120 problems

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…

Let $\bar{M}$ be the curve under consideration, and let $\cal{L}$ and $\cal{M}$ be two bundles on $\bar{M}$ given by gluing conditions. Log-convexity conjecture. If both…

Let $A$ be an abelian surface, and let $F$ be a simple vector bundle on $A$ with Mukai vector $v:=v(F)$. Suppose that $c_1(F)$ is symmetric. For a positive integer $d$, define … wh…

Orthogonal splitting type dimension conjecture. Its dimension is

Bruzzo's conjecture. The following conditions are equivalent:

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 $d$ be a degree and let $\vec{e}=(e_1,\ldots,e_{d-1})$ be a $(d-1)$-tuple of scrollar invariants. A cover is primitive if it does not factor through a nontrivial proper subcove…

A BN-triple is a triple $(g,r,d)$ specifying a Brill–Noether curve of genus $g$, degree $d$, and dimension $r$. For a general BN-curve $X\subset\mathbb{P}^r$ of type $(g,r,d)$, con…

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…

Hard Lefschetz and Hodge–Riemann conjecture. The class

Semistability–cohomological semistability conjecture. $E$ is semistable if and only if $E$ is cohomologically semistable.

Higher-rank Mistretta–Stoppino conjecture. The following assertions hold: (i) $E$ is linearly (semi)stable; and (ii) $E$ is linearly (semi)stable if and only if $M_E$ is (semi)stab…

The Tschirnhausen realization conjecture. The tuple $(e_1,\dots,e_{d-1})$ is achieved as the scrollar invariants of a smooth irreducible primitive cover $C \to \mathbb{P}^1$ of deg…