52 problems

Let $F$ be an elliptic space with minimal model $\Lambda V$, and let $F\to X\to B$ be a fibration. The condition that $\dim V^{\mathrm{odd}}=\dim V^{\mathrm{even}}$ is imposed on t…

Let $f:\mathcal{X}\to\mathcal{Y}$ be a fibration in varieties over a number field $K$. Let $R$ be the finite set of excluded prime ideals, let $q_{\mathfrak p}$ denote the residue-…

Halperin's conjecture. The fibration is TNCZ, meaning that $j$ induces a surjection on rational cohomology.

Let $Y^\partial$ be a smooth or e-admissible effective orbifold, let $g^\partial:Y^\partial\rightarrow Z$ be a fibration, and let $Y_z^\partial$ be its orbifold general fiber. Orbi…

Let $f:X\to C$ be a fibration over a function field with $K$-core factorization $f=g_f\circ c_f$, and suppose the orbifold fibration … is not split. Conjecture IV{FF}. There should…

Let $g:(Y/H)\to Z$ be a prepared and high holomorphic fibration between manifolds, with $Y\in\mathcal C$. Let $\Delta(g,H)$ be its base orbifold, let $z\in Z$ be general, and write…

Height comparison conjecture. There exist a family of $v$-adic metrics on $K_Y$ and constants $c_2>c_1>0$ such that

Asymptotic arithmetic fibration conjecture. The set of ${\cal L}$-targets of $V$ forms an asymptotic arithmetic ${\cal L}$-fibration.

Let $Y$ be a complex analytic manifold and let … be a projective morphism to the complex disc. A homotopy fiber bundle is a morphism whose fibers are homotopy equivalent in the cor…

Isotriviality conjecture for simple fibres. If $X_y$ is simple and $a(Y)=0$, then

Let $X$ be a compact Kähler manifold with nef anti-canonical bundle $-K_X$. A locally constant fibration is a fibration that is locally constant in the sense of Definition 2.3 of M…

Let $f:X\dashrightarrow C$ be a canonically fibered surface, meaning that $X$ and $C$ are irreducible, smooth and projective over $\mathbb{C}$ with dimensions $2$ and $1$, respecti…

Let $f:X\to B$ be a cellular map of finite CW complexes. Let $P$ be the finite-space map from the three-point diagram with one distinguished point and two indistinguishable points…

A canonical threefold on the Noether line is a canonical threefold whose canonical volume and geometric genus satisfy … A simple fibration in $(1,2)$-surfaces means the type of fib…

Let $F\to E\overset{p}\longrightarrow B$ be a fibration of simply connected spaces of finite type. Suppose that $\Omega p$ has a homotopy section and that $E$ is coformal over a ri…

Let $\pi:X\to\mathbb{P}^1$ be a standard fibration, and let $\theta:\mathbb{P}^1\to(\mathbb{P}^1,\partial_\pi)$ be a finite étale orbifold morphism that is a $G$-torsor under a fin…

Orbifold Fano bound conjecture. If the $\mathbb{Q}$-divisor $-(K_{\mathbb{P}^1}+\partial_\pi)$ is ample, then, for every $\varepsilon>0$,

Bobadilla–Kollár's conjecture. If $f$ is a $\mathbb Z$-homology fiber bundle, then $f$ is smooth.

Let $X$ be a smooth projective variety, and let $f:X\to\mathbb{P}^n$ be a surjective morphism. Let $\Delta(f)$ denote the discriminant locus of $f$. Higher-dimensional uniruledness…

Let $X$ be a smooth projective variety, and let $f:X\to\mathbb{P}^1$ be a surjective morphism with at most $2$ singular fibers. Two-singular-fiber uniruledness conjecture. Then $X$…

Let $f:X\to Y$ be a smooth projective morphism with connected fibers between smooth quasi-projective varieties. Write $\bar\kappa$ for logarithmic Kodaira dimension, and let $F$ be…

Relative McKernan–Prokhorov conjecture. The set of base-polarized fibrations $X\to(Z,A)$ such that there exists a $(d,r,\epsilon)$-log Calabi–Yau rationally connected fibration…

Let $f:(X,H)\to(B,L)$ be a polarized smooth fibration such that every fiber $(X_b,H_b)$ is K-stable. Assume that $\operatorname{Aut}(X_b,H_b)=0$ for every $b\in B$. Dervan–Sektnan'…

Let $f:(X,H)\to(B,L)$ be a smooth polarized fibration. The fibration is fibration stable, and $(B,L)$ has a twisted constant scalar curvature Kähler metric with respect to the Weil…

Conjectured estimate. Estimate (2) is well-defined and holds.