Gross, Mark; Wilson, Pelham M. H. Large complex structure limits of K3 surfaces, J. Differ. Geom., Volume 55 (2000) no. 3, pp. 475-546 | DOI | MR | Zbl

[25] He, Weiyong Scalar curvature and properness on Sasaki manifolds (2018) (https://arxiv.org/abs/1802.03841)

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

[27] Jian, Wangjian; Shi, Yalong; Song, Jian A remark on constant scalar curvature Kähler metrics on minimal models, Proc. Am. Math. Soc., Volume 147 (2019) no. 8, pp. 3507-3513 | DOI | MR | Zbl

[28] Kobayashi, Ryoichi Einstein–Kähler V-metrics on open Satake V-surfaces with isolated quotient singularities, Math. Ann., Volume 272 (1985) no. 3, pp. 385-398 | DOI | MR | Zbl

[29] Koł odziej, Sł awomir The complex Monge–Ampère equation, Acta Math., Volume 180 (1998) no. 1, pp. 69-117 | DOI | MR | Zbl

[30] Li, Haozhao; Shi, Yalong A criterion for the properness of the K-energy in a general Kähler class (II), Commun. Contemp. Math., Volume 18 (2016) no. 6, 1550071, 15 pages | DOI | MR | Zbl

[31] Phong, Duong H.; Song, Jian; Sturm, Jacob Complex Monge–Ampère equations, Surveys in differential geometry. Vol. XVII (Surveys in Differential Geometry), Volume 17, International Press, 2012, pp. 327-410 | DOI | MR | Zbl

[32] Phong, Duong H.; Sturm, Jacob Lectures on stability and constant scalar curvature, Handbook of geometric analysis, No. 3 (Advanced Lectures in Mathematics), Volume 14, International Press, 2010, pp. 357-436 | MR | Zbl

[33] 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 | DOI | MR | Zbl

[34] Rubinstein, Yanir A. Tian’s properness conjectures: an introduction to Kähler geometry (2018) (https://arxiv.org/abs/1807.00928)

[35] Shen, Xi Sisi Estimates for metrics of constant Chern scalar curvature, Math. Res. Lett., Volume 28 (2021) no. 5, pp. 1525-1550 | DOI | MR | Zbl

[36] Sjöström Dyrefelt, Zakarias Existence of cscK metrics on smooth minimal models, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 23 (2022) no. 1, pp. 223-232 | DOI | MR | Zbl

[37] Song, Jian Nakai-Moishezon criterions for complex Hessian equations (2020) (https://arxiv.org/abs/2012.07956)

[38] Song, Jian; Tian, Gang Bounding scalar curvature for global solutions of the Kähler–Ricci flow, Am. J. Math., Volume 138 (2016) no. 3, pp. 683-695 | DOI | MR | Zbl

[39] Song, Jian; Tian, Gang The Kähler–Ricci flow through singularities, Invent. Math., Volume 207 (2017) no. 2, pp. 519-595 | DOI | MR | Zbl

[40] Song, Jian; Weinkove, Ben On the convergence and singularities of the J-flow with applications to the Mabuchi energy, Commun. Pure Appl. Math., Volume 61 (2008) no. 2, pp. 210-229 | DOI | MR | Zbl

[41] Song, Jian; Weinkove, Ben An introduction to the Kähler–Ricci flow, An introduction to the Kähler–Ricci flow (Lecture Notes in Mathematics), Volume 2086, Springer, 2013, pp. 89-188 | DOI | MR | Zbl

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

[43] Strominger, Andrew; Yau, Shing-Tung; Zaslow, Eric Mirror symmetry is T-duality, Nucl. Phys., B, Volume 479 (1996) no. 1-2, pp. 243-259 | DOI | MR | Zbl

[44] Székelyhidi, Gábor An introduction to extremal Kähler metrics, Graduate Studies in Mathematics, 152, American Mathematical Society, 2014, xvi+192 pages | DOI | MR

[45] Tian, Gang On Kähler–Einstein metrics on certain Kähler manifolds with C 1 (M)>0, Invent. Math., Volume 89 (1987) no. 2, pp. 225-246 | DOI | MR | Zbl

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

[47] Tian, Gang Canonical metrics in Kähler geometry. Notes taken by Meike Akveld, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2000, vi+101 pages | DOI | MR | Zbl

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

[49] Tian, Gang; Zhang, Zhou On the Kähler–Ricci flow on projective manifolds of general type, Chin. Ann. Math., Ser. B, Volume 27 (2006) no. 2, pp. 179-192 | DOI | MR | Zbl

[50] Tosatti, Valentino KAWA lecture notes on the Kähler–Ricci flow, Ann. Fac. Sci. Toulouse, Math., Volume 27 (2018) no. 2, pp. 285-376 | DOI | Numdam | MR | Zbl

[51] Tsuji, Hajime Existence and degeneration of Kähler–Einstein metrics on minimal algebraic varieties of general type, Math. Ann., Volume 281 (1988) no. 1, pp. 123-133 | DOI | MR | Zbl

[52] Weinkove, Ben On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energy, J. Differ. Geom., Volume 73 (2006) no. 2, pp. 351-358 | DOI | MR | Zbl

[53] Yau, Shing Tung On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I, Commun. Pure Appl. Math., Volume 31 (1978) no. 3, pp. 339-411 | DOI | MR | Zbl

[54] Zhang, Zhou Degenerate Monge–Ampère equations over projective manifolds, Ph. D. Thesis, Massachusetts Institute of Technology (2006)

[55] Zheng, Kai Existence of constant scalar curvature Kähler cone metrics, properness and geodesic stability (2018) (https://arxiv.org/abs/1803.09506)

Cité par Sources :