The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality

Song, Jian¹ ; Wang, Xiaowei²

¹ Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA
² Department of Mathematics and Computer Sciences, Rutgers University, Newark, NJ 07102, USA

Geometry & topology, Tome 20 (2016) no. 1, pp. 49-102.

Résumé

We partially confirm a conjecture of Donaldson relating the greatest Ricci lower bound $R\left(X\right)$ to the existence of conical Kähler–Einstein metrics on a Fano manifold $X$. In particular, if $D\in |-K_X|$ is a smooth divisor and the Mabuchi $K$–energy is bounded below, then there exists a unique conical Kähler–Einstein metric satisfying $Ric\left(g\right)=\beta g+\left(1-\beta \right)\left[D\right]$ for any $\beta \in \left(0,1\right)$. We also construct unique conical toric Kähler–Einstein metrics with $\beta =R\left(X\right)$ and a unique effective $\mathbb{Q}$–divisor $D\in \left[-K_X\right]$ for all toric Fano manifolds. Finally we prove a Miyaoka–Yau-type inequality for Fano manifolds with $R\left(X\right)=1$.

Détail

DOI : 10.2140/gt.2016.20.49

Classification : 32Q20, 53C55
Keywords: Kähler–Einstein metric, conic Kähler metric, toric variety

Bibliographie

[1] M Atiyah, C Lebrun, Curvature, cones and characteristic numbers, Math. Proc. Cambridge Philos. Soc. 155 (2013) 13

[2] R J Berman, A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics, (2012)

[3] R J Berman, K–polystability of Q–Fano varieties admitting Kähler–Einstein metrics, preprint (2015)

[4] R J Berman, S Boucksom, V Guedj, A Zeriahi, A variational approach to complex Monge–Ampère equations, preprint (2009)

[5] S Brendle, Ricci flat Kähler metrics with edge singularities, Int. Math. Res. Not. 2013 (2013) 5727

[6] X Chen, S Donaldson, S Sun, Kähler–Einstein metrics on Fano manifolds, II : Limits with cone angle less than 2π, J. Amer. Math. Soc. 28 (2015) 199

[7] V Datar, B Guo, J Song, X Wang, Connecting toric manifolds by conical Kähler–Einstein metrics, preprint (2013)

[8] J P Demailly, On Tian’s invariant and log canonical thresholds, (2008)

[9] W Y Ding, Remarks on the existence problem of positive Kähler–Einstein metrics, Math. Ann. 282 (1988) 463

[10] S K Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002) 289

[11] S K Donaldson, Extremal metrics on toric surfaces: a continuity method, J. Differential Geom. 79 (2008) 389

[12] S K Donaldson, Kähler geometry on toric manifolds, and some other manifolds with large symmetry, from: "Handbook of geometric analysis" (editors L Ji, P Li, R Schoen, L Simon), Adv. Lect. Math. 7, Int. Press (2008) 29

[13] S K Donaldson, Kähler metrics with cone singularities along a divisor, from: "Essays in mathematics and its applications" (editors P M Pardalos, T M Rassias), Springer (2012) 49

[14] P Eyssidieux, V Guedj, A Zeriahi, Singular Kähler–Einstein metrics, J. Amer. Math. Soc. 22 (2009) 607

[15] V Guillemin, Kähler structures on toric varieties, J. Differential Geom. 40 (1994) 285

[16] T Jeffres, R Mazzeo, Y A Rubinstein, Kähler–Einstein metrics with edge singularities, Ann. Math. 183 (2016) 95

[17] S Kołodziej, The complex Monge–Ampère equation, Acta Math. 180 (1998) 69

[18] C Li, Greatest lower bounds on Ricci curvature for toric Fano manifolds, Adv. Math. 226 (2011) 4921

[19] C Li, On the limit behavior of metrics in the continuity method for the Kähler–Einstein problem on a toric Fano manifold, Compos. Math. 148 (2012) 1985

[20] C Li, Remarks on logarithmic K–stability, Commun. Contemp. Math. 17 (2015) 1450020, 17

[21] C Li, S Sun, Conical Kähler–Einstein metrics revisited, Comm. Math. Phys. 331 (2014) 927

[22] H Li, On the lower bound of the K–energy and F–functional, Osaka J. Math. 45 (2008) 253

[23] F Luo, G Tian, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116 (1992) 1119

[24] T Mabuchi, K–energy maps integrating Futaki invariants, Tohoku Math. J. 38 (1986) 575

[25] R C Mcowen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988) 222

[26] Y Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977) 225

[27] O Munteanu, G Székelyhidi, On convergence of the Kähler–Ricci flow, Comm. Anal. Geom. 19 (2011) 887

[28] Y Odaka, S Sun, Testing log K–stability by blowing up formalism, Ann. Fac. Sci. Toulouse Math. 24 (2015) 505

[29] D H Phong, N Sesum, J Sturm, Multiplier ideal sheaves and the Kähler–Ricci flow, Comm. Anal. Geom. 15 (2007) 613

[30] D H Phong, J Song, J Sturm, B Weinkove, The Moser–Trudinger inequality on Kähler–Einstein manifolds, Amer. J. Math. 130 (2008) 1067

[31] Y A Rubinstein, On energy functionals, Kähler–Einstein metrics, and the Moser–Trudinger–Onofri neighborhood, J. Funct. Anal. 255 (2008) 2641

[32] J Song, The α–invariant on certain surfaces with symmetry groups, Trans. Amer. Math. Soc. 357 (2005) 45

[33] S Sun, Note on K–stability of pairs, Math. Ann. 355 (2013) 259

[34] G Székelyhidi, Greatest lower bounds on the Ricci curvature of Fano manifolds, Compos. Math. 147 (2011) 319

[35] G Tian, On Kähler–Einstein metrics on certain Kähler manifolds with C1(M) > 0, Invent. Math. 89 (1987) 225

[36] G Tian, On stability of the tangent bundles of Fano varieties, Internat. J. Math. 3 (1992) 401

[37] G Tian, Kähler–Einstein metrics on algebraic manifolds, from: "Transcendental methods in algebraic geometry", Lecture Notes in Math. 1646, Springer (1996) 143

[38] G Tian, Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997) 1

[39] G Tian, Canonical metrics in Kähler geometry, Birkhäuser (2000)

[40] G Tian, B Wang, On the structure of almost Einstein manifolds, J. Amer. Math. Soc. 28 (2015) 1169

[41] G Tian, S T Yau, Existence of Kähler–Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, from: "Mathematical aspects of string theory" (editor S T Yau), Adv. Ser. Math. Phys. 1, World Scientific (1987) 574

[42] G Tian, X Zhu, A nonlinear inequality of Moser–Trudinger type, Calc. Var. Partial Differential Equations 10 (2000) 349

[43] M Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793

[44] H Tsuji, Stability of tangent bundles of minimal algebraic varieties, Topology 27 (1988) 429

[45] X J Wang, X Zhu, Kähler–Ricci solitons on toric manifolds with positive first Chern class, Adv. Math. 188 (2004) 87

[46] S T Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. USA 74 (1977) 1798

[47] S T Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I, Comm. Pure Appl. Math. 31 (1978) 339

[48] S T Yau, Open problems in geometry, from: "Differential geometry: partial differential equations on manifolds" (editor R Greene), Proc. Sympos. Pure Math. 54, Amer. Math. Soc. (1993) 1

[49] Y Zhang, Miyaoka–Yau inequality for minimal projective manifolds of general type, Proc. Amer. Math. Soc. 137 (2009) 2749

[50] Z Zhang, On degenerate Monge–Ampère equations over closed Kähler manifolds, Int. Math. Res. Not. 2006 (2006)