13 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…

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 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 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…

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:

Iskovskikh's stability conjecture. The tangent bundle $\Theta_X$ is (semi)stable.

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

Generically nef tangent bundle conjecture.