Where This Track Leads

Book 5A ends where the two-dimensional Ricci flow begins. This last chapter is short, and it is for later. It explains where the complex point of view leads in higher dimensions: to the Kähler–Ricci flow, a branch of the subject that runs in parallel to the road to Poincaré and meets algebraic geometry. Nothing in Books 6A to 12C depends on it. Read it now for orientation, and come back to it after Book 11B, when you know the Ricci flow well enough for it to be useful.

The one idea to take away is this. In complex dimension one, the Ricci flow is a scalar heat equation for the conformal factor (5A.5 Uniformization and the Two-Dimensional Ricci Flow). On a Kähler manifold of any dimension the same thing happens: the Ricci flow, a system of PDE for a tensor, collapses to one scalar equation for a potential function, a parabolic complex Monge–Ampère equation. That is why the Kähler case is both easier and deeper than the general one: easier because scalar equations have stronger maximum principles, deeper because the scalar equation is governed by the cohomology and algebraic geometry of the manifold.

By the end of this chapter you will be able to:

• say what a Kähler metric is, and why every conformal metric on a Riemann surface is one;
• explain why the Ricci form is a complex Hessian of a log-determinant, and how that makes the Kähler–Ricci flow a scalar equation;
• describe what the flow does to the cohomology class of the metric, and why its existence time can be read off from cohomology;
• name the main results and open problems, and know what to read and when.

6.1Kähler metrics

A complex manifold of complex dimension $n$ is a manifold covered by charts to $\mathbb{C}^n$ whose transition maps are holomorphic (8A.1 Smooth Structures, with "smooth" replaced by "holomorphic"). A Riemann surface is the case $n = 1$. A Riemannian metric on it is Hermitian if multiplication by $i$ on tangent vectors is an isometry; in local coordinates it is described by a positive definite Hermitian matrix $(g_{j\bar k})$, and its Kähler form is

The metric is Kähler if $d\omega = 0$. Then, locally, $\omega = i\partial\bar\partial\psi$ for a real function $\psi$, a Kähler potential: the whole metric is the complex Hessian of a single function (Exercise 6.1).

In complex dimension one every Hermitian metric is Kähler, since a $2$-form on a surface is automatically closed. A conformal metric $e^{2u}|dz|^2$ has Kähler form $\omega = e^{2u}\,\frac i2dz\wedge d\bar z = e^{2u}\,dx\wedge dy$, its area form. So everything in 5A.5 Uniformization and the Two-Dimensional Ricci Flow was Kähler geometry in dimension one.

The key formula is for the curvature. For a Kähler metric the Ricci curvature, written as a form like $\omega$, is

The Ricci form is a complex Hessian of the log of the volume density. In dimension one, $\det(g_{j\bar k})$ is $e^{2u}$ up to a constant, $i\partial\bar\partial f = \frac12\Delta f\,dx\wedge dy$, and the formula reads $\operatorname{Ric} = -\Delta u\,dx\wedge dy = K\omega$ with $K = -e^{-2u}\Delta u$: the curvature formula of 5A.5 Uniformization and the Two-Dimensional Ricci Flow (Exercise 6.2).

6.2The flow becomes a scalar equation

The Kähler–Ricci flow is the Ricci flow started from a Kähler metric. It stays Kähler, and it is usually written for the Kähler form as

which is Hamilton's flow $\partial_tg = -2\operatorname{Ric}$ with time rescaled by a factor of $2$.

Two facts turn this into a scalar equation. First, $\operatorname{Ric}(\omega)$ is closed, and its de Rham cohomology class does not depend on $\omega$: it is a multiple of the first Chern class $c_1(M)$, a topological invariant of the complex manifold. So the class of $\omega(t)$ moves along a straight line:

Second, by the $\partial\bar\partial$-lemma, two Kähler forms in the same class differ by $i\partial\bar\partial$ of a function. So one can choose a simple reference path $\hat\omega_t$ in the moving class and write $\omega(t) = \hat\omega_t + i\partial\bar\partial\varphi(t)$. Then the flow is equivalent to

for a fixed volume form $\Omega$ chosen to match the reference path. This is a parabolic complex Monge–Ampère equation: the determinant of the complex Hessian of $\varphi$ (plus a background) appears inside a logarithm.

In dimension one the "determinant" of a $1\times1$ matrix is just its entry, and the equation becomes the logarithmic diffusion $\partial_tv = \Delta\log v$ of 5A.5 Uniformization and the Two-Dimensional Ricci Flow, written for a potential (Exercise 6.3). So the two-dimensional Ricci flow you met in the last chapter is the first case of the Kähler–Ricci flow, and its higher-dimensional version keeps the same structure: a scalar quantity, the log of a volume ratio, diffusing.

6.3What is known, and what is open

Read this section as a map; each statement is checkable against the topic guide and the papers it cites.

Existence time from cohomology. The flow exists exactly as long as the class $[\omega_0] - t[\operatorname{Ric}(\omega_0)]$ remains a Kähler class (Tian and Zhang, after Tsuji). The singular time is determined by cohomology alone, before solving anything. Nothing like this is true for the Ricci flow on a general Riemannian manifold (Figure 6.1).

Convergence to canonical metrics. Huai-Dong Cao (1985) proved that when $c_1(M) < 0$ or $c_1(M) = 0$ the suitably normalised flow exists for all time and converges to a Kähler–Einstein metric, respectively a Ricci-flat one. This gave a parabolic proof of the theorems of Aubin and Yau on the Calabi conjecture, the higher-dimensional analogue of the flat and hyperbolic cases of uniformization in 5A.5 Uniformization and the Two-Dimensional Ricci Flow.

Fano manifolds, where $c_1(M) > 0$, are the analogue of the sphere, and they are harder, just as the sphere was the hard case of 11A.7 Ricci Flow on Surfaces. Perelman proved uniform bounds on the scalar curvature and diameter along the normalised flow, written up by Sesum and Tian. The Hamilton–Tian conjecture, that the flow converges to a possibly singular Kähler–Ricci soliton, was proved by Xiuxiong Chen and Bing Wang, and independently by Richard Bamler.

The analytic minimal model program. Jian Song and Gang Tian proposed that on a projective variety the flow should carry out the birational surgeries of the minimal model program of algebraic geometry (contracting curves, flips) as its finite-time singularities, and then collapse onto the canonical model. Much of this is understood for surfaces and in special cases; the general program, and the precise form of singularities in higher dimensions, are active research areas.

6.4When to come back

The Kähler–Ricci flow needs three things this book doesn't provide:

• Kähler geometry: complex manifolds, the $\partial\bar\partial$-lemma, Chern classes and line bundles. Song and Weinkove's notes summarise what is needed.
• The Ricci flow itself: the evolution equations, maximum principles and solitons of Books 11A and 11B. For Fano manifolds, Perelman's entropy of Book 12A.
• The elliptic theory: Yau's estimates for the complex Monge–Ampère equation, which the parabolic estimates imitate. Book 6A's Schauder theory (6A.6 Parabolic Regularity) is the background.

So the order is: finish Book 11B, read Song–Weinkove, then the Boucksom–Eyssidieux–Guedj volume. The site's topic guide on the Kähler–Ricci flow and the concept card "Kähler–Ricci flow and the minimal model program" collect the foundational papers and follow new ones as they appear.

6.5History

Erich Kähler introduced the metrics named after him in 1933. Eugenio Calabi conjectured in the 1950s that the Ricci form can be prescribed within its cohomology class; Thierry Aubin and Shing-Tung Yau proved the existence of Kähler–Einstein metrics when $c_1 < 0$, and Yau proved the Calabi conjecture, including the Ricci-flat case, in work announced in 1977 and published in 1978. Cao's parabolic proof by the Kähler–Ricci flow appeared in 1985. Song and Tian's analytic minimal model program dates from the mid-2000s, and the Hamilton–Tian conjecture was settled in the late 2010s.

6.6Exercises

Solution

$-i\partial\bar\partial\log v = -\frac12\Delta(2u)\,dx\wedge dy = -\Delta u\,dx\wedge dy$, and $K\omega = -e^{-2u}\Delta u\cdot e^{2u}\,dx\wedge dy = -\Delta u\,dx\wedge dy$.

Solution

(a) $\partial_t\omega = \partial_tv\,dx\wedge dy$ and $-\operatorname{Ric}(\omega) = i\partial\bar\partial\log v = \frac12\Delta\log v\,dx\wedge dy$. (b) $\partial_tv = \frac12\Delta\partial_t\psi = \frac12\Delta\log v$.

Solution

$\det(cg) = c^n\det g$, and $\log c^n$ is constant, so $i\partial\bar\partial$ kills it. Then $\partial_t\omega = -\lambda\omega_0$ and $-\operatorname{Ric}(\omega(t)) = -\operatorname{Ric}(\omega_0) = -\lambda\omega_0$.

Solution

Let $m(t) = \max_xu(x, t)$, attained at $x_t$. At $x_t$ the Hessian of $u$ is negative semidefinite, so $\Delta u(x_t, t) \leq 0$ and $\partial_tu(x_t, t) \leq 0$. A standard argument (6A.4 Maximum Principles) turns this into $m(t_2) \leq m(t_1)$ for $t_2 > t_1$: for instance, apply it to $u - \varepsilon t$, which would have $\partial_t > 0$ at a first time its maximum exceeds $m(t_1)$, a contradiction; then let $\varepsilon \to 0$.