12 problems

Positive Chern character conjecture. If ${\rm ch}_k(X)$ is positive for every $k\le n$, then

Let $n$ be a positive integer, let $\mathbb{P}^n$ be complex projective space, and let $L^{2j}(\mathbb{P}^n)\in H^{4j}(\mathbb{P}^n,\mathbb{Q})$ denote its degree-$4j$ cohomology…

Hartshorne's unstable rank-two bundle conjecture. There exists $n_0'\geq 5$ such that any unstable rank $2$ vector bundle $E$ on$\mathbb{P}^n$, with $n\geq n_0'$, is decomposable.

Let $0\le s\le k$ be an integer. Let $f:\mathbb{P}^k\to\mathbb{P}^k$ be a holomorphic endomorphism of degree $d\geq 2$, and let $T$ be its Green $(1,1)$-current. For an analytic su…

Let $X$ be a Fano manifold of Picard number $1$, meaning that $-K_X$ is ample and the Picard number of $X$ is $1$. Projective-space conjecture. If $X$ admits a non-isomorphic surje…

Let $X$ be a smooth projective variety, and let $f:X\to\mathbb{P}^n$ be a surjective morphism. Let $\Delta(f)$ denote the discriminant locus of $f$. Higher-dimensional uniruledness…

Let $X$ be an $n$-dimensional smooth $l$-Fano variety, meaning that $X$ is smooth and Fano and its Chern characters $\text{ch}_i(X)$ are positive for every…

Let $X:=\mathbb{P}^n$, let $f:X\to X$ be an endomorphism with $\deg(f)\geq 2$, and let $Y$ be a closed subvariety of $X$. The linear-subspace conjecture. If $Y$ is $f^{-1}$-invaria…

Let $X$ be a smooth Fano variety of Picard number one, meaning that $-K_X$ is ample and $\rho(X)=1$. Suppose that $X$ admits a non-isomorphic surjective endomorphism $f$. The proje…

Let $X$ be a smooth Fano variety of dimension $n$ over an algebraically closed field, and let $C\subseteq X$ be a rational curve. Mori–Mukai's characterization conjecture. If … for…

Cohomological criterion for syzygy bundles. The two cohomology groups are isomorphic if and only if they are both zero; consequently, the syzygy bundles $F_i$ are Steiner if and on…

Let $E$) be a globally generated vector bundle on $\mathbb{P}^n$, with $n\geq 2$ and $1\leq c_1(E)\leq n-1$, such that … Write $P(B)$ for the bundle associated with a direct sum…