K-stability
The notion of K-stability was first introduced by Gang Tian[1] and reformulated more algebraically later by Simon Donaldson.
In the special case of Fano varieties, K-stability precisely characterises the existence of Kähler–Einstein metrics.
attention turned to the loosely related problem of finding canonical metrics on vector bundles over complex manifolds.
[9] As proved by Donaldson, the theorem states that a holomorphic vector bundle over a compact Riemann surface is stable if and only if it corresponds to an irreducible unitary Yang–Mills connection.
That is, a unitary connection which is a critical point of the Yang–Mills functional On a Riemann surface such a connection is projectively flat, and its holonomy gives rise to a projective unitary representation of the fundamental group of the Riemann surface, thus recovering the original statement of the theorem by M. S. Narasimhan and C. S.
[14] In 1997, Tian suggested such a stability condition, which he called K-stability after the K-energy functional introduced by Toshiki Mabuchi.
[1] This definition is differential geometric and is directly related to the existence problems in Kähler geometry.
It is possible to describe the Donaldson-Futaki invariant in terms of intersection theory, and this was the approach taken by Tian in defining the CM-weight.
Initially it was presumed one should just ignore trivial test configurations as defined above, whose Donaldson-Futaki invariant always vanishes, but it was observed by Li and Xu that more care is needed in the definition.
Let us write its expansion as The norm of a test configuration is defined by the expression According to the analogy with the Hilbert-Mumford criterion, once one has a notion of deformation (test configuration) and weight on the central fibre (Donaldson-Futaki invariant), one can define a stability condition, called K-stability.
is: K-stability was originally introduced as an algebro-geometric condition which should characterise the existence of a Kähler–Einstein metric on a Fano manifold.
The conjecture was resolved in the 2010s in works of Xiuxiong Chen, Simon Donaldson, and Song Sun,[24][25][26][27][28][29] The strategy is based on a continuity method with respect to the cone angle of a Kähler–Einstein metric with cone singularities along a fixed anticanonical divisor, as well as an in-depth use of the Cheeger–Colding–Tian theory of Gromov–Hausdorff limits of Kähler manifolds with Ricci bounds.
is K-polystable.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.
[35] Robert Berman, Sébastien Boucksom, and Mattias Jonsson also provided a proof from the variational approach.
[36] It is expected that the Yau–Tian–Donaldson conjecture should apply more generally to cscK metrics over arbitrary smooth polarised varieties.
Donaldson built on the conjecture of Yau and Tian from the Fano case after his definition of K-stability for arbitrary polarised varieties was introduced.
[37][38][39] For arbitrary polarised varieties it was proven by Stoppa, also using work of Arezzo and Pacard, that the existence of a cscK metric implies K-polystability.
The significant challenge is to prove the reverse direction, that a purely algebraic condition implies the existence of a solution to a PDE.
It has been known since the original work of Pierre Deligne and David Mumford that smooth algebraic curves are asymptotically stable in the sense of geometric invariant theory, and in particular that they are K-stable.
It was shown by Donaldson that for toric surfaces, it suffices to test convex functions of a particularly simple form.
Donaldson showed that for toric surfaces it is enough to test K-stability only on simple rational piecewise-linear functions.
, the first Hirzebruch surface, which is the blow up of the complex projective plane at a point, with respect to the polarisation given by
It is possible to use geometric invariant theory directly to obtain other notions of stability for varieties that are closely related to K-stability.
However, from the expression one observes that and so K-stability is in some sense the limit of Chow stability as the dimension of the projective space
By taking the support of the ideals this corresponds to blowing up along a flag of subschemes inside the copy
In the special case where this flag of subschemes is of length one, the Donaldson-Futaki invariant can be easily computed and one arrives at slope K-stability.
Despite this, slope K-stability can still be used to identify K-unstable varieties, and therefore by the results of Stoppa, give obstructions to the existence of cscK metrics.
For example, Ross and Thomas use slope K-stability to show that the projectivisation of an unstable vector bundle over a K-stable base is K-unstable, and so does not admit a cscK metric.
[50] Work of Apostolov–Calderbank–Gauduchon–Tønnesen-Friedman shows the existence of a manifold which does not admit any extremal metric, but does not appear to be destabilised by any test configuration.
