15 problems

Let $d$ be a positive integer, and let $\Gamma\subset [0,1]$ be a DCC set of rational numbers whose closure is contained in $\mathbb Q$. A projective $\mathbb R$-complementary norm…

Let a polarized variety mean a projective variety equipped with a polarization, and consider the moduli space of K-polystable polarized varieties. K-moduli conjecture. A set of K-p…

Let $\mathcal{G}$ be a log bounded set of couples that is complete under dominant small modifications, and let $\widehat{\mathcal{G}}$ be its K-moduli completion: a couple belongs…

For each of the seven specified Noether--Lefschetz divisors in the degree-$22$ moduli space, let $X$ be the Gorenstein canonical Fano threefold constructed from a general polarized…

Let $(S,L)$ be an ADE K3 surface in the degree-$22$ moduli space $\text{cusefont{U}{dutchcal}{m}{n}F}_{22}$, and let $\text{cusefont{U}{dutchcal}{m}{n}P}^{\mathrm{plt}}$ denote the…

Let $\mathscr{X}\rightarrow U$ be the family of Fano threefolds of Family textnumero 2.16, let $\lambda_{\operatorname{CM}}$ be its CM line bundle, and let $U^{\operatorname{ss}}$…

Let $X$ be a K-semistable $\mathbb{Q}$-Gorenstein degeneration of a smooth Fano threefold of Family textnumero 2.16. Degeneration conjecture. $X$ is isomorphic to the blow-up of a…

Let $C$ be an irreducible pseudo-stable genus $2$ curve with one node and normalization $\widetilde{C}$, and let $V_0$ be the associated K-polystable $(2,2)$-complete intersection.…

K-moduli conjecture for polarized Fano fibrations. Given a positive integer $n$ and $\epsilon>0$, there exists a projective moduli space parametrizing K-polystable polarized Fano f…

Let $S$ be a finite type $\mathbbm{k}$-scheme and let $\pi\colon\mathcal{X}\to S$ be a $\mathbb{Q}$-Gorenstein family whose $h$-invariants and volumes are constant on closed fibers…

Fix $n$, $V$, $T$, and a Reeb vector field $\xi$, and let $M_V^{\rm ss}(\xi)$ be the good moduli space of $n$-dimensional K-semistable Fano cones of volume density $V$, with descen…

Ascher–DeVleming–Liu's conjecture. At $\varepsilon=0$, there exists a moduli space of Calabi–Yau pairs that is the common ample model of the Hodge line bundles on $M^{\rm K}$ and…

Let ${\mathcal F}^\ast$ be the locally symmetric variety, let $\lambda$ be its Hodge line bundle, and for $s\in[0,1]\cap{\mathbb Q}$ define … Let $P^\ast$ denote the relevant open…

Let a universal family be given over one of the weak K-moduli stacks under consideration, including Shah’s or Alexeev–Engel–Thompson’s compactification of the moduli stack of degre…

Properness conjecture. The good moduli space $X^{\rm Kps}_{n,V}$ is proper.