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.
The first attempt to define K-stability for Fano manifolds was made 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. It was defined in reference to the K-energy functional previously introduced by Toshiki Mabuchi. Tian's definition of K-stability was later replaced by a purely algebro-geometric refinement that was first formulated by Simon Donaldson in 2001.
K-stability has become an important notion in the study and classification of Fano varieties. 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. (Tian then announced a nearly identical proof, under circumstances that resulted in a bitter priority dispute.) This theorem was later generalised to singular K-polystable Fano varieties due to the work of Berman–Boucksom–Jonsson, Li and Liu-Xu-Zhuang. 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 good moduli space of K-polystable Fano varieties. Due to the reformulations of the K-stability condition by Fujita–Li, 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.
Definition and characterisations
The notion of K-stability for Fano manifolds was originally specified using differential geometry by Tian, who extended the purely analytical notion of the Futaki invariant of a vector field to the case of certain normal varieties with orbifold singularities. This was later reformulated in a purely algebro-geometric form by Donaldson, but this general definition lost a direct link to the geometry of Fano varieties, instead making sense for the broader class of all projective varieties. Work of Tian shows that the Donaldson–Futaki invariant specifying the weight of the -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). In the Fano case these intersection numbers, which involve the anticanonical divisor of the variety and its test configuration, can be given powerful alternative characterisations in terms of the algebraic and birational geometry of the Fano variety.
Thus in the case of Fano varieties, there are many different but equivalent characterisations of K-stability, and some of these characterisations lend themselves to explicit calculation or easier proofs of results.
In this section all definitions are stated in the generality of a -Fano variety, which is a Fano variety with ample -Cartier anticanonical divisor and at worst Kawamata log terminal (klt) singularities. The definitions of K-stability can be made for any -Gorenstein Fano variety (that is, any Fano variety where the anticanonical divisor is -Cartier), however it was proven by Odaka that every K-semistable Fano variety has at worst klt singularities, so for the purpose of studying K-stability it suffices to assume at worst klt singularities. Every definition can be extended in a straightforward way to -log Fano pairs, a pair of a klt variety X and klt divisor such that is ample and -Cartier.
Traditional definition
The definition of K-stability for a Fano manifold, or more generally a -Fano variety, can be given in many forms. The general definition of K-stability in terms of test configurations (see K-stability for more details) can be simplified if the type of test configuration one considers can be simplified. For example, in the case of toric varieties, one may always take test configurations which are also toric, and this leads to a recharacterisation of K-stability in terms of convex functions on the moment polytope of the toric variety, as was observed by Donaldson in his first paper on K-stability. In the case of Fano manifolds, it was already implicit in the work of Tian that one may restrict to test configurations with a simplified central fibre, in that case where the central fibre is a normal variety.
In this case there exists an intersection-theoretic formula for the Donaldson–Futaki invariant of a normal test configuration for . Explicitly, one extends the test configuration to a test configuration over the complex projective line trivially at the point , one has a formula
With respect to this invariant, if is a -Fano variety, we say is
K-semistable if for any normal test configuration . K-stable if for any normal test configuration which is not isomorphic to the trivial test configuration outside a set of codimension 2. Uniformly K-stable if for any normal test configuration , where is the minimum norm of the test configuration and is a uniform constant depending only on . K-unstable if is not K-semistable. According to the above definitions, there are implications Uniformly K-stable K-stable K-semistable The above definitions are not well-suited to the situation where the Fano variety has automorphisms. When the space of automorphisms is positive-dimensional , it was observed by Akito Futaki that there are certain test configurations constructed out of the automorphisms of which are "trivial" for the perspective of testing K-stability. In this case one should restrict to those test configurations which are equivariant with respect to the action of a maximal torus , and this leads to the notion of K-polystability or reduced K-stability. We say is
K-polystable if for any test configuration and equality holds precisely when is isomorphic to outside a set of codimension 2. Reduced uniformly K-stable if where is the reduced -norm of the test configuration. As for the case where the automorphism group is not positive-dimensional, we have implications
Reduced uniformly K-stable K-polystable K-semistable The condition of uniform stability is a priori stronger than stability, because it assumes a uniform bound above zero for the Donaldson–Futaki invariant of the test configuration. However it turns out in the case of -Fano varieties uniform stability is actually equivalent to stability. Many results can be proved more easily for uniform K-stability because a uniform bound is stronger than a non-uniform bound, so often one works with this definition as opposed to the traditional K-(poly)stability. In the more general case of a polarised variety considered in the article on K-stability it is still an open and important problem to characterise how (reduced) uniform K-stability relates to K-(poly)stability.
Special test configurations
As mentioned above, sometimes the type of test configuration to be considered can be simplified. In the case of Fano varieties, a special test configuration is a test configuration such that we have a rational equivalence of divisors for some , and the central fibre is also a -Fano variety.
It is proved by Li-Xu that given any test configuration , there exists a special test configuration such that
This implies that for the purposes of testing the K-stability of , it suffices restrict to just looking at the above definitions of K-stability for special test configurations. 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.
The main alternative characterisation is in terms of a different notion of Ding stability, which is a variation of the K-stability condition for the Ding invariant
where one adds on the log canonical threshold of the test configuration. The Ding invariant can only be defined in the setting of Fano varieties. Using this new invariant instead of , one can define every notion of Ding stability exactly as above, leading to Ding (semi/poly)stability and uniform versions. The Ding invariant has better formal properties with respect to algebraic geometry than the Donaldson–Futaki invariant. It is known that when a test configuration is special, the Ding invariant agrees with the Donaldson–Futaki invariant up to a constant factor, and so for Fano varieties Ding stability is equivalent to K-stability.
Alpha invariant The first known effective criteria to test for K-stability was developed by Tian. Originally Tian's work was designed to directly provide a criterion for the existence of a Kähler–Einstein metric on a Fano manifold, and by later work it is known that every Kähler–Einstein Fano manifold is K-polystable. 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.
The alpha invariant of Tian can be defined relative to a group of automorphisms , and the alpha invariant corresponds to the concept of reduced K-stability or K-polystability above. Fix a -invariant Kähler metric on a Fano manifold. Define a special class of Kähler potentials by
Then the alpha invariant is defined by
The importance of this invariant is as follows:
It was observed by Demailly that the alpha invariant can be given a purely algebro-geometric definition in terms of an infimum of the log canonical threshold over all -divisors in the linear systems . Precisely, Demailly showed This allows purely algebro-geometric proofs of the existence of Kähler–Einstein metrics. Zhuang later gave an algebraic proof of Tian's theorem for all Fano varieties.
Beta invariant The beta invariant makes close contact with birational geometry. 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 . 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 . Whilst it is not possible to show that this "slope" K-stability is equivalent to K-stability, by passing not just to divisors inside but divisors inside any birational model over , one obtains "enough" objects to accurately test for K-stability.
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. By contract the -invariant integrates up to the pseudoeffective threshold where the natural divisor has positive volume (since every ample divisor has positive volume, the pseudoeffective threshold goes beyond the Seshadri constant). This extra information gives Fujita and Li's valuative criterion enough information to fully characterise K-stability.
Suppose is a normal variety with -Cartier canonical divisor . One says is a divisor over if is a divisor contained inside some normal variety such that there exists a proper birational morphism (for example given by a blow up of ). One defines the log discrepancy of a divisor over as