Kähler geometry

What it is

A Kähler manifold is a complex manifold with a Hermitian metric $g$ whose associated 2-form $\omega = \sqrt{-1}\,g_{i\bar j}\,dz^i\wedge d\bar z^j$ is closed. In holomorphic coordinates, $d\omega = 0$ says $\partial_k g_{i\bar j} = \partial_i g_{k\bar j}$. Equivalently, the complex structure $J$ is parallel for the Levi-Civita connection, so the holonomy lies in $U(n)$. Locally $\omega = \sqrt{-1}\,\partial\bar\partial\phi$ for a real potential $\phi$, which means $g_{i\bar j} = \partial_i\partial_{\bar j}\phi$.

The Levi-Civita connection then has only the Christoffel symbols $\Gamma^k_{ij} = g^{k\bar l}\partial_i g_{j\bar l}$ and their conjugates, and the Ricci curvature collapses to a single second derivative:

Since $\det(g_{k\bar l})$ defines a Hermitian metric on the anticanonical bundle, $\operatorname{Ric}(\omega)$ is a curvature form of that bundle. It is closed and represents $2\pi c_1(M)$. Curvature questions therefore become scalar equations for potentials, and they are tied to topology and algebraic geometry through $c_1$.

#Conventions on this page

With $\omega$ as above, $\omega^n/n!$ is the Riemannian volume form and $[\operatorname{Ric}(\omega)] = 2\pi c_1(M)$. We write $S = g^{i\bar j}R_{i\bar j}$, which is half the Riemannian scalar curvature $R$. Other normalizations are common: Song and Weinkove's lecture notes put $\sqrt{-1}/2\pi$ in front of $\omega$ and of the Ricci form, which removes the $2\pi$ from $c_1$. Check the convention before copying a constant from a paper.

#The central problems

The central problems concern canonical metrics, a best metric in a given Kähler class:

• Kähler–Einstein metrics, with $\operatorname{Ric}(\omega) = \lambda\omega$. Yau solved the Calabi conjecture, giving Ricci-flat metrics when $c_1 = 0$ and, with Aubin, negative Kähler–Einstein metrics when $c_1 < 0$. On Fano manifolds existence is equivalent to K-polystability.
• Extremal and constant scalar curvature Kähler (cscK) metrics, which make sense in every Kähler class. Chen and Cheng proved a priori estimates and existence results.
• Kähler–Ricci solitons, the replacement when no Kähler–Einstein metric exists.

Examples

Flat space and tori. $\mathbb C^n$ has potential $\lvert z\rvert^2$, and the metric descends to every complex torus $\mathbb C^n/\Lambda$. These are flat, and a generic torus of dimension at least two is not projective.

Projective space. The Fubini–Study form is $\omega_{FS} = \sqrt{-1}\,\partial\bar\partial\log(1+\lvert z\rvert^2)$ in an affine chart. Here $\det(g_{i\bar j}) = (1+\lvert z\rvert^2)^{-(n+1)}$, so the Ricci formula gives $\operatorname{Ric}(\omega_{FS}) = (n+1)\,\omega_{FS}$ in one line. It is an Einstein metric of positive scalar curvature.

Projective manifolds. The restriction of a closed form is closed, so every complex submanifold of a Kähler manifold is Kähler, and in particular every smooth projective variety is. Smooth hypersurfaces show all three signs of $c_1$. For degree $d$ in $\mathbb{CP}^{n+1}$, adjunction gives $K = \mathcal O(d-n-2)$, so the hypersurface is Fano for $d < n+2$, has $c_1 = 0$ for $d = n+2$ (the quintic threefold, the quartic K3 surface) and has $c_1 < 0$ for $d > n+2$.

Curves. In complex dimension one every Hermitian metric is Kähler, because $d\omega$ is a 3-form on a real surface. Kähler geometry there is the uniformization theorem, and Kähler–Ricci flow is the Ricci flow on surfaces.

A non-example. The Hopf surface $S^1\times S^3$ is a compact complex manifold with $b_1 = 1$. Odd Betti numbers of a compact Kähler manifold are even, so it has no Kähler metric. Flows of Hermitian metrics on such manifolds are the subject of generalized and non-Kähler flows.

Why it matters

Kähler geometry is where Ricci flow meets algebraic geometry most directly. Cao's Kähler–Ricci flow reproved the Calabi–Yau theorem. On Fano manifolds the Hamilton–Tian conjecture, proved by Chen–Wang and by Bamler, says that the flow converges along subsequences to a Kähler–Ricci soliton on a possibly singular Fano variety. On projective varieties the flow is expected to carry out the minimal model program (Song–Tian). It is also a laboratory where singularities are predicted by algebra, such as contractions of curves and divisors, rather than by analysis alone.

Cohomology, the Kähler cone and potentials

A compact Kähler manifold has more structure than a general complex manifold. Its cohomology splits as $H^k(M,\mathbb C) = \bigoplus_{p+q=k} H^{p,q}(M)$, wedging with $\omega$ satisfies hard Lefschetz, and the $\partial\bar\partial$-lemma holds. Three consequences organize everything below.

Potentials parametrize a class. Two cohomologous Kähler forms differ by $\sqrt{-1}\,\partial\bar\partial\varphi$, with $\varphi$ unique up to a constant. A Kähler class is therefore the space of potentials

and any tensor equation for $\omega_\varphi$ in that class can be rewritten for one function. The basic identity is $\operatorname{Ric}(\omega_\varphi) = \operatorname{Ric}(\omega) - \sqrt{-1}\,\partial\bar\partial\log(\omega_\varphi^n/\omega^n)$.

The Kähler cone is computable. The Kähler classes form an open convex cone in $H^{1,1}(M,\mathbb R)$. Demailly and Paun proved it is one connected component of the set of classes $\alpha$ with $\int_Y \alpha^{\dim Y} > 0$ for every irreducible analytic subset $Y$. By Kodaira's embedding theorem, a manifold with a rational Kähler class is projective.

Integrals are topological. Volume $\int_M\omega^n$ and average scalar curvature $\bar S = 2\pi n\,c_1(M)\cdot[\omega]^{n-1}/[\omega]^n$ depend only on the classes involved. This is why so much of what follows can be predicted before solving a PDE.

Canonical metrics: the equations

Calabi proved uniqueness and Yau proved existence. Choosing $F$ with $\sqrt{-1}\,\partial\bar\partial F = \operatorname{Ric}(\omega) - \rho$ and $\int_M e^F\omega^n = \int_M\omega^n$, the theorem is the solvability of the complex Monge–Ampère equation $\omega_\varphi^n = e^F\omega^n$. When $c_1(M) = 0$ in real cohomology, taking $\rho = 0$ gives a Ricci-flat metric in every Kähler class: the Calabi–Yau metrics that string theory needs and that have no known formula, which is why they are computed numerically.

#Kähler–Einstein metrics

Take $\lambda \in \{-1, 0, 1\}$ matching the sign of $c_1(M)$, and a Kähler form with $\operatorname{Ric}(\omega) = \lambda\omega + \sqrt{-1}\,\partial\bar\partial F$, which is possible when $\lambda\ne 0$ exactly if $[\omega] = 2\pi\lambda c_1(M)$. Then $\operatorname{Ric}(\omega_\varphi) = \lambda\omega_\varphi$ becomes $\omega_\varphi^n = e^{F-\lambda\varphi}\omega^n$. The sign in the exponent decides everything.

$c_1(M)$	Equation	Answer
$< 0$	$\omega_\varphi^n = e^{F+\varphi}\omega^n$	Always a unique solution (Aubin; Yau)
$= 0$	$\omega_\varphi^n = e^{F}\omega^n$	A unique Ricci-flat metric in each Kähler class (Yau)
$> 0$	$\omega_\varphi^n = e^{F-\varphi}\omega^n$	Exactly when K-polystable

On Fano manifolds there are obstructions. The automorphism group of a Kähler–Einstein Fano manifold is reductive (Matsushima, 1957), and the Futaki invariant must vanish (1983). Tian (1990) settled Fano surfaces: they have Kähler–Einstein metrics exactly when the automorphism group is reductive, which excludes only $\mathbb{CP}^2$ blown up at one or two points. Tian (1997) then found Fano manifolds with no holomorphic vector fields and no Kähler–Einstein metric, which showed that a further obstruction exists.

That obstruction is K-stability. A test configuration is a $\mathbb C^*$-equivariant degeneration of the polarized manifold to a possibly singular central fibre, and its Donaldson–Futaki invariant is a number computed from the action on that fibre. K-polystability asks that the invariant be non-negative, and zero only for product configurations (with a normality condition on the total space). The full story, including the proofs by the continuity method and by the flow, is in Kähler–Einstein and K-stability.

#Extremal and cscK metrics

Calabi (1982) proposed minimizing $\int_M S^2\,\omega^n$ over a Kähler class. Critical points are extremal metrics, characterized by $g^{i\bar j}\partial_{\bar j}S\,\partial_i$ being a holomorphic vector field. If that field vanishes, $S$ is constant and the metric is cscK. Calabi's own examples include extremal metrics with non-constant $S$ on the one-point blow-up of $\mathbb{CP}^2$, the same surface that has no Kähler–Einstein metric.

cscK generalizes Kähler–Einstein. If $[\omega]$ is a multiple of $c_1(M)$ and $S$ is constant, then $\operatorname{Ric}(\omega)$ has constant trace, so it is harmonic, and it must equal the harmonic representative $\lambda\omega$. As a PDE, the cscK equation is fourth order in $\varphi$. Chen and Cheng write it as a coupled second-order system:

The space of Kähler metrics and the K-energy

Mabuchi (1986–87) gave $\mathcal H$ a Riemannian structure, $\langle\psi_1,\psi_2\rangle_\varphi = \int_M\psi_1\psi_2\,\omega_\varphi^n$, and introduced the K-energy $\mathcal M$, whose derivative along $\psi$ is, up to normalization,

So cscK metrics are the critical points of $\mathcal M$. The Chen–Tian formula writes $\mathcal M$ as the entropy $\int_M\log(\omega_\varphi^n/\omega^n)\,\omega_\varphi^n$ plus lower-order energy terms, which is one reason entropy bounds appear in the estimates.

A geodesic in $\mathcal H$ solves $\ddot\varphi = g_\varphi^{i\bar j}\,\partial_i\dot\varphi\,\partial_{\bar j}\dot\varphi$. Semmes and Donaldson observed that this is a homogeneous complex Monge–Ampère equation on $M$ times an annulus. Donaldson's programme treats $\mathcal H$ as a non-positively curved symmetric-space analogue, on which $\mathcal M$ is convex along geodesics and a destabilizing test configuration behaves like a geodesic ray at infinity with non-positive K-energy slope.

Chen (2000) showed that any two potentials are joined by a weak geodesic with bounded mixed second derivatives, that $\mathcal H$ is a genuine metric space, and that the cscK metric in a class is unique when $c_1 < 0$ or $c_1 = 0$. Berman and Berndtsson later proved uniqueness up to automorphisms in general, a result that Berman, Darvas and Lu use.

The first statement is Donaldson's geodesic stability conjecture. Berman, Darvas and Lu supply a converse direction. A cscK metric forces properness of the K-energy, modulo automorphisms, and for a polarized manifold it forces K-polystability.

How the proofs work

The template is Yau's. Deform along a continuity path such as $\omega_t^n = e^{tF+c_t}\omega^n$. Openness follows because the linearization is the invertible Laplacian $\Delta_{\omega_t}$. Closedness needs a priori estimates, which come in a fixed order.

The $C^0$ estimate. When $c_1 < 0$ the term $e^{\varphi}$ gives a bound from the maximum principle at once. When the exponent has no such help, Yau used Moser iteration. Kołodziej (1998) extended the bound to densities $e^F$ in $L^p$ with $p > 1$ using pluripotential theory. Guo, Phong and Tong (2021) found a PDE proof of the sharp estimate by comparison with an auxiliary Monge–Ampère equation, an idea they credit to Chen and Cheng.

The second-order estimate. Apply the maximum principle to $\log\operatorname{tr}_\omega\omega_\varphi - A\varphi$ for large $A$, using a lower bound on the bisectional curvature of $\omega$. Together with the equation, which fixes the determinant, this bounds $\omega_\varphi$ above and below by $\omega$ (Aubin; Yau).

Higher order. Calabi's third-order estimate, or the complex Evans–Krylov theorem, gives $C^{2,\alpha}$ control, and Schauder theory does the rest. Cao's proof runs the same ladder for the parabolic equation, and Song and Weinkove's notes write out each rung for the flow.

Where the Fano case differs. With $e^{F-\varphi}$ the $C^0$ estimate can fail, and K-stability measures exactly how. Chen–Donaldson–Sun and Tian replace it by an analysis of Gromov–Hausdorff limits along a path of metrics with cone singularities, showing that a limit is algebraic and so produces a destabilizing test configuration if the path fails.

Where cscK differs. A fourth-order equation has no maximum principle, so Chen and Cheng work with the coupled system. Their first paper shows that all higher derivatives are controlled by a $C^0$ bound on $\varphi$, and that this bound in turn follows from an upper bound on the entropy $\int_M F e^F\omega^n$. Properness of $\mathcal M$ bounds the entropy along a continuity path proposed by Chen, which closes the argument.

Kähler geometry inside Ricci flow

On a Kähler manifold $\operatorname{Ric}(\omega)$ is a closed $(1,1)$-form, so Hamilton's equation $\partial_t g = -2\operatorname{Ric}$ reads $\partial_t\omega = -2\operatorname{Ric}(\omega)$. Its class moves on a straight line, $[\omega(t)] = [\omega_0] - 4\pi t\,c_1(M)$. Fix a volume form $\Omega$, put $\chi = \sqrt{-1}\,\partial\bar\partial\log\Omega$ (which represents $-2\pi c_1(M)$) and $\hat\omega_t = \omega_0 + 2t\chi$. Then $\omega(t) = \hat\omega_t + \sqrt{-1}\,\partial\bar\partial\varphi$ solves the flow when

This scalar equation is strictly parabolic, and by Hamilton's uniqueness theorem its solution is the Ricci flow. So the Kähler condition is preserved. Most Kähler papers use $\partial_s\omega = -\operatorname{Ric}(\omega)$, which is the same flow with $s = 2t$.

Tian and Zhou Zhang state this in the time $s = 2t$, after earlier work of Tsuji. The formula is exact: for $\omega_0 = \omega_{FS}$ on $\mathbb{CP}^n$ the solution is $(1 - 2(n+1)t)\,\omega_{FS}$, so $T = 1/(2(n+1))$.

The same picture drives the long-time theory. Cao showed the suitably normalized flow converges to the Kähler–Einstein or Ricci-flat metric when $c_1 < 0$ or $c_1 = 0$. The analytic MMP concept card summarizes what happens at finite-time singularities in general.

Two other flows belong here. The Calabi flow $\partial_t\varphi = S(\omega_\varphi) - \bar S$ is the fourth-order gradient flow of the K-energy, filed with the other curvature flows. Chern–Ricci and pluriclosed flows extend the idea to Hermitian metrics without the Kähler condition. The odd-dimensional counterpart is Sasakian geometry, where the Riemannian cone over the manifold is Kähler.

Key results

Year	Result	Who
1957	Automorphisms of a Fano Kähler–Einstein manifold are reductive	Matsushima
1978	Calabi conjecture; Kähler–Einstein metrics for $c_1 < 0$	Yau; Aubin
1982	Extremal metrics introduced	Calabi
1983	The Futaki invariant obstructs Kähler–Einstein metrics	Futaki
1985	Kähler–Ricci flow converges when $c_1 \le 0$	Cao
1990	Kähler–Einstein metrics on Fano surfaces with reductive automorphisms	Tian
1997	K-stability; Fano manifolds with no vector fields and no such metric	Tian
2000	Metric space structure of $\mathcal H$ and weak geodesics	Chen
2002	Algebraic definition of K-stability	Donaldson
2004	Numerical description of the Kähler cone	Demailly–Paun
2006	Maximal existence time of the flow from cohomology	Tian–Zhang
2009	Singular Kähler–Einstein metrics on klt pairs	Eyssidieux–Guedj–Zeriahi
2012	Fano Kähler–Einstein metrics from K-polystability	Chen–Donaldson–Sun; Tian
2016	cscK implies properness and K-polystability	Berman–Darvas–Lu
2018	cscK estimates; existence from properness; geodesic stability	Chen–Cheng
2021	PDE proof of sharp $L^\infty$ estimates for Monge–Ampère	Guo–Phong–Tong

Where it stands

Very active. Current work covers singular Kähler–Einstein metrics on varieties (following Guedj and coauthors, who solve degenerate Monge–Ampère equations for klt pairs), the cscK version of the Yau–Tian–Donaldson conjecture, Kähler–Ricci flow and birational geometry, Calabi–Yau degenerations and the SYZ conjecture, and non-Kähler (Hermitian) generalizations. In the feed below, Kähler geometry is one of the most common co-tags on Ricci flow papers.

Some recent preprints on this site, which as far as we know are not yet refereed, show the pace:

• Type I bounds. Jian and Song prove a Type I scalar curvature estimate at finite-time singularities on compact Kähler manifolds, with geometric consequences for the limits. On Kähler surfaces, Cifarelli, Conlon, Hallgren and Zhang and Xu and Zhou give Type I curvature bounds for volume-collapsing singularities.
• Through a singularity. Chen, Hallgren and Lavoyer describe a shrinker–cone–expander transition for noncollapsed flows in complex dimension two.
• Complete shrinkers. Conlon and Deruelle prove uniqueness of the soliton vector field, Collins, Huang, Tong and Xu construct shrinkers on toric Fano fibrations, and Esparza and Zhang prove a Matsushima theorem and K-polystability.
• Generalized Kähler. Huang and Zhu prove convergence of generalized Kähler–Ricci flow to a soliton on toric Fano manifolds, a conjecture of Apostolov, Streets and Ustinovskiy.

#Open problems

• cscK and stability. Existence implies K-polystability. Whether a suitable form of K-stability implies existence, in a general polarization, is the open half of the Yau–Tian–Donaldson conjecture.
• Calabi flow. Calabi and Chen conjectured that the flow exists for all time from any smooth potential. Chen and Cheng show it continues while the scalar curvature stays bounded. Li, Wang and Zheng prove continuation on Kähler surfaces under a Sobolev bound, and study the limits of such flows.
• Analytic MMP. Gromov–Hausdorff convergence at every finite-time singularity, and continuation through flips, are open in general. So are the collapsed cases.
• Degenerations. How Calabi–Yau metrics behave as the complex structure degenerates, and what this says about SYZ fibrations, is known only in part.