# Questions tagged [kahler-ricci-flow]

Kähler-Ricci flow is a Kähler version of Ricci-flow for Kähler manifolds

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### Vanishing components of Kähler metric

Let $(X, \omega) $ be a $n$-dimensional complex Kähler manifold such that $\omega^{n-1}=d\alpha $.
Does $\partial\alpha^{n-1,n-2} =0$ (resp. $\bar\partial\alpha^{n-2,n-1} =0$)
Where $\alpha^{n-1,n-2}$ ...

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### Reference request: a PDE related to Kahler–Ricci flow

I was reading the survey by Imbert and Silvestre where I noticed the PDE
$$
\frac {\partial u} {\partial t} = \ln(\det (D^2u))
$$
for the study of the Kahler–Ricci flow (Eq (2.2) at page 10 in ...

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### Upper bound on the bisectional curvature

This is a follow-up to the question Schwarz lemma and bisectional curvature lower bound. Looking at the same note Song and Weinkove - Lecture notes on the Kähler–Ricci flow, page 24, the first line ...

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### Schwarz lemma and bisectional curvature lower bound

Reading a proof of the Schwarz lemma for the Kähler-Ricci flow from p22 of these lecture notes. I am confused as to what they mean by taking $$\inf _{x \in M} \{\hat{R}_{i \bar i j \bar j}(x) \mid \{\...

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### Reading material for an analytical aspect of Kähler Geometry

This question was originally posted on MSE.
But I would like to post it here to see whether anyone could recommend some reference for me.
I am currently reading the paper "Three-circle theorem ...

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### Ricci flow preserves almost Kahler condition?

I have been unable to find a reference to the following (perhaps too naive) question.
Suppose we have an almost Kahler manifold $(M^{2n},\omega,J,g)$ i.e. the almost complex structure $J$ is non-...

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### reference for the weak compactness of currents

I am trying to follow the arguments in page 22 of the following paper k\"{a}hler currents and null loci
It quotes the weak compactness of currents, I wonder if there is any reference about it. My ...

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### Ricci flow preserves locally symmetry along the flow

Let $(M,g_0)$ be a closed locally symmetric Riemannian manifold and let $g(t)_{t\in[0,T)}$ be a solution to the Ricci flow on $M$ with $g(0)=g_0$. How one can prove that Ricci flow preserves locally ...

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### Extending Kahler metric across a divisor

Let $(X,\omega)$ be a complete noncompact Kahler manifold of finite volume. Suppose $X$ is can be compactified to a compact projective manifold $M$ so that $D=M-X$ is a divisor of simple normal ...

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### Tian's approach for solving the conjecture of invariance of plurigenera in Kahler setting

Let $f:X\to Y$ be a smooth holomorphic fibre space whose fibres $f^{-1}(y)$ have pseudoeffective canonical bundles. suppose that
$$\frac{\partial \omega(t)}{\partial t}=-Ric_{X/Y}(\omega(t))-\omega(...

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### Ricci flow on Kähler manifold

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!

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### Curvatures preserved under the Kahler-Ricci flow

Maybe it is a trivial question. Is there any obvious reason that non-negative holomorphic bisectional curvature is preserved by (normalized) Kahler-Ricci flow, but non-negative Ricci curvature is not ...