94 problems

A fibration has trivial fibre transport when its fibre transport action is trivial, and a fibre is positively rationally elliptic when its rational homotopy and cohomology groups a…

Loughran–Smeets conjecture. Under some mild assumptions, the upper bound of Loughran and Smeets for $N_{\mathrm{loc}}(\pi,B)$ is sharp.

Let $f:S\to B$ be a fibration of a smooth projective surface over a smooth projective curve, with general fibre of genus $g$, and let $q_f:=q(S)-b$ denote its relative irregularity…

Let a Fano fibration be a fibration whose fibres are Fano varieties, possibly with singular fibres in a degeneration. Fano-fibration degeneration conjecture. A Fano fibration is po…

Log Iitaka conjecture. One has

Let $f:S\to\mathbb P^1$ be a nontrivial fibration of semistable curves of genus $g\geq 1$, and let $s$ be the number of its singular fibers. Beauville's conjecture. If $g\geq 2$, t…

Loughran–Rome–Sofos conjecture. There exists a constant $c(\pi)>0$ such that, as $B\to+\infty$,

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

Let $f:S\to B$ be a locally non-trivial fibration, and write $\omega_{S/B}$ for its relative canonical sheaf and $\lambda_f$ for its slope. A locally free quotient of a vector bund…

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-…

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…

A Morse parameterization is the simultaneous continuous parameterization of the members of a family of manifolds, generalizing Whitney's continuous choice of a point in curves. Shc…

Let $M$ be a $G_2$-manifold and consider calibrated fibrations on $M$, analogous to special Lagrangian or holomorphic fibrations on a Calabi–Yau threefold. Conjectural characteriza…

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…

Let $X$ be a special compact complex manifold in Fujiki's class $\mathcal C$. A special manifold should be built from manifolds that are either rationally connected or have Kodaira…

Let $F$ be the stated $\mathrm U(1)$-invariant model for a codimension-two singularity in a special Lagrangian fibration, in which two codimension-one $T^2$-cones come together and…

Let $F:\mathbb C^3\to\mathbb R\times\mathbb C$ be the piecewise-smooth special Lagrangian fibration whose fibres are $T^2$-cones at $a=0$ and nonsingular $S^1\times\mathbb R^2$ for…

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.

Existence-of-fibrations conjecture. There exist a resolution of singularities

Weak saturation conjecture. The complex polarized variety $V({\bf C})$ admits an ${\cal L}$-primitive fibration having infinitely many fibers $W$ defined over $F$ which form an asy…

Let $V$ be a variety over $\mathbb{C}$, let $B$ be an effective $\mathbb{Q}$-Weil divisor with coefficients at most $1$ such that $(V,B)$ is log terminal, and let $L$ be a big…

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…

Fibrational internal-category conjecture. A fibrational generalization of the notion of internal category should be suitable to provide a generalization of the equivalence between…