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Original Article

The Uniform Version of Yau–Tian–Donaldson Conjecture for Singular Fano Varieties

Department of Mathematics, Purdue University, West Lafayette, IN 47907-2067, USA
Present Address: Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA
School of Mathematical Sciences and BICMR, Peking University, Yiheyuan Road 5, Beijing 100871, China
School of Mathematical Sciences, Zhejiang University, Zheda Road 38, Hangzhou 310027, Zhejiang, China
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Abstract

We prove the following result: if a Q-Fano variety is uniformly K-stable, then it admits a Kähler–Einstein metric. This proves the uniform version of Yau–Tian–Donaldson conjecture for all (singular) Fano varieties with discrete automorphism groups. We achieve this by modifying Berman–Boucksom–Jonsson's strategy in the smooth case with appropriate perturbative arguments. This perturbation approach depends on the valuative criterion and non-Archimedean estimates, and is motivated by our previous paper.

References

1

Berman, R.: K-polystability of Q-Fano varieties admitting Kähler–Einstein metrics. Invent. Math. 203(3), 973–1025 (2016)

2
Berman, R.: Convexity of the Mabuchi functional on singular Fanos, notes through private communication
3

Berman, R., Berndtsson, R.: Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics. J. Am. Math. Soc. 30, 1165–1196 (2017)

4

Berman, R., Boucksom, S., Eyssidieux, P., Guedj, V., Zeriahi, A.: Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties. J. Reine Angew. Math. 751, 27–89 (2019)

5

Berman, R., Boucksom, S., Guedj, V., Zeriahi, A.: A variational approach to complex Monge–Ampère equations. Publ. Math. Inst. Hautes Études Sci. 117, 179–245 (2013)

6
Berman, R., Boucksom, S., Jonsson, M.: A variational approach to the Yau–Tian–Donaldson conjecture, arXiv:1509.04561v1-v2
7
Berman, R., Boucksom, S., Jonsson, M.: A variational approach to the Yau–Tian–Donaldson conjecture, arXiv:1509.04561v3
8

Berman, R., Darvas, T., Lu, C.H.: Convexity of the extended K-energy and the large time behaviour of the weak Calabi flow. Geom. Topol. 21, 2945–2988 (2017)

9

Berndtsson, B.: A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry. Invent. Math. 200(1), 149–200 (2015)

10
Blum, H., Jonsson, M.: Thresholds, valuations, and K-stability. Adv. Math. 365, 107062, 57 pp. (2020)
11

Blum, H., Xu, C.Y.: Uniqueness of K-polystable degenerations of Fano varieties. Ann. Math. (2) 190(2), 609–656 (2019)

12

Boucksom, S., Favre, C., Jonsson, M.: Valuations and plurisubharmonic singularities. Publ. RIMS 44, 449–494 (2008)

13

Boucksom, S., Hisamoto, T., Jonsson, M.: Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs. Ann. Inst. Fourier (Grenoble) 67, 743–841 (2017)

14

Boucksom, S., Hisamoto, T., Jonsson, M.: Uniform K-stability and asymptotics of energy functionals in Kähler geometry. J. Eur. Math. Soc. (JEMS) 21(9), 2905–2944 (2019)

15
Boucksom, S., Jonsson, M.: Singular semipositive metrics on line bundles on varieties over trivially valued fields. arXiv:1801.08229
16
Boucksom, S., Jonsson, M.: A non-Archimedean approach to K-stability. arXiv:1805.11160v1
17

Chen, X.X.: The space of Kähler metrics. J. Differ. Geom. 56(2), 189–234 (2000)

18
Chen, X.X., Donaldson, S.K., Sun, S.: Kähler-Einstein metrics on Fano manifolds, I-III. J. Amer. Math. Soc. 28, 183–197, 199–234, 235–278 (2015)
19
Chu, J.C., Tosatti, V., Weinkove, B.: On the C1,1 regularity of geodesics in the space of Kähler metrics. Ann. PDE 3(2), Paper No. 15, 12 pp. (2017)
20

Coman, D., Guedj, V., Zeriahi, A.: Extension of plurisubharmonic functions with growth control. J. Reine Angew. Math. 676, 33–49 (2013)

21

Coman, D., Ma, X.N., Marinescu, G.: Equidistribution for sequences of line bundles on normal Kähler spaces. Geom. Topol. 21, 923–962 (2017)

22

Darvas, T.: The Mabuchi geometry of finite energy classes. Adv. Math. 285, 182–219 (2015)

23

Darvas, T.: Metric geometry of normal Kähler spaces, energy properness, and existence of canonical metrics. Int. Math. Res. Not. (IMRN) 2017(22), 6752–6777 (2017)

24

Darvas, T., He, W.Y.: Geodesic rays and Kähler–Ricci trajectories on Fano manifolds. Trans. Am. Math. Soc. 369, 5069–5085 (2017)

25

Darvas, T., Rubinstein, Y.: Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics. J. Am. Math. Soc. 30, 347–387 (2017)

26

Demailly, J.-P.: Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1, 361–409 (1992)

27

Demailly, J.-P., Pali, N.: Degenerate complex Monge–Ampère equations over compact Kähler manifolds. Int. J. Math. 21(3), 357–405 (2010)

28

Dervan, R.: Uniform stability of twisted constant scalar curvature Kähler metrics. Int. Math. Res. Not. (IMRN) 2016(15), 4728–4783 (2016)

29

Di Nezza, E., Guedj, V.: Geometry and topology of the space of Kähler metrics on singular varieties. Compos. Math. 154, 1593–1632 (2018)

30
Donaldson, S.: Symmetric spaces, Kähler geometry and Hamiltonian dynamics. In: Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, Vol. 196, Amer. Math. Soc., Providence, RI, pp. 13–33 (1999)
31

Donaldson, S.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 62(2), 289–349 (2002)

32

Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler–Einstein metrics. J. Am. Math. Soc. 22, 607–639 (2009)

33

Fujita, K.: A valuative criterion for uniform K-stability of Q-Fano varieties. J. Reine Angew. Math. 751, 309–338 (2019)

34

Fujita, K.: Uniform K-stability and plt blowups of log Fano pairs. Kyoto J. Math. 59(2), 399–418 (2019)

35

Fujita, K., Odaka, Y.: On the K-stability of Fano varieties and anticanonical divisors. Tohoku Math. J. (2) 70(4), 511–521 (2018)

36

Guedj, V., Zeriahi, A.: The weighted Monge–Ampère energy of quasiplurisubharmonic functions. J. Funct. Anal. 250(2), 442–482 (2007)

37
Guedj, V., Zeriahi, A.: Degenerate Complex Monge–Ampère Equations, EMS Tracts in Mathematics, Vol. 26. EMS, Zürich (2017)
38

Guenancia, H., Pǎun, M.: Conic singularities metrics with prescribed Ricci curvature: general cone angles along normal crossing divisors. J. Differ. Geom. 103(1), 15–57 (2016)

39
Han, J.Y., Li, C.: On the Yau–Tian–Donaldson conjecture for generalized Kähler–Ricci soliton equations. arXiv:2006.00903
40
Hisamoto, T.: Mabuchi’s soliton metric and relative D-stability. arXiv:1905.05948
41

Jeffres, T., Mazzeo, R., Rubinstein, Y.: Kähler–Einstein metrics with edge singularities, with an appendix by C. Li and Y. Rubinstein. Ann. Math. (2) 183(1), 95–176 (2016)

42
Kollár, J.: Lectures on Resolution of Singularities, Annals of Mathematics Studies, Vol. 166. Princeton University Press, Princeton, NJ (2007)
43

Kołodziej, S.: The complex Monge–Ampère equation. Acta Math. 180(1), 69–117 (1998)

44

Li, C.: K-semistability is equivariant volume minimization. Duke Math. J. 166(16), 3147–3218 (2017)

45
Li, C.: G-uniform stability and Kähler–Einstein metrics on Fano varieties. arXiv:1907.09399
46
Li, C., Tian, G., Wang, F.: On the Yau–Tian–Donaldson conjecture for singular Fano varieties. arXiv:1711.09530v3
47

Li, C., Wang, X.W., Xu, C.Y.: On the proper moduli spaces of smoothable Kähler–Einstein Fano varieties. Duke Math. J. 168(8), 1387–1459 (2019)

48

Li, C., Xu, C.Y.: Special test configuration and K-stability of Fano varieties. Ann. Math. (2) 180(1), 197–232 (2014)

49

Odaka, Y.: The GIT stability of polarized varieties via discrepancy. Ann. Math. (2) 177(2), 645–661 (2013)

50

Spotti, C., Sun, S., Yao, C.J.: Existence and deformations of Kähler–Einstein metrics on smoothable Q-Fano varieties. Duke Math. J. 165(16), 3043–3083 (2016)

51

Tian, G.: Kähler–Einstein metrics with positive scalar curvature. Invent. Math. 130, 1–37 (1997)

52

Tian, G.: K-stability and Kähler–Einstein metrics. Commun. Pure Appl. Math. 68(7), 1085–1156 (2015)

53

Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I. Commun. Pure Appl. Math. 31, 339–411 (1978)

Peking Mathematical Journal
Pages 383-426
Cite this article:
Li, C., Tian, G. & Wang, F. The Uniform Version of Yau–Tian–Donaldson Conjecture for Singular Fano Varieties. Peking Math J 5, 383-426 (2022). https://doi.org/10.1007/s42543-021-00039-5
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