K-stability of Fano varieties

This article discusses specifically the K-stability of Fano varieties. For K-stability more generally, see K-stability.

In mathematics, and in particular algebraic geometry, K-stability is an algebro-geometric stability condition for projective algebraic varieties and complex manifolds. K-stability is of particular importance for the case of Fano varieties, where it is the correct stability condition to allow the formation of moduli spaces, and where it precisely characterises the existence of Kähler–Einstein metrics.

K-stability was first defined for Fano manifolds by Gang Tian in 1997 in response to a conjecture of Shing-Tung Yau from 1993 that there should exist a stability condition which characterises the existence of a Kähler–Einstein metric on a Fano manifold.[1][2] It was defined in reference to the K-energy functional previously introduced by Toshiki Mabuchi.[3] Tian's definition of K-stability was reformulated by Simon Donaldson in 2001 in a purely algebro-geometric way.[4]

K-stability has become an important notion in the study and classification of Fano varieties. In 2012 Xiuxiong Chen, Donaldson, and Song Sun and independently Gang Tian[5] proved that a smooth Fano manifold is K-polystable if and only if it admits a Kähler–Einstein metric.[6][7][8] This was later generalised to singular K-polystable Fano varieties due to the work of Berman–Boucksom–Jonsson and others. K-stability is important in constructing moduli spaces of Fano varieties, where observations going back to the original development of geometric invariant theory show that it is necessary to restrict to a class of stable objects to form good moduli. It is now known through the work of Chenyang Xu and others that there exists a projective coarse moduli space of K-polystable Fano varieties of finite type. This work relies on Caucher Birkar's proof of boundedness of Fano varieties, for which he was awarded the 2018 Fields medal. Due to the reformulations of the K-stability condition by Fujita–Li and Odaka, the K-stability of Fano varieties may be explicitly computed in practice. Which Fano varieties are K-stable is well understood in dimension one, two, and three.

  1. ^ Tian, Gang (1997). "Kähler–Einstein metrics with positive scalar curvature". Inventiones Mathematicae. 130 (1): 1–37. Bibcode:1997InMat.130....1T. doi:10.1007/s002220050176. MR 1471884. S2CID 122529381.
  2. ^ Yau, Shing-Tung (1993). "Open problems in geometry". Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990). Proceedings of Symposia in Pure Mathematics. Vol. 54. pp. 1–28. doi:10.1090/pspum/054.1/1216573. ISBN 9780821814949. MR 1216573.
  3. ^ Mabuchi, Toshiki (1986). "K-energy maps integrating Futaki invariants". Tohoku Mathematical Journal. 38 (4). doi:10.2748/tmj/1178228410. S2CID 122723602.
  4. ^ Donaldson, Simon K. (2002). "Scalar curvature and stability of toric varieties". Journal of Differential Geometry. 62 (2): 289–349. doi:10.4310/jdg/1090950195.
  5. ^ Cite error: The named reference :9 was invoked but never defined (see the help page).
  6. ^ Chen, Xiuxiong; Donaldson, Simon; Sun, Song (2014). "Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities". Journal of the American Mathematical Society. 28: 183–197. arXiv:1211.4566. doi:10.1090/S0894-0347-2014-00799-2. S2CID 119641827.
  7. ^ Chen, Xiuxiong; Donaldson, Simon; Sun, Song (2014). "Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π". Journal of the American Mathematical Society. 28: 199–234. arXiv:1212.4714. doi:10.1090/S0894-0347-2014-00800-6. S2CID 119140033.
  8. ^ Chen, Xiuxiong; Donaldson, Simon; Sun, Song (2014). "Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof". Journal of the American Mathematical Society. 28: 235–278. arXiv:1302.0282. doi:10.1090/S0894-0347-2014-00801-8. S2CID 119575364.