20 problems

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…

Minimal-rank conjecture. Every uniform vector bundle $\mathcal{F}$ on $\mathbb{P}^n$ of rank $<2n$ is homogeneous.

Linear subvariety conjecture. If $Y$ is $f^{-1}$-invariant or $f^{-1}$-periodic, then $Y$ is a linear subspace of $X$.

Let $M_{n,c}$ be the statement that, for a globally generated vector bundle $E$ on $\mathbb{P}^n$ with first Chern class at most $\min\{c,n-1\}$ and $H^0(E^*)=H^1(E^*)=0$, one of t…

Let $X$ be a smooth compactification of $\mathbb{C}^n$ with $b_2(X)=1$, and suppose that $X$ is Kähler and its divisor at infinity is smooth. Folklore conjecture. Then $X$ must be…

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…

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…

Let $M$ be a hyperkähler manifold and let … be a fibration with no multiple fibers. For a general curve $C\subset\mathbb{C}P^n$, a continuous section is a continuous map…

Let $\mathbb{RP}^n$ be real projective space equipped with a metric, and consider the systolic inequality relating its volume and systole. Extremal metric conjecture. It is conject…

Let $N_{0,d}(p)$ be the number of degree-$d$ rational curves through $2d-p$ points and $2p$ lines in general position in $\mathbb P^3$. The projective three-space growth conjecture…

Let $f$ be an endomorphism of the $k$-analytic projective space, and let $F_{\mathrm{norm}}(f)$ and $F_{\mathrm{harm}}(f)$ denote its normal and harmonic Fatou sets, respectively.…

For curves in $\mathbb{P}^3$, the mobility count measures the largest number of general points that a curve can pass through relative to its degree raised to the power $3/2$. Perri…

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 $\gamma$ be an integral cohomology table satisfying the numerical condition of Corollary and suppose that some multiple of $\gamma$ is the cohomology table of a vector bundle.…