40 problems

Movable Bend and Break conjecture. There exists a constant $Q(\mathcal X)$ such that, whenever

Let $V\to\mathbb{P}^1$ be a Fano fibration satisfying the $K$-condition, which is weaker than the $K^2$-condition. K-condition conjecture. The $K$-condition is sufficient for birat…

Let $V\to S$ be a fibration into Fano varieties satisfying (i) $V$ is smooth and $\mathop{\rm dim}V\geq 4$; (ii) $\mathop{\rm dim}S\geq 1$ and the anticanonical class $-K_V$ is rel…

Twisted fibration rigidity conjecture. If the fibration $V/S$ as above is sufficiently twisted over the base $S$, then $V/S$ is birationally rigid.

Let $\pi:\mathcal X\to B$ be a good Fano fibration, let $\lambda$ be an intersection profile, and let $\mathsf{S}_{\lambda,\mathbb Z}$ be the corresponding integral section classes…

Let $\pi:\mathcal X\to B$ be a good Fano fibration, let $\lambda$ be an intersection profile, and let $c(\mathcal X,\lambda)$ be the leading constant in the all-height Manin conjec…

Let $\pi:\mathcal X\to B$ be a good Fano fibration, let $\lambda$ be an intersection profile, let $M_{\alpha}$ be the unique Manin component for sufficiently large…

Let $\pi:\mathcal X\to B$ be a good Fano fibration and let $\mathsf Z\subset\mathcal X_{\eta}(K(B))$ be the exceptional set. A properly constructible thin set is a finite union of…

Let $\pi:\mathcal X\to B$ be a good Fano fibration, let $q$ be the cardinality of the constant field, and suppose that $\mathcal X_{\eta}(K(B))$ is not thin. With…

Let $\pi:\mathcal X\to B$ be a good Fano fibration, let $K(B)$ be the function field of $B$, and let $\mathcal X_{\eta}(\mathbb A_{K(B)})^{\operatorname{Br}(\mathcal X_{\eta})}$ be…

Let $\pi:\mathcal{X}\to B$ be a good Fano fibration, let $\mathcal{X}_{\eta}$ be its generic fiber, and let $\mathsf Z$ be the union of the images of all breaking thin maps into…

Let $X$ be a compact Kähler manifold undergoing the normalized Kähler–Ricci flow, and suppose that, in the finite-time singularity case, there is a proper holomorphic map with conn…

Sun–Zhang's conjecture. A polarized Fano fibration $(\pi:X\to Y,\xi)$ admits a Kähler–Ricci shrinker $(\omega,\xi)$, unique up to automorphisms of $X$ preserving $\xi$, if and only…

Let $\mathcal{M}^o$ be a finite-type algebraic $\mathbbm{k}$-stack underlying a $\mathbb{Q}$-Gorenstein faithfully flat affine family of $\mathbb{Q}$-Fano fibrations, equipped with…

Consider $T$-equivariant Fano fibrations $\pi\colon X\to Y$, fixing the multi-Hilbert function of $T\curvearrowright Y$ and the Hilbert polynomial of $-K_X$ on the fibers of $\pi$.…

Let $T$ be a torus and let $c>0$ be a real number. Consider isomorphism classes of $T$-equivariant $(\mathbb{Q})$-Fano fibrations $T\curvearrowright(X\xrightarrow{\pi}Y)$, with wei…

Let $X\xrightarrow{\pi}Y\ni p$ be a Fano fibration over an affine pointed variety. Let $v$ be a quasi-monomial valuation of $K(X)$ whose center is supported inside $\pi^{-1}(p)$, a…

Tangent flows at infinity conjecture. Given a non-collapsed ancient Kähler-Ricci flow $(X,\omega(t))$, $t\in(-\infty,0]$: 1. $X$ admits a non-collapsing weak Fano fibration structu…

Finite-time two-step degeneration conjecture. For any $p\in Y$ and $q\in\pi^{-1}(p)$: 1. The Kähler-Ricci flow $\omega(t)$ naturally induces a weighted-volume-minimizing, hence K-s…

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…

Two-step degeneration conjecture for weighted volume. 1. There is a unique minimizer $v_*$ of $\mathbb W$ on $\operatorname{Val}_{\pi,p}^*$, which is quasi-monomial. 2. The graded…

K-polystability conjecture. A polarized Fano fibration $(\pi:X\rightarrow Y,\xi)$ with $X$ smooth admits a Kähler-Ricci shrinker, which is unique up to the action of…

Bounded relative-global klt complement conjecture. For every positive integer $d$ and positive real number $\epsilon$, there exists a positive integer $N$, depending only on $d$ an…

Let $f\colon X\to Z$ be a Fano fibration, with $X$ $\epsilon$-lc of dimension $d$, and let $Z$ be a curve. Shokurov's bounded-complements conjecture. The existence of a bounded klt…

Let $d\in\mathbb N$ and let $\epsilon\in\mathbb R^{>0}$. A Fano fibration is a contraction $f\colon X\to Z$ for which $X$ has good singularities and $-K_X$ is ample over $Z$. McKer…