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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.