25 problems

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…

Colliot-Thélène's conjecture. Let $X$ be a rationally connected smooth variety over $k$. Then $X(k)$ is dense in $X(k_{\Omega})^{\operatorname{Br}_{\mathrm{nr}}}$.

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…

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…

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

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…

Let $X$ be a rationally connected variety of dimension $3$, and let $G\subset \mathrm{Bir}(X)$ be a finite abelian group. By the third type, we mean the case in which $G$ faithfull…

Let $K$ be a henselian defectless valued field with divisible value group and residue field $k$. A field has the separably rationally connected rational-point property if every sep…

Let $K$ be a tame valued field with divisible value group and residue field $k$. Let $V$ be a separably rationally connected variety over $K$, and set $F=K(V)$. Valuation conjectur…

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…

A complex manifold $X$ is rationally connected if every pair of points in $X$ can be connected by a rational curve $\mathbb{CP}^1\to X$. A compact Kähler manifold is a compact comp…

Let $X$ be a smooth projective variety over a finite field, and let $d=\dim X$. Say that $X$ is separably rationally connected in codimension $1$ if it admits a morphism…

Kollár–Szabó conjecture. The group

Let $d$ be a positive integer. Consider all $d$-dimensional klt Calabi–Yau rationally connected projective varieties. Rationally connected Calabi–Yau boundedness conjecture modulo…

Let $d$ be a positive integer. Consider any $d$-dimensional klt Calabi–Yau rationally connected projective variety $X$. Rationally connected Calabi–Yau index conjecture. There exis…

Let $d$ be a positive integer. Consider the minimal log discrepancies of all normal projective Calabi–Yau rationally connected varieties of dimension $d$. Minimal-log-discrepancy n…

Let $X$ be a projective rationally connected manifold and let $\mathcal{F}$ be a regular foliation on $X$. Touzet's conjecture. The foliation $\mathcal{F}$ is induced by a smooth m…

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 geometrically rationally connected smooth projective variety of dimension $n$, let $L$ be a big and nef $\mathbb Q$-divisor on $X$, let … and let $\delta(X,L)$ be the…

Let $X$ be a smooth proper variety over a number field $k$. Let $\Omega_f$ be the set of finite places of $k$ and $\Omega_\infty$ the set of infinite places. For each infinite plac…

Hassett–Tschinkel's weak approximation conjecture. Proper rationally connected varieties defined over $F$ satisfy weak approximation.

Rationality–Cox ring conjecture. $E^1(X)$ is rational if and only if

Let $k$ be an algebraically closed field of characteristic zero, let $B$ be a curve over $k$, and let $k(B)$ be its function field. Let $X$ be a rationally connected variety over…