The most important problem for this area is the existence of Kähler–Einstein metrics for compact Kähler manifolds.
This problem can be split up into three cases dependent on the sign of the first Chern class of the Kähler manifold: When first Chern class is not definite, or we have intermediate Kodaira dimension, then finding canonical metric remained as an open problem, which is called the algebrization conjecture via analytical minimal model program.
should curve due to the existence of mass or energy, a quantity encapsulated by the stress–energy tensor
It can be proven using the Bianchi identities that, in any larger dimension, the scalar curvature of any connected Einstein manifold must be constant.
There are many equivalent ways of formulating this compatibility condition, and one succinct interpretation is to ask that
A Kähler–Einstein manifold is one which combines the above properties of being Kähler and admitting an Einstein metric.
The combination of these properties implies a simplification of the Einstein equation in terms of the complex structure.
The solutions of this equation appear as critical points of the K-energy functional introduced by Toshiki Mabuchi on the space of Kähler potentials in the class
The existence problem for Kähler–Einstein metrics can be split up into three distinct cases, dependent on the sign of the topological constant
When the Kähler manifold is compact, the problem of existence has been completely solved.
[6][7] The existence of a Kähler metric satisfying the topological assumption is a consequence of Yau's proof of the Calabi conjecture.
These manifolds are of special significance in physics, where they should appear as the string background in superstring theory in 10 dimensions.
It was observed by Akito Futaki that there are possible obstructions to the existence of a solution given by the holomorphic vector fields of
[9] Indeed, much earlier it had been observed by Matsushima and Lichnerowicz that another necessary condition is that the Lie algebra of holomorphic vector fields
[10][11] It was conjectured by Yau in 1993, in analogy with the similar problem of existence of Hermite–Einstein metrics on holomorphic vector bundles, that the obstruction to existence of a Kähler–Einstein metric should be equivalent to a certain algebro-geometric stability condition similar to slope stability of vector bundles.
[12] In 1997 Tian Gang proposed a possible stability condition, which came to be known as K-stability.
[13] The conjecture of Yau was resolved in 2012 by Chen–Donaldson–Sun using techniques largely different from the classical continuity method of the case
[4][14] Chen–Donaldson–Sun have disputed Tian's proof, claiming that it contains mathematical inaccuracies and material which should be attributed to them.
[15] Donaldson was awarded the 2015 Breakthrough Prize in Mathematics in part for his contribution to the proof,[16] and the 2021 New Horizons Breakthrough Prize was awarded to Sun in part for his contribution.
is K-polystable.A proof based along the lines of the continuity method which resolved the case
In higher dimensions, one seeks a minimal model which has nef canonical bundle.
In this sense, the minimal model program can be viewed as an analogy of the Ricci flow in differential geometry, where regions where curvature concentrate are expanded or contracted in order to reduce the original Riemannian manifold to one with uniform curvature (precisely, to a new Riemannian manifold which has uniform Ricci curvature, which is to say an Einstein manifold).
In the case of 3-manifolds, this was famously used by Grigori Perelman to prove the Poincaré conjecture.
In settings where there are only mild (orbifold) singularities to this canonical model, it is possible to ask whether the Kähler–Ricci flow of
A precise result along these lines was proven by Cascini and La Nave,[21] and around the same time by Tian–Zhang.
of general type exists for all time, and after at most a finite number of singularity formations, if the canonical model
Later, Jian Song and Tian studied the case where the projective variety
[23] It is possible to give an alternative proof of the Chen–Donaldson–Sun theorem on existence of Kähler–Einstein metrics on a smooth Fano manifold using the Kähler-Ricci flow, and this was carried out in 2018 by Chen–Sun–Wang.
Many techniques from the Kähler–Einstein case carry on to the cscK setting, albeit with added difficulty, and it is conjectured that a similar algebro-geometric stability condition should imply the existence of solutions to the equation in this more general setting.
Instead of asking the Ricci curvature of the Levi-Civita connection on the tangent bundle of a Kähler manifold