20 problems

Let $X$ be a projective manifold, meaning a smooth projective variety, whose tangent bundle $T_X$ is nef. Campana–Peternell conjecture. The variety $X$ should be homogeneous. This…

Let $X$ be a projective manifold with nef anticanonical bundle $-K_X$. For ample divisors $H_1,\ldots,H_{n-1}$, define generic $(H_1,\ldots,H_{n-1})$-semipositivity of $T_X$ by req…

Let $X$ be a Fano manifold with Picard rank one. The tangent bundle $T_X$ is said to be $(-K_X)$-slope stable if every subsheaf has strictly smaller $(-K_X)$-slope than $T_X$. Isko…

Let $X$ be a smooth complex projective $n$-dimensional variety, let $\mathcal{F}$ be an ample vector bundle on $X$ that is a subsheaf of $\wedge^pT_X$, and let $p>0$. Assume that…

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…

Borisenko's conjecture. The zero vector field is the unique vector field whose image $\xi(M^n)$ is a totally geodesic submanifold of $TM^n$; equivalently, the base manifold is the…

Dual-number manifold conjecture. In analogy with the Newlander–Nirenberg theorem, $TM$ should already be a manifold defined over the ring

Main Conjecture. The subset of points $x$ in $J(n,m,k)$ such that the map $\Psi_k^0$ is non-transversal to the stratum containing the point $\Psi_k^0(x)\in Mat^0(n,m,k)$ has codime…

Foliation splitting conjecture. There exists a foliation $\mathcal{G}$ on $\mathsf{X}$ such that

Let $X$ be a Fano manifold of Picard number $1$ with dimension at least $3$. The normalized tangent bundle of $X$ is pseudoeffective if and only if $X$ is one of the following vari…

Let $X$ be a compact Kähler manifold, and let $\alpha_1,\ldots,\alpha_{n-1}$ be Kähler classes. For a coherent sheaf, generically nef means generically nef with respect to the indi…

Anticanonical-degree conjecture. There is a constant $C_0$ depending only on the Picard number $\rho(X)$ such that $T_X$ is big if and only if

Let $X$ be a Fano manifold with Picard number one. A rational curve $f:\mathbb P^1\to X$ is unbendable when $f^*TX$ decomposes as … for some nonnegative integers $p,q$. The unbenda…

Let $X$ be a Fano manifold with Picard number one. A vector bundle is simple if its only endomorphisms are scalar multiplications. The simplicity conjecture. The tangent bundle…

Let $X$ be a Fano manifold with Picard number one. The semistability conjecture. The tangent bundle $TX$ is semistable. This conjecture was disproved by Kanemitsu, who found smooth…

Let $X$ be a smooth projective variety and $A$ an ample divisor on $X$. Let … where $\xi=c_1(\mathcal{O}_{\mathbb{P}(T_X)}(1))$ is the tautological class on $\mathbb{P}(T_X)$ and…

Exterior-power ampleness conjecture. The following assertions should hold:

RC-positive tangent bundle conjecture. If $X$ is rationally connected, then $T_X$ is RC-positive.

Li's conjecture. Every such variety $X$ is an abelian variety.

Generically nef tangent bundle conjecture.