50 problems

Colliot-Thélène's conjecture. Weak approximation with Brauer–Manin obstruction is the only obstruction to weak approximation for smooth rationally connected varieties over number f…

Let $B$ be a smooth affine connected curve over a field of characteristic $0$, and let $X_B$ be a $B$-flat projective variety whose geometric generic fiber…

Batyrev–Manin conjecture. It is conjectured that this number grows as

Let $X$ be a projective manifold that is rationally connected, meaning that any two points of $X$ can be joined by a rational curve. Let Oka-1 denote the property that, for every o…

Let $X$ be a rationally connected manifold, and let $\mathcal{F}\subset T_X$ be a regular foliation. Touzet's conjecture. The leaves of $\mathcal{F}$ are algebraic. This conjecture…

Let $k$ be a $C_{1}$ field, meaning that every hypersurface of degree at most $n$ in $\mathbb{P}^{n}_{k}$ has a $k$-rational point. A variety is separably rationally connected if t…

Let $X$ be a projective rationally connected variety of dimension $d$ with $\epsilon$-lc singularities, where $\epsilon>0$, and suppose that $-K_X$ is nef. McKernan–Prokhorov's bou…

Let $X$ be a rationally connected projective variety over an algebraically closed field of characteristic zero, and let $\mathcal F$ be a regular foliation on $X$. Algebraic integr…

Let $X$ be a smooth projective separably rationally connected variety of dimension $d$ over an algebraically closed field. The groups … and, when $X$ is defined over the complex nu…

Let $X$ be a projective manifold with split tangent bundle … A subbundle $V\subset T_X$ is integrable if it is closed under the Lie bracket of local sections. Integrability conject…

Let ${\mathbb P}$ be a projective space, let $W_{\mathbb P}\subset{mathbb P}$ be a sufficiently general hypersurface, and let $Y_{\mathbb P}$ be the double space branched over…

Let $X$ be a smooth complex projective variety. The numerical characterization conjecture. $X$ is rationally connected if and only if … for every $m\geq 1$. The “only if” direction…

Let $X$ be a smooth projective rationally connected variety over the function field of a curve. Let $B$ be the base curve and let $\pi:\mathcal X\rightarrow B$ be a regular model o…

Let $(X,\omega)$ be a rational algebraic manifold. Suppose there exists a surjective morphism $\pi:X\to\mathbb{CP}^n$ such that $\pi|_{X\setminus S}$ is one-to-one for some subvari…

Batyrev–Manin counting conjecture. There exist $r\in\mathbb{N}$, $0<\beta\in\mathbb{Q}$, and a number field $F'\supset F$ such that

Loginov's conjecture. The third case does not produce any further groups: every such group is of product type or of K3 type.

Let $X$ be a rationally connected smooth complex variety, and let $\mathcal F$ be a regular foliation on $X$. The known conclusion for regular foliations on weak Fano complex varie…

Let $X$ be a projective manifold with a splitting … Assume that $X$ is rationally connected. Integrability conjecture. The subbundles $V_1$ and $V_2$ are integrable. The conjecture…

A surjective endomorphism of a projective algebraic variety $X$ is a morphism $f\colon X\to X$. It is int-amplified if there exists an ample Cartier divisor $A$ on $X$ such that…

A complex manifold $Y$ is an Oka-1 manifold if it enjoys the Oka property with approximation and jet interpolation for maps $X\to Y$ from any open Riemann surface $X$. A projective…

Touzet's fibration conjecture. The foliation $\mathcal{F}$ is induced by a fibration $X\rightarrow Y$ onto a projective manifold.

Meng–Zhang's conjecture. $X$ is a toric variety.

Let $R$ be a DVR with fraction field $K$ and residue field $k$. Let $X$ be a flat, projective $R$-scheme such that the generic fiber $X_K$ is smooth and separably rationally connec…

Colliot-Thélène–Sansuc analogue. The surface $X$ verifies (BM).

Let $X$ be as in the cited theorem of Kollár--Tian, with generic fiber $X_K$ and integer $n$ as in that theorem. Write $C\!H_2(X)$ and $C\!H_1(X_K)$ for the relevant Chow groups, a…