122 problems

For every smooth complex projective variety $X$ and every slope-stable ample holomorphic vector bundle $E\to X$, does there exist a smooth strongly pseudoconvex Finsler metric on…

For every projective variety $X\subseteq\mathbb{P}^N$ over a field, equipped with its given embedding, does there exist a nonzero coherent sheaf $\mathcal{E}$ on $X$ that is Ulrich…

Let $(E,h)\to X$ be a Griffiths semipositive Hermitian holomorphic vector bundle. Let $P$ be a non-negative combination of Schur polynomials, and let $P\big(c_{\bullet}(E,h)\big)$…

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 $(E,V)$ be a generated coherent system of type $(n,d,v)$ on a general curve $X$ of genus $g\ge3$, where $n,d,v$ are positive integers and $v>n$. Write $D_{E,V}$ for its dual sp…

Let $X$ be a smooth projective variety, and let $A$ and $P$ be line bundles on $X$. Assume that $A$ is very ample, and set $L_d:=dA+P$ for every positive integer $d$. Ein–Lazarsfel…

Let $C$ be a general canonical curve of genus $g\geq 7$, embedded by its canonical linear system in ${\mathbf P}^{g-1}$, and let $N_{C/{\mathbf P}^{g-1}}$ denote its normal bundle.…

Let $k$ be an algebraically closed field, and let $X$ be a smooth affine $k$-scheme of dimension $d\geq2$. Suslin's cancellation conjecture. If $\mathcal{E}$ and $\mathcal{E}'$ are…

Let $C$ be a curve of genus $g$, and let $E$ be a semistable vector bundle on $C$ of rank $n$ and degree $d$. Write $\mu(E)=d/n$, and let $\gamma_1$ denote the Clifford index of…

Let $X\subset \mathbb{P}^{N}$ be a smooth projective variety, and call a vector bundle on $X$ Ulrich when it satisfies the Ulrich conditions for the given embedding. Ulrich's conje…

Set $\rho_{g,k}:=3g-3-\binom{k+1}{2}$. Let $C$ be a general curve of genus $g\geq 2$, and consider the moduli space of stable rank-two vector bundles on $C$ with canonical determin…

Degree-independence conjecture. For every integer $d$, the sum

Let $E$ be an object carrying a $Z$-critical equation, and let $h$ be a metric on $E$. A strong subsolution is a metric satisfying the positivity conditions on all formal matrix de…

Let $E\to(X,\omega)$ be a Hermitian vector bundle, and let $h$ be a Hermitian metric on $E$. A strong subsolution for the $Z$-critical equation is a metric satisfying … as an…

Let $S=mathbbm{k}[x_1,\ldots,x_n]$ and $A=mathbbm{k}[t]$. Let $\operatorname{D^b_{DM}}(S)$ and $\operatorname{D^b_{DM}}(A)$ denote the relevant bounded derived categories of differe…

Let $k\geq 2$, and consider the complex … Assume that $\beta$ is surjective and that $\alpha$ degenerates in codimension $2$; equivalently, the cokernel in degree $0$ is…

Radicality conjecture. The ideal above is radical. This stronger assertion includes cases involving certain unions of splitting-type strata. The source records the case $r=2$ as as…

Fulton–Lazarsfeld's connectivity conjecture. If

Let $E$ be a rank $r\geq 2n$ locally-free instanton sheaf of trivial splitting type on $\mathbb{P}^{2n}$ and charge $c$. Lifting conjecture. There is a rank $r$ locally-free instan…

Let $K_{\sigma}$ be the intermediate variety in the factorization of the morphism from the Kirwan desingularization $K$ to the desingularization $S$. Narasimhan–Ramanan desingulari…

Let $X$ be a smooth projective surface, let $C\subset X$ be the framing divisor, and let $\pi:\tilde X\to X$ be the blowup of a point $x_0\notin C$. Let…

Let $X$ be a fixed smooth projective variety and let $n$ be a positive integer. For a commutative differential graded algebra $A$, let $\mathbb{R}\underline{Vect}_{n}(X)(A)$ be the…

Let $\mathbb{P}^N$ be a projective space of dimension at least six, and let $\mathcal{E}$ be a vector bundle of rank two on it. The rank two bundle conjecture. The bundle…

Poisson vector bundle conjecture. (1) Every homogeneous Hermitian Poisson vector bundle over ${\mathfrak g}^*$ is Poisson-trivial. (2) There is a one-to-one correspondence between…

Let $X$ be the genus-two curve in characteristic $2$, let $X_1$ be its Frobenius twist, let $F\colon X\to X_1$ be the relative Frobenius morphism, and let $\mathrm{M}_X$ and…