22 problems

Let $f_U:U\to V$ be a smooth projective family, and suppose that $V$ has a morphism $\mu:V\to P_h$, where $P_h$ is the coarse moduli scheme associated with the moduli functor of po…

Let $X$ be a compact Kähler manifold that is special, and let $p>0$. For every rank-one coherent subsheaf $L\subset\Omega_X^p$, the numerical cotangent bound conjecture. One has ……

Let $X$ be a projective manifold defined over a finitely generated field $K\subset\mathbb C$. Call $X$ arithmetically special when $X(K')$ is Zariski dense for some finite extensio…

Let $X\in\mathcal C$, and call $X$ $H$-special when its Kobayashi pseudometric satisfies $d_X\equiv 0$. Conjecture IIIH. The manifold $X$ is special if and only if it is $H$-specia…

Let $X$ be a special manifold of dimension $n\geq 2$. A manifold is strongly irreducible special when every covering family with special general member is trivial in the sense defi…

Let $X\in\mathcal C$ be smooth, and let $\operatorname{ess}(X)$ denote the specialness invariant introduced in the paper. Deformation conjecture. The invariant…

Let $f:X\dashrightarrow Y$ be a special fibration with $\kappa(Y,f)=-\infty$. Let $\Delta(f)$ be its base orbifold, and let $\kappa$-rationally generated mean that the general orbi…

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…

Weakly Special Conjecture. If $X$ is a weakly special variety over a number field $K$, then $X$ has a potentially dense set of integral points.

Lang–Vojta–Chevalley–Weil conjecture. If $X$ satisfies potential density of integral points, then $X$ is weakly special.

Nilpotency conjecture. The fundamental group $\pi_1(X)$ is virtually nilpotent.

Virtually nilpotent fundamental-group conjecture. The fundamental group $\pi_1(X)$ is virtually nilpotent.

Let $K$ be a finitely generated field and let $X$ be a smooth projective variety over $K$. The variety $X$ has potential WHP if $X_L$ has the weak Hilbert property for some finite…

Let $(X,\Delta)$ be a smooth proper orbifold pair. Call it Kobayashi-special when its Kobayashi pseudometric vanishes identically, Picard-special when it admits the specified dense…

Let $X$ be a smooth projective variety over a number field $K$. A variety is weakly special if no finite étale cover of $X_{\overline{K}}$ rationally dominates a positive-dimension…

Let $f:X\to Y$ be a smooth projective morphism with connected fibres between two quasi-projective manifolds. Assume that $K_{X_y}$ is pseudo-effective for general $y\in Y$, and tha…

Let $K$ be an algebraically closed, complete, non-Archimedean valued field of characteristic zero, and let $X/K$ be a smooth projective variety. Write $X^{\operatorname{an}}$ for i…

Let $X$ be a projective variety over … to $X^{\operatorname{an}}$ whose image is analytically dense. Campana's conjecture. $X$ has a dense entire curve if and only if $X$ is specia…

A log pair $(X,D)$ is special if $\kappa(L)<p$ for every $p$ and every invertible sheaf $L\subset\Omega^p_X(\log D)$. A quasi-projective manifold is special if it admits a smooth p…

Let $X$ be a projective manifold, let $X_1:=\mathbb P(T_X)$ be the projectivized tangent bundle, and for every entire curve $f:\mathbb C\to X$ let $f_{[1]}:\mathbb C\to X_1$ be its…

Let $X$ be a smooth complex algebraic variety and $r$ a natural number. Let $\mathcal{M}_B(X,r)$ be the moduli space of semisimple complex local systems of rank $r$. An absolute…

Let $X$ be a smooth complex algebraic variety. Let $\M_B(X)$ denote the moduli space of semisimple local systems of all ranks, and call a subset of its complex points absolute when…