19 problems

Virtual Vafa–Witten formula. $\overline{\chi}_{-y}^{\mathrm{vir}}(M)$ equals the coefficient of $x^{\operatorname{vd}(M)}$ in

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…

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

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

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 $D\in\mathbb D^S$,

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

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

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…

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

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…

For $d\in\mathbb Z_{>0}$ and $\chi\in\mathbb Z$, let $M_{d,\chi}$ be the moduli space of one-dimensional Gieseker semistable sheaves, and let…

Let $S$ be the smooth projective surface and let $\CM$ denote the relevant smooth moduli space of sheaves. For each $n\in\BN$, write $e_{(0,...,0)}$ for the functor associated with…

Finite Quot tautological Chern-class conjecture. One has

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

Let $Y$ be a bielliptic surface, let $f\colon X\to Y$ be its canonical covering of degree $n=2$, and let $w$ be a sheaf class on $Y$. Write $K_{X;f^*H,f^*A}(f^*w)$ and…

Let $\Sigma_{g,\ell}$ be a ruled surface over a Riemann surface $C_g$ of genus $g>0$, with fibre $f$. Let $r\geq 1$, $c_1$ be a first Chern class, and let…

Let $\Sigma_\ell$ be a Hirzebruch surface with generic fibre $f$, let $r\geq 1$, and let $c_1\in H_2(\Sigma_\ell,\mathbb{Z})$. Write $H_{r,c_1}(z,\tau;f)$ for the generating functi…