Kähler–Einstein metric

In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The most important special case of these are the Calabi–Yau manifolds, which are Kähler and Ricci-flat.
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.
Definition.
Einstein manifolds.
Suppose formula_1 is a Riemannian manifold. In physics the Einstein field equations are a set of partial differential equations on the metric tensor formula_2 which describe how the manifold formula_3 should curve due to the existence of mass or energy, a quantity encapsulated by the stress–energy tensor formula_4. In a vacuum where there is no mass or energy, that is formula_5, the Einstein Field Equations simplify. Namely, the Ricci curvature of formula_2 is a symmetric formula_7-tensor, as is the metric formula_2 itself, and the equations reduce to
where formula_10 is the scalar curvature of formula_2. That is, the Ricci curvature becomes proportional to the metric. A Riemannian manifold formula_1 satisfying the above equation is called an Einstein manifold.
Every two-dimensional Riemannian manifold is Einstein. It can be proven using the Bianchi identities that, in any larger dimension, the scalar curvature of any connected Einstein manifold must be constant. For this reason, the Einstein condition is often given as
for a real number formula_14
Kähler manifolds.
When the Riemannian manifold formula_1 is also a complex manifold, that is it comes with an integrable almost-complex structure formula_16, it is possible to ask for a compatibility between the metric structure formula_2 and the complex structure formula_18. There are many equivalent ways of formulating this compatibility condition, and one succinct interpretation is to ask that formula_18 is orthogonal with respect to formula_2, so that formula_21 for all vector fields formula_22, and that formula_18 is preserved by the parallel transport of the Levi-Civita connection formula_24, captured by the condition formula_25. Such a triple formula_26 is called a Kähler manifold.
Kähler–Einstein metrics.
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. Namely, on a Kähler manifold one can define the Ricci form, a real formula_27-form, by the expression
where formula_29 are any tangent vector fields to formula_3.
The almost-complex structure formula_18 forces formula_32 to be antisymmetric, and the compatibility condition formula_25 combined with the Bianchi identity implies that formula_32 is a closed differential form. Associated to the Riemannian metric formula_2 is the Kähler form formula_36 defined by a similar expression formula_37. Therefore the Einstein equations for formula_2 can be rewritten as
the Kähler–Einstein equation.
Since this is an equality of closed differential forms, it implies an equality of the associated de Rham cohomology classes formula_40 and formula_41. The former class is the first Chern class of formula_3, formula_43. Therefore a necessary condition for the existence of a solution to the Kähler–Einstein equation is that formula_44, for some formula_45. This is a topological necessary condition on the Kähler manifold formula_26.
Note that since the Ricci curvature formula_47 is invariant under scaling formula_48, if there is a metric such that formula_44, one can always normalise to a new metric with formula_50, that is formula_51. Thus the Kähler–Einstein equation is often written
depending on the sign of the topological constant formula_53.
Transformation to a complex Monge–Ampere equation.
The situation of compact Kähler manifolds is special, because the Kähler–Einstein equation can be reformulated as a complex Monge–Ampere equation for a smooth Kähler potential on formula_3. By the topological assumption on the Kähler manifold, we may always assume that there exists some Kähler metric formula_55. The Ricci form formula_56 of formula_57 is given in local coordinates by the formula
By assumption formula_57 and formula_56 are in the same cohomology class formula_43, so the formula_62-lemma from Hodge theory implies there exists a smooth function formula_63 such that formula_64.
Any other metric formula_65 is related to formula_57 by a Kähler potential formula_67 such that formula_68. It then follows that if formula_32 is the Ricci form with respect to formula_36, then
Thus to make formula_72 we need to find formula_73 such that
This will certainly be true if the same equation is proven after removing the derivatives formula_62, and in fact this is an equivalent equation by the formula_62-lemma up to changing formula_73 by the addition of a constant function. In particular, after removing formula_62 and exponentiating, the equation is transformed into 
This partial differential equation is similar to a real Monge–Ampere equation, and is known as a complex Monge–Ampere equation, and subsequently can be studied using tools from convex analysis. Its behaviour is highly sensitive to the sign of the topological constant formula_51. 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 formula_43.
Existence.
The existence problem for Kähler–Einstein metrics can be split up into three distinct cases, dependent on the sign of the topological constant formula_53. Since the Kähler form formula_36 is always a positive differential form, the sign of formula_53 depends on whether the cohomology class formula_43 is positive, negative, or zero. In algebraic geometry this is understood in terms of the canonical bundle of formula_3: formula_87 if and only if the canonical bundle formula_88 is an ample line bundle, and formula_89 if and only if formula_90 is ample. If formula_88 is a trivial line bundle, then formula_92.
When the Kähler manifold is compact, the problem of existence has been completely solved.
The case "c1(X)&lt;0".
When the Kähler manifold formula_3 satisfies the topological assumption formula_87, the canonical bundle is ample and so formula_53 must be negative. If the necessary topological assumption is satisfied, that is there exists a Kähler metric formula_36 such that formula_97, then it was proven by Aubin and Yau that a Kähler–Einstein always exists. The existence of a Kähler metric satisfying the topological assumption is a consequence of Yau's proof of the Calabi conjecture.
Theorem (Aubin, Yau): A compact Kähler manifold with formula_87 always admits a Kähler–Einstein metric.
The case "c1(X)=0".
When the canonical bundle formula_88 is trivial, so that formula_92, the manifold is said to be Calabi–Yau. These manifolds are of special significance in physics, where they should appear as the string background in superstring theory in 10 dimensions. Mathematically, this corresponds to the case where formula_101, that is, when the Riemannian manifold formula_1 is Ricci flat.
The existence of a Kähler–Einstein metric was proven in this case by Yau, using a continuity method similar to the case where formula_87. The topological assumption assumption formula_92 introduces new difficulties into the continuity method. Partly due to his proof of existence, and the related proof of the Calabi conjecture, Yau was awarded the Fields medal.
Theorem (Yau): A compact Kähler manifold with trivial canonical bundle, a Calabi–Yau manifold, always admits a Kähler–Einstein metric, and in particular admits a Ricci-flat metric.
The case "c1(X)&gt;0".
When the anticanonical bundle formula_90 is ample, or equivalently formula_89, the manifold is said to be Fano. In contrast to the case formula_107, a Kähler–Einstein metric does not always exist in this case. It was observed by Akito Futaki that there are possible obstructions to the existence of a solution given by the holomorphic vector fields of formula_3, and it is a necessary condition that the Futaki invariant of these vector fields is non-negative. Indeed, much earlier it had been observed by Matsushima and Lichnerowicz that another necessary condition is that the Lie algebra of holomorphic vector fields formula_109 must be reductive.
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. In 1997 Tian Gang proposed a possible stability condition, which came to be known as K-stability.
The conjecture of Yau was resolved in 2012 by Chen–Donaldson–Sun using techniques largely different from the classical continuity method of the case formula_107, and at the same time by Tian. Chen–Donaldson–Sun have disputed Tian's proof, claiming that it contains mathematical inaccuracies and material which should be attributed to them. Tian has disputed these claims. The 2019 Veblen prize was awarded to Chen–Donaldson–Sun for their proof. Donaldson was awarded the 2015 Breakthrough Prize in Mathematics in part for his contribution to the proof, and the 2021 New Horizons Breakthrough Prize was awarded to Sun in part for his contribution.
Theorem: A compact Fano manifold formula_3 admits a Kähler–Einstein metric if and only if the pair formula_112 is K-polystable.
A proof based along the lines of the continuity method which resolved the case formula_107 was later provided by Datar–Székelyhidi, and several other proofs are now known. See the Yau–Tian–Donaldson conjecture for more details.
Kähler–Ricci flow and the minimal model program.
A central program in birational geometry is the minimal model program, which seeks to generate models of algebraic varieties inside every birationality class, which are in some sense "minimal", usually in that they minimize certain measures of complexity (such as the arithmetic genus in the case of curves). In higher dimensions, one seeks a minimal model which has "nef" canonical bundle. One way to construct minimal models is to contract certain curves formula_114 inside an algebraic variety formula_3 which have negative self-intersection. These curves should be thought of geometrically as subvarieties on which formula_3 has a concentration of negative curvature.
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 the setting of Kähler manifolds, the Kähler–Ricci flow was first written down by Cao. Here one fixes a Kähler metric formula_117 with Ricci form formula_118, and studies the geometric flow for a family of Kähler metrics formula_119 parametrised by formula_120:
When a projective variety formula_3 is of general type, the minimal model formula_123 admits a further simplification to a "canonical model" formula_124, with ample canonical bundle. In settings where there are only mild (orbifold) singularities to this canonical model, it is possible to ask whether the Kähler–Ricci flow of formula_3 converges to a (possibly mildly singular) Kähler–Einstein metric on formula_124, which should exist by Yau and Aubin's existence result for formula_127.
A precise result along these lines was proven by Cascini and La Nave, and around the same time by Tian–Zhang.
Theorem: The Kähler–Ricci flow on a projective variety formula_3 of general type exists for all time, and after at most a finite number of singularity formations, if the canonical model formula_124 of formula_3 has at worst orbifold singularities, then the Kähler–Ricci flow on formula_3 converges to the Kähler–Einstein metric on formula_124, up to a bounded function which is smooth away from an analytic subvariety of formula_3.
In the case where the variety formula_3 is of dimension two, so is a surface of general type, one gets convergence to the 
Kähler–Einstein metric on formula_124.
Later, Jian Song and Tian studied the case where the projective variety formula_3 has log-terminal singularities.
Kähler–Ricci flow and existence of Kähler–Einstein metrics.
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. Namely, if the Fano manifold is K-polystable, then the Kähler-Ricci flow exists for all time and converges to a Kähler–Einstein metric on the Fano manifold.
Generalizations and alternative notions.
Constant scalar curvature Kähler metrics.
When the canonical bundle formula_88 is not trivial, ample, or anti-ample, it is not possible to ask for a Kähler–Einstein metric, as the class formula_43 cannot contain a Kähler metric, and so the necessary topological condition can never be satisfied. This follows from the Kodaira embedding theorem.
A natural generalisation of the Kähler–Einstein equation to the more general setting of an arbitrary compact Kähler manifold is to ask that the Kähler metric has constant scalar curvature (one says the metric is cscK). The scalar curvature is the total trace of the Riemannian curvature tensor, a smooth function on the manifold formula_1, and in the Kähler case the condition that the scalar curvature is constant admits a transformation into an equation similar to the complex Monge–Ampere equation of the Kähler–Einstein setting. 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.
When the compact Kähler manifold satisfies the topological assumptions necessary for the Kähler–Einstein condition to make sense, the constant scalar curvature Kähler equation reduces to the Kähler–Einstein equation.
Hermite–Einstein metrics.
Instead of asking the Ricci curvature of the Levi-Civita connection on the tangent bundle of a Kähler manifold formula_3 is proportional to the metric itself, one can instead ask this question for the curvature of a Chern connection associated to a Hermitian metric on "any" holomorphic vector bundle over formula_3 (note that the Levi-Civita connection on the holomorphic tangent bundle is precisely the Chern connection of the Hermitian metric coming from the Kähler structure). The resulting equation is called the Hermite–Einstein equation, and is of special importance in gauge theory, where it appears as a special case of the Yang–Mills equations, which come from quantum field theory, in contrast to the regular Einstein equations which come from general relativity.
In the case where the holomorphic vector bundle is again the holomorphic tangent bundle and the Hermitian metric is the 
Kähler metric, the Hermite–Einstein equation reduces to the Kähler–Einstein equation. In general however, the geometry of the Kähler manifold is often fixed and only the bundle metric is allowed to vary, and this causes the Hermite–Einstein equation to be easier to study than the Kähler–Einstein equation in general. In particular, a complete algebro-geometric characterisation of the existence of solutions is given by the Kobayashi–Hitchin correspondence.