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11. Other problems


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      Problem 11.1.

      (Z. Patakfalvi) Consider $f: \mathfrak{X} \to T$(smooth projective curve), assume the general fiber is unifrom K-stable. Let $c_{nef}:=$ minimum of $c$ such that $-K_{\mathfrak{X}/T}+c \cdot f^* (t)$ is nef. Find an effective upper bound of $c_{nef}$ in terms of $\delta, \mathrm{vol}$ of general fiber.


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          Problem 11.2.

          (I. Cheltsov, A. Castravet, S. Donaldson) Let $X$ be a smooth Fano variety, $\mathrm{Pic}(X)=\mathbb{Z}(-K_X)$. Is $X$ always K-semistable?

          (S. Donaldson) Is the moduli spcae of vector bundles on a curve K-(semi)stable?

          (C. Araujo) What about moduli space of parabolic bundles over $\mathbb{P}^1$ with half weight?


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              Problem 11.3.

              (C. Araujo) Let $X$ be a smooth Fano variety. Let $V$ denote the variety of minimal degree rational curves through a general point. Does the K-(semi)stability of $X$ implies special properties of $V$?


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                  Problem 11.4.

                  (Y. Liu) Explicitly describe the K-moduli compactification of $(\mathbb{P}^1 \times \mathbb{P}^1, c \cdot C)$, where $C \in |\mathscr{O}(3,3)|, 0 < c < 2/3$.

                  $\cdot$ Birational model of $M_4$,

                  $\cdot$ When $c=1/2$, a divisor of this K-moduli space isomorphic with K-moduli space of $dP_1$.

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