In differential geometry, a KĂ€hlerâEinstein metric on a complex manifold is a Riemannian metric that is both a KĂ€hler metric and an Einstein metric. A manifold is said to be KĂ€hlerâEinstein if it admits a KĂ€hlerâEinstein metric. The most important special case of these are the CalabiâYau manifolds, which are KĂ€hler and Ricci-flat.
The most important problem for this area is the existence of KĂ€hlerâEinstein metrics for compact KĂ€hler manifolds. This problem can be split up into three cases dependent on the sign of the first Chern class of the KĂ€hler 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.
Suppose is a Riemannian manifold. In physics the Einstein field equations are a set of partial differential equations on the metric tensor which describe how the manifold should curve due to the existence of mass or energy, a quantity encapsulated by the stressâenergy tensor . In a vacuum where there is no mass or energy, that is , the Einstein Field Equations simplify. Namely, the Ricci curvature of is a symmetric -tensor, as is the metric itself, and the equations reduce to
where is the scalar curvature of . That is, the Ricci curvature becomes proportional to the metric. A Riemannian manifold 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
When the Riemannian manifold is also a complex manifold, that is it comes with an integrable almost-complex structure , it is possible to ask for a compatibility between the metric structure and the complex structure . There are many equivalent ways of formulating this compatibility condition, and one succinct interpretation is to ask that is orthogonal with respect to , so that for all vector fields , and that is preserved by the parallel transport of the Levi-Civita connection , captured by the condition . Such a triple is called a KĂ€hler manifold.
A KĂ€hlerâEinstein manifold is one which combines the above properties of being KĂ€hler 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 KĂ€hler manifold one can define the Ricci form, a real -form, by the expression
where are any tangent vector fields to .
The almost-complex structure forces to be antisymmetric, and the compatibility condition combined with the Bianchi identity implies that is a closed differential form. Associated to the Riemannian metric is the KĂ€hler form defined by a similar expression . Therefore the Einstein equations for can be rewritten as
the KĂ€hlerâEinstein equation.
Since this is an equality of closed differential forms, it implies an equality of the associated de Rham cohomology classes and . The former class is the first Chern class of , . Therefore a necessary condition for the existence of a solution to the KĂ€hlerâEinstein equation is that , for some . This is a topological necessary condition on the KĂ€hler manifold .
Note that since the Ricci curvature is invariant under scaling , if there is a metric such that , one can always normalise to a new metric with , that is . Thus the KĂ€hlerâEinstein equation is often written
depending on the sign of the topological constant .
The situation of compact KĂ€hler manifolds is special, because the KĂ€hlerâEinstein equation can be reformulated as a complex MongeâAmpere equation for a smooth KĂ€hler potential on . [5] By the topological assumption on the KĂ€hler manifold, we may always assume that there exists some KĂ€hler metric . The Ricci form of is given in local coordinates by the formula
By assumption and are in the same cohomology class , so the -lemma from Hodge theory implies there exists a smooth function such that .
Any other metric is related to by a KĂ€hler potential such that . It then follows that if is the Ricci form with respect to , then
Thus to make we need to find such that
This will certainly be true if the same equation is proven after removing the derivatives , and in fact this is an equivalent equation by the -lemma up to changing by the addition of a constant function. In particular, after removing and exponentiating, the equation is transformed into
This partial differential equation is similar to a real MongeâAmpere equation, and is known as a complex MongeâAmpere equation, and subsequently can be studied using tools from convex analysis. Its behaviour is highly sensitive to the sign of the topological constant . The solutions of this equation appear as critical points of the K-energy functional introduced by Toshiki Mabuchi on the space of KĂ€hler potentials in the class .
The existence problem for KĂ€hlerâEinstein metrics can be split up into three distinct cases, dependent on the sign of the topological constant . Since the KĂ€hler form is always a positive differential form, the sign of depends on whether the cohomology class is positive, negative, or zero. In algebraic geometry this is understood in terms of the canonical bundle of : if and only if the canonical bundle is an ample line bundle, and if and only if is ample. If is a trivial line bundle, then . When the KĂ€hler manifold is compact, the problem of existence has been completely solved.
When the KĂ€hler manifold satisfies the topological assumption , the canonical bundle is ample and so must be negative. If the necessary topological assumption is satisfied, that is there exists a KĂ€hler metric such that , then it was proven by Aubin and Yau that a KĂ€hlerâEinstein metric always exists. [6] [7] The existence of a KĂ€hler metric satisfying the topological assumption is a consequence of Yau's proof of the Calabi conjecture.
Theorem (Aubin, Yau): A compact KĂ€hler manifold with always admits a KĂ€hlerâEinstein metric.
When the canonical bundle is trivial, so that , the manifold is said to be CalabiâYau. 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 , that is, when the Riemannian manifold is Ricci flat.
The existence of a KĂ€hlerâEinstein metric was proven in this case by Yau, using a continuity method similar to the case where . [8] The topological assumption assumption 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 KĂ€hler manifold with trivial canonical bundle, a CalabiâYau manifold, always admits a KĂ€hlerâEinstein metric, and in particular admits a Ricci-flat metric.
When the anticanonical bundle is ample, or equivalently , the manifold is said to be Fano. In contrast to the case , a KĂ€hlerâEinstein 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 , and it is a necessary condition that the Futaki invariant of these vector fields is non-negative. [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 must be reductive. [10] [11]
It was conjectured by Yau in 1993, in analogy with the similar problem of existence of HermiteâEinstein metrics on holomorphic vector bundles, that the obstruction to existence of a KĂ€hlerâEinstein 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 Chenâ Donaldsonâ Sun using techniques largely different from the classical continuity method of the case , [1] [2] [3] and at the same time by Tian. [4] [14] ChenâDonaldsonâSun have disputed Tian's proof, claiming that it contains mathematical inaccuracies and material which should be attributed to them. [a] Tian has disputed these claims. [b] The 2019 Veblen prize was awarded to ChenâDonaldsonâSun for their proof. [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. [17]
Theorem: A compact Fano manifold admits a KĂ€hlerâEinstein metric if and only if the pair is K-polystable.
A proof based along the lines of the continuity method which resolved the case was later provided by DatarâSzĂ©kelyhidi, and several other proofs are now known. [18] [19] See the YauâTianâDonaldson conjecture for more details.
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 inside an algebraic variety which have negative self-intersection. These curves should be thought of geometrically as subvarieties on which 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 KĂ€hler manifolds, the KĂ€hlerâRicci flow was first written down by Cao. [20] Here one fixes a KĂ€hler metric with Ricci form , and studies the geometric flow for a family of KĂ€hler metrics parametrised by :
When a projective variety is of general type, the minimal model admits a further simplification to a canonical model , with ample canonical bundle. In settings where there are only mild ( orbifold) singularities to this canonical model, it is possible to ask whether the KĂ€hlerâRicci flow of converges to a (possibly mildly singular) KĂ€hlerâEinstein metric on , which should exist by Yau and Aubin's existence result for .
A precise result along these lines was proven by Cascini and La Nave, [21] and around the same time by TianâZhang. [22]
Theorem: The KĂ€hlerâRicci flow on a projective variety of general type exists for all time, and after at most a finite number of singularity formations, if the canonical model of has at worst orbifold singularities, then the KĂ€hlerâRicci flow on converges to the KĂ€hlerâEinstein metric on , up to a bounded function which is smooth away from an analytic subvariety of .
In the case where the variety is of dimension two, so is a surface of general type, one gets convergence to the KĂ€hlerâEinstein metric on .
Later, Jian Song and Tian studied the case where the projective variety has log-terminal singularities. [23]
It is possible to give an alternative proof of the ChenâDonaldsonâSun theorem on existence of KĂ€hlerâEinstein metrics on a smooth Fano manifold using the KĂ€hler-Ricci flow, and this was carried out in 2018 by ChenâSunâWang. [24] Namely, if the Fano manifold is K-polystable, then the KĂ€hler-Ricci flow exists for all time and converges to a KĂ€hlerâEinstein metric on the Fano manifold.
When the canonical bundle is not trivial, ample, or anti-ample, it is not possible to ask for a KĂ€hlerâEinstein metric, as the class cannot contain a KĂ€hler metric, and so the necessary topological condition can never be satisfied. This follows from the Kodaira embedding theorem.
A natural generalisation of the KĂ€hlerâEinstein equation to the more general setting of an arbitrary compact KĂ€hler manifold is to ask that the KĂ€hler 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 , and in the KĂ€hler case the condition that the scalar curvature is constant admits a transformation into an equation similar to the complex MongeâAmpere equation of the KĂ€hlerâEinstein setting. Many techniques from the KĂ€hlerâEinstein 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 KĂ€hler manifold satisfies the topological assumptions necessary for the KĂ€hlerâEinstein condition to make sense, the constant scalar curvature KĂ€hler equation reduces to the KĂ€hlerâEinstein equation.
Instead of asking the Ricci curvature of the Levi-Civita connection on the tangent bundle of a KĂ€hler manifold 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 (note that the Levi-Civita connection on the holomorphic tangent bundle is precisely the Chern connection of the Hermitian metric coming from the KĂ€hler structure). The resulting equation is called the HermiteâEinstein equation, and is of special importance in gauge theory, where it appears as a special case of the YangâMills 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 KĂ€hler metric, the HermiteâEinstein equation reduces to the KĂ€hlerâEinstein equation. In general however, the geometry of the KĂ€hler manifold is often fixed and only the bundle metric is allowed to vary, and this causes the HermiteâEinstein equation to be easier to study than the KĂ€hlerâEinstein equation in general. In particular, a complete algebro-geometric characterisation of the existence of solutions is given by the KobayashiâHitchin correspondence.
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