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[1] and reformulated more algebraically later by Simon Donaldson.[2] The definition was inspired by a comparison to geometric invariant theory (GIT) 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 (cscK metrics).

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.[10] 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.[11][12][13] 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.[14]

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.

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.

Generic fibres of a test configuration are all isomorphic to the variety X, whereas the central fibre may be distinct, and even singular.

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.[24][25] One elegant way of defining K-stability is given by Székelyhidi using the norm of a test configuration, which we first describe.[26]

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.

K-stability was originally introduced as an algebro-geometric condition which should characterise the existence of a Kähler–Einstein metric on a Fano manifold. This came to be known as the Yau–Tian–Donaldson conjecture (for Fano manifolds), and was resolved in the affirmative in 2012 by Xiuxiong Chen, Simon Donaldson, and Song Sun,[27][28][29][30] following a strategy 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. A proof based on the same techniques was provided at the same time by Tian.[31][32] Shortly thereafter in 2015, a proof based on the "classical" continuity method was provided by Datar and Székelyhidi,[33][34] followed by another one by Chen–Sun–Wang [35] based on the Kähler–Ricci flow. Berman-Boucksom-Jonsson also provided a proof[36] from the variational approach. The 2019 Veblen Prize was awarded to Chen, Donaldson, and Sun for their work.[37] Donaldson was awarded the 2015 Breakthrough Prize in Mathematics in part for his contribution to the proof,[38] and Song Sun was awarded the 2021 New Horizons Breakthrough Prize in part for his contributions.[39] They have alleged that Tian's work, published at the same time as their own, contains some mathematical errors and material which should be attributed to them; Tian has disputed their claims.[a][b]

Theorem (Chen–Donaldson–Sun, Tian, Datar–Székelyhidi, Chen–Sun–Wang, and Berman–Boucksom–Jonsson)

It is expected that the Yau–Tian–Donaldson conjecture should apply more generally to cscK metrics over arbitrary smooth polarised varieties. In fact, the Yau–Tian–Donaldson conjecture refers to this more general setting, with the case of Fano manifolds being a special case, which was conjectured earlier by Yau and Tian. 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.[2]

As discussed, the Yau–Tian–Donaldson conjecture has been resolved in the Fano setting. It was proven by Donaldson in 2009 that the Yau–Tian–Donaldson conjecture holds for toric varieties of complex dimension 2.[40][41][42] 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.[43][44] This is in some sense the easy direction of the conjecture, as it assumes the existence of a solution to a difficult partial differential equation, and arrives at the comparatively easy algebraic result. The significant challenge is to prove the reverse direction, that a purely algebraic condition implies the existence of a solution to a PDE.

K-stability arises from an analogy with the Hilbert-Mumford criterion for finite-dimensional geometric invariant theory. It is possible to use geometric invariant theory directly to obtain other notions of stability for varieties that are closely related to K-stability.

One may similarly define asymptotic Chow semistability and asymptotic Hilbert semistability, and the various notions of stability are related as follows:

It is however not know whether K-stability implies asymptotic Chow stability.[47]

It was shown by Ross and Thomas that K-semistability implies slope K-semistability.[49] However, unlike in the case of vector bundles, it is not the case that slope K-stability implies K-stability. In the case of vector bundles it is enough to consider only single subsheaves, but for varieties it is necessary to consider flags of length greater than one also. 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. This is a converse to results of Hong, which show that the projectivisation of a stable bundle over a base admitting a cscK metric, also admits a cscK metric, and is therefore K-stable.[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.[51] This suggests that the definition of K-stability as given here may not be precise enough to imply the Yau–Tian–Donaldson conjecture in general. However, this example is destabilised by a limit of test configurations. This was made precise by Székelyhidi, who introduced filtration K-stability.[52][26] A filtration here is a filtration of the coordinate ring

We say that a filtration is finitely generated if its Rees algebra is finitely generated. It was proven by David Witt Nyström that a filtration is finitely generated if and only if it arises from a test configuration, and by Székelyhidi that any filtration is a limit of finitely generated filtrations.[53] Combining these results Székelyhidi observed that the example of Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman would not violate the Yau–Tian–Donaldson conjecture if K-stability was replaced by filtration K-stability. This suggests that the definition of K-stability may need to be edited to account for these limiting examples.