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.
In 2012 Xiuxiong Chen, Donaldson, and Song Sun proved that a smooth Fano manifold is K-polystable if and only if it admits a Kähler–Einstein metric.
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.
-action on the central fibre of a test configuration can be computed in terms of certain intersection numbers (corresponding to the weight of an action on the so-called CM line bundle).
The fact that one may assume the central fibre of the test configuration is also Fano leads to strong links with birational geometry and the minimal model program, providing a number of alternative characterisations of K-stability described in the following sections.
Tian's original definition of the alpha invariant was analytical in nature, but can be used to verify the existence of a Kähler–Einstein metric in practice.
It was later observed by Odaka–Sano that the alpha invariant can be given a purely algebro-geometric definition in terms of an infimum of the log canonical threshold over all
This invariant was developed by Fujita and Li in an attempt to discover a characterisation of K-stability in terms of divisors or valuations of the Fano variety
[21][25] This work was inspired by earlier ideas of Ross–Thomas which attempted to describe K-stability in terms of algebraic invariants coming out of subschemes of the variety
In particular Fujita realised that Ross–Thomas's notion of slope K-stability was limited by only integrating up to the Seshadri constant of the subscheme, where the natural divisor on the blow-up becomes ample.
The quantized delta invariants can be computed in terms of m-basis type divisors which are given by choices of bases in the fixed finite-dimensional vector space
Interpreted using the delta invariant (and indeed using earlier results), one concludes that a toric Fano manifold is K-stable if and only if the barycentre of its polytope
[8] Chen, Donaldson, and Sun have alleged that Tian's claim to equal priority for the proof is incorrect, and they have accused him of academic misconduct.
[b] Chen, Donaldson, and Sun were recognized by the American Mathematical Society's prestigious 2019 Veblen Prize as having had resolved the conjecture.
The proofs of Chen–Donaldson–Sun and Tian were based on a delicate study of Gromov–Hausdorff limits of Fano manifolds with Ricci curvature bounds.
[34] Robert Berman, Sébastien Boucksom, and Mattias Jonsson also provided a proof from a new variational approach, which interprets K-stability in terms of Non-Archimedean geometry.
[35] Of particular interest is that the proof of Berman–Boucksom–Jonsson also applies to the case of a smooth log Fano pair, and does not use the notion of K-polystability but of uniform K-stability as introduced by Dervan and Boucksom–Hisamoto–Jonsson.
Building on the variational techniques Berman–Boucksom–Jonsson and the so-called quantized delta invariants of Fujita–Odaka, Zhang produced a short quantization-based proof of the YTD conjecture for smooth Fano manifolds.
[36] Using other techniques entirely, Berman has also produced a proof of a YTD-type conjecture using a thermodynamic approach called uniform Gibbs stability, where a Kähler–Einstein metric is constructed through a random point process.
[37][38] The new proof of the Yau–Tian–Donaldson conjecture by Berman–Boucksom–Jonsson using variational techniques opened up the possible study of K-stability and Kähler–Einstein metrics for singular Fano varieties.
By the YTD conjecture these are alternatively moduli spaces of smooth K-polystable Fano varieties with discrete automorphism groups.
Typically to apply Mumford's geometric invariant theory to construct moduli, one must embed a family of varieties inside a fixed finite-dimensional projective space.
It is not always possible to embed a family of varieties inside a fixed projective space and therefore describe their moduli with geometric invariant theory, and this special property is called boundedness.
A fundamental property of Fano varieties is that they fail to be bounded, and thus their stability cannot be reasonably captured by any finite-dimensional geometric invariant theory.
In order to find a genuine moduli space as a projective variety or scheme, one must prove certain properties about S-completeness and
For example, either by explicit construction or the use the Tian's alpha invariant, all smooth Kähler–Einstein manifolds of dimension 1 and 2 were known before the definition of K-stability was introduced.
Using the alpha invariant, Tian showed that a smooth Fano surface admits a Kähler–Einstein metric and is K-polystable if and only if it is not the blow up of the complex projective plane
In dimension 3 purely algebraic techniques can be used to find examples of K-stable Fano varieties which are not a priori known to admit Kähler–Einstein metrics.
admits a Kähler–Einstein metric and is therefore K-polystable by work of Donaldson, who computed Tian's alpha invariant explicitly using the criterion above.