Ricci flow on Kähler manifold

Question

Knowing the Ricci flow on Riemann surfaces, see e.g.

Ricci flow on Riemann surfaces

How could we write the Ricci flow on Kähler manifold? Thanks for the reply!

Answer 1

Read all about it in J. Song's notes from 2012.

Answer 2

Let $X$ be a Kähler manifold, with Kähler form $\omega$, then Kähler ricci flow introduced by S.T.Yau as

$$\frac{\partial \omega}{\partial t}=-Ric(\omega)+\lambda\omega$$ If the initial metric $\omega_0$ be Kähler, then all metrics along Kähler Ricci flow are Kähler metrics.

In fact Kähler resolves the singularities and could be seen as PDE surgery. This is the main philisophy of Kähler Ricci flow. For pair $(X,D)$ where $D$ is a divisor with conic singularities, then we can replace Kähler-Ricci flow with the following equation as conical Kähler Ricci flow

$$\frac{\partial \omega}{\partial t}=-Ric(\omega)+\lambda\omega+[D]$$ where $[D]$ is the current of integration

see https://www.maths.cam.ac.uk/system/files/canonicalmetrics_0.pdf