13 problems

Douglas–Reinbacher–Yau conjecture. There exists a rank-$r$ stable reflexive sheaf $\mathcal E$ on $X$ with respect to some polarization such that $c_1(\mathcal E)=0$,…

Existence conjecture. There exists a stable reflexive sheaf $V$, with respect to some ample class, such that

Let Setup be fixed, let $v \colon P \to \mathbb{R}_{>0}$ be a weight, and let $\mathcal{E}$ be a reflexive sheaf of rank $r$ that is $v$-stable. Bogomolov–Gieseker conjecture. The…

Let $X$ be a normal compact Kähler threefold with only isolated non-quotient singularities, let $b1$ be a Kähler class on $X$, and let ${\mathcal F}$ be an $b1$-stable non-zero tor…

Götsche–Kool conjecture. The Segre numbers of moduli spaces of stable sheaves on surfaces are related to the Verlinde numbers of these moduli spaces by an explicit correspondence.

Let $X$ be the local Calabi–Yau threefold associated with the surface $S$, let $v=(\beta,n)\in N_{\leq 1}(S)$, and let $\sigma_1=B_1+iH_1$ and $\sigma_2=B_2+iH_2$ be stability cond…

Let $X$ be a smooth projective Calabi–Yau $4$-fold, let $\beta\in H_2(X,\mathbb{Z})$ be an irreducible curve class, and let $\gamma_1,\ldots,\gamma_l$ be integral cohomology classe…

Let $S$ be a del Pezzo surface with a smooth anticanonical divisor, and let $\beta$ be a curve class. The higher genus relative Gromov–Witten invariants of $S$ relative to the divi…

Let $S$ be a K3 surface, let $v=(r,\xi,a)\in H^{2*}(S,\mathbb{Z})$ be a Mukai vector, and let $\omega$ be a $v$-generic Kähler class satisfying the hypotheses under which the modul…

Strong Bogomolov–Gieseker conjecture. One has

Stable-sheaf inequality conjecture. Then

Let $X$ be an abelian surface with $\operatorname{NS}(X)=\mathbb{Z} H$ and let $v$ be a Mukai vector with … Suppose that $v$ has at least one solution of the numerical equation. Am…

Let $\pi:S\to\mathbb{P}^1$ be a complex elliptic K3 surface with a section $\sigma$, let $f$ be its generic fibre, and let $M(t)$ denote the moduli space of rank $2$ stable sheaves…