264 problems

Let $X$ be a smooth subvariety of $\mathbb{P}^n$ over $\mathbb{C}$. An Ulrich bundle on $X$ is a nonzero vector bundle satisfying the equivalent linear-resolution, cohomological-va…

Let $E$ be an ample vector bundle over a compact complex manifold $M$. A Hermitian metric on $E$ is Griffiths-positive when its curvature is positive in the Griffiths sense. Griffi…

Let $C$ be a Petri curve, let $L$ be a line bundle on $C$, and let $(L,V)$ be a linear system with $V\subset H^0(L)$; write $D_{L,V}$ for its dual span bundle. Butler's stability c…

Let $C$ be an integral curve of genus $g$, and for each positive integer $n$ define … Here ${\rm Cliff}_1(C)$ is the rank-one Clifford index. Mercat's conjecture. … The conjecture…

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…

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 $X$ be a general canonical curve of genus $g\geq7$, embedded by its canonical linear series, and let $N_{X/\mathbb{P}^{g-1}}$ be its normal bundle. Aprodu–Farkas–Ortega's conje…

Let $C$ be a smooth projective curve of genus $g\geq 1$, and let $r$ and $d$ be coprime integers. Consider the moduli space of stable vector bundles on $C$ of rank $r$ and degree…

Let $k$ be a number field and let $X$ be a smooth projective variety over $k$. The cotangent bundle $\Omega^1_X$ is nef when it is a nef vector bundle. Nef-cotangent conjecture. If…

Let $X$ be a complex projective manifold and let $E$ be a holomorphic vector bundle on $X$ that admits a holomorphic connection. Atiyah conjecture. The bundle $E$ always admits a f…

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 $f\colon X\to Y$ be a finite morphism between smooth irreducible projective curves, and let $V$ be a general vector bundle on $X$. Beauville's conjecture. The pushforward…

Mistretta–Stoppino's conjecture. If

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 $k$ 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 $X\subset \mathbb{P}^{N}$ 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…

Let $E$ be a vector bundle on $\mathbb{P}^1\times\mathbb{P}^1$. It has natural cohomology if, for every pair of integers $(n,m)$, the cohomology of…

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

Let $(E,\theta)$ be a nilpotent semistable Higgs bundle of exponent at most $p$. Strong Higgs semistability conjecture. The Higgs bundle $(E,\theta)$ is strongly Higgs semistable.…

Classification conjecture. Every such bundle is one of the explicitly constructed $3$-Buchsbaum rank $2$ bundles that is not an instanton.

Let $C$ be a smooth curve, let ${\cal L}$ be a line bundle on $C$, and let $V\subseteq H^0(C,{\cal L})$ define a linear series. Write $M_{V,{\cal L}}$ for the dual span bundle asso…

Petri-type injectivity conjecture. The Petri map $\mu_N$ is injective.

Let ${\mathcal E}_B$ be one of the rank $6$ vector bundles on ${\mathbb P}^3$ constructed in the paper from a tensor $B$. The bundles have Chern polynomial $1+3t+6t^2+4t^3$ and min…

Generalized lower-bound conjecture. One should have

Let $X$ be a smooth projective curve over $\mathbb{F}_q$, let $n$ be a positive integer, and let $E$ be a smooth geometrically irreducible $\mathbb{\bar Q}_\ell$-sheaf of rank $n$…