K-stability and related topics

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2. Distribution of invariants

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Problem 2.1.
(C. Li) Give an algebraic proof of stronger version of K.Fujita’s result that for smooth Fano $X$ , if $\alpha(X)=\frac{n}{n+1}$ , then $X$ is uniform K-stable.
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Problem 2.2.
(I. Cheltsov, Y. Liu, Z. Zhuang) Find as large $\alpha$ invariant as possible for K-unstable smooth Fanos.
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Problem 2.3.
(C. Jiang) For K-semistable klt Fano varieties, in a fixed dimension, does the set of all $\alpha$ -invariants sastisfy the DCC? Is there a gap above $\frac{1}{n+1}$ ? If so, how large?
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Problem 2.4.
(L. Qi) Does the set of $\alpha$ -invariants of K-semistable Fanos satisfy the ACC? What about $\delta$ -invariant?
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Problem 2.5.
(C. Jiang) For smooth Fano varieties, is it true that $\alpha \leq 1$ ? Conjecture of Kawamata that $|-K_X| \neq \varnothing$ implies this one.
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Problem 2.6.
(C. Li) How large is the gap of global volumes of K-semistable Fano varieties from $(n+1)^n$ ?

(C. Xu) Is the second largest volume achieved by quadratic?
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Problem 2.7.
(C. Jiang) Find volume upper bound of birational superrigid Fano varieties.
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Problem 2.8.
(C. Xu) What about similar questions for $\widehat{\mathrm{vol}}$ ?

Cite this as: AimPL: K-stability and related topics, available at http://aimpl.org/kstability.

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