K-stability


In mathematics, and especially differential and algebraic geometry, K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The notion of K-stability was first introduced by Gang Tian and reformulated more algebraically later by Simon Donaldson. The definition was inspired by a comparison to geometric invariant theory stability. In the special case of Fano varieties, K-stability precisely characterises the existence of Kähler–Einstein metrics. More generally, on any compact complex manifold, K-stability is [|conjectured] to be equivalent to the existence of constant scalar curvature Kähler metrics.

History

In 1954, Eugenio Calabi formulated a conjecture about the existence of Kähler metrics on compact Kähler manifolds, now known as the Calabi conjecture. One formulation of the conjecture is that a compact Kähler manifold admits a unique Kähler–Einstein metric in the class. In the particular case where, such a Kähler–Einstein metric would be Ricci flat, making the manifold a Calabi–Yau manifold. The Calabi conjecture was resolved in the case where by Thierry Aubin and Shing-Tung Yau, and when by Yau. In the case where, that is when is a Fano manifold, a Kähler–Einstein metric does not always exist. Namely, it was known by work of Yozo Matsushima and André Lichnerowicz that a Kähler manifold with can only admit a Kähler–Einstein metric if the Lie algebra is reductive. However, it can be easily shown that the blow up of the complex projective plane at one point, is Fano, but does not have reductive Lie algebra. Thus not all Fano manifolds can admit Kähler–Einstein metrics.
After the resolution of the Calabi conjecture for attention turned to the loosely related problem of finding canonical metrics on vector bundles over complex manifolds. In 1983, Donaldson produced a new proof of the Narasimhan–Seshadri theorem. 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. Seshadri. During the 1980s this theorem was generalised through the work of Donaldson, Karen Uhlenbeck and Yau, and Jun Li and Yau to the Kobayashi–Hitchin correspondence, which relates stable holomorphic vector bundles to Hermitian–Einstein connections over arbitrary compact complex manifolds. A key observation in the setting of holomorphic vector bundles is that once a holomorphic structure is fixed, any choice of Hermitian metric gives rise to a unitary connection, the Chern connection. Thus one can either search for a Hermitian–Einstein connection, or its corresponding Hermitian–Einstein metric.
Inspired by the resolution of the existence problem for canonical metrics on vector bundles, in 1993 Yau was motivated to conjecture the existence of a Kähler–Einstein metric on a Fano manifold should be equivalent to some form of algebro-geometric stability condition on the variety itself, just as the existence of a Hermitian–Einstein metric on a holomorphic vector bundle is equivalent to its stability. Yau suggested this stability condition should be an analogue of slope stability of vector bundles.
In 1997, Tian suggested such a stability condition, which he called K-stability after the K-energy functional introduced by Toshiki Mabuchi. The K originally stood for kinetic due to the similarity of the K-energy functional with the kinetic energy, and for the German kanonisch for the canonical bundle. Tian's definition was analytic in nature, and specific to the case of Fano manifolds. Several years later Donaldson introduced an algebraic condition described in this article called K-stability, which makes sense on any polarised variety, and is equivalent to Tian's analytic definition in the case of the polarised variety where is Fano.

Definition

In this section we work over the complex numbers, but the essential points of the definition apply over any field. A polarised variety is a pair where is a complex algebraic variety and is an ample line bundle on. Such a polarised variety comes equipped with an embedding into projective space using the Proj construction,
where is any positive integer large enough that is very ample, and so every polarised variety is projective. Changing the choice of ample line bundle on results in a new embedding of into a possibly different projective space. Therefore a polarised variety can be thought of as a projective variety together with a fixed embedding into some projective space.

Hilbert–Mumford criterion

K-stability is defined by analogy with the Hilbert–Mumford criterion from finite-dimensional geometric invariant theory. This theory describes the stability of points on polarised varieties, whereas K-stability concerns the stability of the polarised variety itself.
The Hilbert–Mumford criterion shows that to test the stability of a point in a projective algebraic variety under the action of a reductive algebraic group, it is enough to consider the one parameter subgroups of. To proceed, one takes a 1-PS of, say, and looks at the limiting point
This is a fixed point of the action of the 1-PS, and so the line over in the affine space is preserved by the action of. An action of the multiplicative group on a one dimensional vector space comes with a weight, an integer we label, with the property that
for any in the fibre over. The Hilbert-Mumford criterion says:
  • The point is semistable if for all 1-PS.
  • The point is stable if for all 1-PS.
  • The point is unstable if for any 1-PS.
If one wishes to define a notion of stability for varieties, the Hilbert-Mumford criterion therefore suggests it is enough to consider one parameter deformations of the variety. This leads to the notion of a test configuration.

Test Configurations

A test configuration for a polarised variety is a pair where is a scheme with a flat morphism and is a relatively ample line bundle for the morphism, such that:
  1. For every, the Hilbert polynomial of the fibre is equal to the Hilbert polynomial of. This is a consequence of the flatness of.
  2. There is an action of on the family covering the standard action of on.
  3. For any , as polarised varieties. In particular away from, the family is trivial: where is projection onto the first factor.
We say that a test configuration is a product configuration if, and a trivial configuration if the action on is trivial on the first factor.

Donaldson–Futaki Invariant

To define a notion of stability analogous to the Hilbert–Mumford criterion, one needs a concept of weight on the fibre over of a test configuration for a polarised variety. By definition this family comes equipped with an action of covering the action on the base, and so the fibre of the test configuration over is fixed. That is, we have an action of on the central fibre. In general this central fibre is not smooth, or even a variety. There are several ways to define the weight on the central fiber. The first definition was given by using Ding-Tian's version of generalized Futaki invariant. This definition is differential geometric and is directly related to the existence problems in Kähler geometry. Algebraic definitions were given by using Donaldson-Futaki invariants and CM-weights defined by intersection formula.
By definition an action of on a polarised scheme comes with an action of on the ample line bundle, and therefore induces an action on the vector spaces for all integers. An action of on a complex vector space induces a direct sum decomposition into weight spaces, where each is a one dimensional subspace of, and the action of when restricted to has a weight. Define the total weight of the action to be the integer. This is the same as the weight of the induced action of on the one dimensional vector space where.
Define the weight function of the test configuration to be the function where is the total weight of the action on the vector space for each non-negative integer. Whilst the function is not a polynomial in general, it becomes a polynomial of degree for all for some fixed integer, where. This can be seen using an equivariant Riemann-Roch theorem. Recall that the Hilbert polynomial satisfies the equality for all for some fixed integer, and is a polynomial of degree. For such, let us write
The Donaldson-Futaki invariant of the test configuration is the rational number
In particular where is the first order term in the expansion
The Donaldson-Futaki invariant does not change if is replaced by a positive power, and so in the literature K-stability is often discussed using -line bundles.
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. Any test configuration admits a natural compactification over , then the CM-weight is defined by
where. This definition by intersection formula is now often used in algebraic geometry.
It is known that coincides with, so we can take the weight to be either or.
The weight can be also expressed in terms of the Chow form and hyperdiscriminant.
In the case of Fano manifolds, there is an interpretation of the weight in terms of new -invariant on valuations found by Chi Li and Kento Fujita.

K-stability

In order to define K-stability, we need to first exclude certain test configurations. 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. One elegant way of defining K-stability is given by Székelyhidi using the norm of a test configuration, which we first describe.
For a test configuration, define the norm as follows. Let be the infinitesimal generator of the action on the vector space. Then. Similarly to the polynomials and, the function is a polynomial for large enough integers, in this case of degree. 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 and weight on the central fibre, one can define a stability condition, called K-stability.
Let be a polarised algebraic variety. We say that is:
  • K-semistable if for all test configurations for.
  • K-stable if for all test configurations for, and additionally whenever.
  • K-polystable if is K-semistable, and additionally whenever, the test configuration is a product configuration.
  • K-unstable if it is not K-semistable.