41 problems

Strange duality conjecture. For any $g\geq6$, this map is an isomorphism.

Toda's chi-independence conjecture. For any $\gamma\in B_S(\alpha)$, the invariant $\Phi(\alpha+n\delta)$ is independent of $n\in\mathbb{Z}$.

Let $X$ be a smooth surface and let $Z_1,Z_2\subset X$ be possibly reducible curves such that $Z_1\cap Z_2$ is zero-dimensional. Write $\mathbf{HA}^T_{X,Z}$ for the $T$-equivariant…

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

Canonical-basis realization conjecture. The algebra

Let $X\to\mathbb P^n$ be an abelian fibration, let $P\to\mathbb P^n$ be its relative Picard, and let $X_0$ be the relative Picard of $P$. Let…

Let $X$ be a smooth projective curve, let $\widehat{\mathfrak{U}}_{\mathbb{A}}$ be the completed convolution algebra, and let $\mathfrak{U}_{\mathbb{A}}^{\operatorname{tor}}$ be th…

Let $X$ be a higher-dimensional hyperkähler manifold, let $\mathcal{F}$ be a simple torsion-free sheaf on $X$, and write $\operatorname{Def}(\mathcal{F})$ for its deformation space…

Let $\mathcal{F}$ be a codimension-one foliation on $\mathbb{P}^3$ with tangent sheaf $T_{\mathcal{F}}$. Calvo-Andrade–Correa–Jardim conjecture. If $T_{\mathcal{F}}$ is not split,…

Let $X$ be the Fano threefold under consideration, let $E$ be a general Gieseker semistable rank two sheaf on $X$ with sufficiently large $c_2(E)$, and let $\mathcal U$ denote the…

Kononov–Moreira–Lim–Pi conjecture. The associated graded group

Let $Z=Z_1\cup\cdots\cup Z_s$ be a union of curves, let $\langle Z\rangle$ be the relevant semigroup of classes, and let $\overline{\mathfrak a}_\gamma$ be the amalgamation map in…

Let $X$ be a smooth surface, let $Z_1$ and $Z_2$ be curves with $Z=Z_1\cup Z_2$, and let $\mathbf{HA}^T_{X,Z}$ denote the corresponding $T$-equivariant cohomological Hall algebra.…

Virasoro conjecture. For every $k\geq -1$ and every $D\in\mathbb D^S$,

Let $M_{r,\chi}$ be the moduli space of one-dimensional stable sheaves on $\mathbb{P}^2$ with support in the class $[rH]$ and Euler characteristic $\chi$, where $H$ is the hyperpla…

Fix coprime integers $r,\chi$ with $r\geq 3$. Let $M_{r,\chi}$ be the moduli space of one-dimensional stable sheaves $F$ on $\mathbb P^2$ with $[\operatorname{supp}(F)]=rH$ and…

Le Potier non-continuity conjecture. The function $\Phi_{S,H}$ is not continuous at $0$.

Let $X$ be a smooth projective variety, let $M=M_\alpha$ be a moduli space of sheaves of topological type $\alpha$, and let $\mathbb{D}^X$ be the descendent algebra. Choose an alge…

Let $X$ be a smooth projective variety, let $M$ be a moduli space of sheaves on $X$ with virtual fundamental class $[M]^{\mathrm{vir}}$, and let $\mathbb{D}^X$ be the descendent al…

Assume the hypotheses of Maulik and Toda hold in this setting, so that higher-genus Gopakumar–Vafa invariants $n^g_\beta$ can be defined from the Hilbert–Chow morphism and perverse…

Let $S$ be a Gorenstein surface with ample anticanonical sheaf, let $D$ be a polarization of $S$, and let $n^D_\gamma$ be the local BPS invariant associated with a curve class…

Let $C$ be an extended ADE curve with type $\tilde D_n$, $\tilde E_6$, $\tilde E_7$, or $\tilde E_8$, and write … Let $J$ be the Jacobian and let $\mathcal M$ be the moduli space o…

Restriction conjecture. The restriction $E|_D$ remains $\ell$-(semi)stable for $a\gg0$.

BFMS conjecture. A generalization of this vanishing equation should remain valid for higher-dimensional constant cycle varieties. The stated vanishing for point classes follows fro…

Let $\Phi$ be the multiplicative class specified in Examples 3 and 3, and let $Z_a$ be the power series defined by the virtual $\chi_y$-genus blowup relation. For the generating se…