36 problems

Let $\mathcal{C}$ be a braided generalized near-group fusion category. Write $R(m,n)$ for the corresponding generalized near-group fusion ring, with $m,n\in\mathbb{Z}_{\geq 1}$, an…

Let $k$ be a field, let $X$ be a variety over $k$, and let ${\mathcal M}_k$ denote the relevant localization of the Grothendieck ring of varieties. Let $Z_{\rm mot}$ be the motivic…

Power-structure compatibility conjecture. For every such $k$, $X$, and $n$,

Let $X$ be a smooth variety over $k$, and let $m\geq 1$. The symmetric power $\operatorname{Sym}^m X=X^m/S_m$ is the quotient by the symmetric group. A variety has $\mathbb{D}$-sin…

Let $K_0(\operatorname{Var}_k^{\dim})$ be the graded Grothendieck ring of varieties, equipped with the symmetric-power operations $\operatorname{Sym}^m$ and the involution…

Várilly-Alvarado and Vakil's density-function conjecture. The quantities $\rho^{\textnormal{geo}}(x)$ and $\rho^{\textnormal{gr}}(x)$ exist and are independent of the choice of…

Várilly-Alvarado and Vakil's Grothendieck-ring measure conjecture. For all $B$, the limit $\lim_{g\to\infty}\pi^{\textnormal{gr}}(B,g)$ converges to a measure…

L-equivalence conjecture for hyperkähler manifolds. If $X$ and $Y$ are $L$-equivalent, then their bounded derived categories of coherent sheaves are equivalent:

Kuznetsov–Shinder conjecture. If $X$ and $Y$ are $D$-equivalent, then they are $L$-equivalent:

Shang's character-formula conjecture. In the Grothendieck ring of $GL(V)$,

Let $\mathcal{A}$ be the infinite-rank cluster algebra and let … be the injective ring morphism described above. Let $\mathcal{O}^{\mathrm{sh,f}}_{\mathbb{Z}}$ be the full subcateg…

Let $k$ be a field of characteristic not $2$. Let $K_0(\operatorname{Var}_k)$ be the Grothendieck ring of $k$-varieties, let $\operatorname{GW}(k)$ be the Grothendieck–Witt ring, a…

Grothendieck-ring growth-dimension conjecture. The function

Let $\mathcal U$ be a geometrically connected and smooth ungraded variety. For $\in\mathbb Z_{\geq 0}^{r}$, let $\mathcal Z_{,n}$ be the locus of ordered $r$-tuples of effective ze…

Let $\mathcal U$ be a geometrically irreducible ungraded variety, and let $\mathcal X_n$ denote its space of effective zero-cycles of degree $n$. Write $\mathbb L$ for the Lefschet…

Let $k$ be a field of characteristic not dividing $6$, let $g$ be the genus, and let $\mathrm{Hur}_{3,g,k}$ be the degree-$3$ Hurwitz stack. Write $\mathbb L$ for the class of the…

Let $\mathcal{F}$ be a complex commutative Grothendieck ring with simultaneously diagonalized fusion matrices having eigenvalues $(\lambda_{i,j})$. Let $(c_j)$ be the eigenvalues o…

Let $S$ and $S'$ be complex projective K3 surfaces. Write $H^2(S,\mathbb{Q})\cong H^2(S',\mathbb{Q})$ for an isomorphism of rational Hodge structures,…

Let $X$ and $Y$ be smooth projective simply connected varieties. They are L-equivalent if $[X]-[Y]$ vanishes in the localization…

Let $X$ and $Y$ be smooth projective simply connected varieties. Write $\mathbb{D}(X)$ for the bounded derived category of coherent sheaves on $X$, and let…

Galkin–Kontsevich weak conjecture. If $\alpha$ is represented by a linear combination of varieties of dimension less than or equal to $2(d-2)$ and

Let $\mathcal{O}^{fin}_{{\mathcal B}^\perp}$ be the category of finite-type modules, and let $K_0(\mathcal{O}^{fin}_{{\mathcal B}^\perp})$ denote its completed Grothendieck ring. F…

Larsen–Lunts correction conjecture. If

Let $\mathcal{M}(p_+,p_-)$ be the singlet vertex operator algebra, and let a suitable quotient category of its singlet modules have Grothendieck ring $K_0$. The relations obtained…

Let $l\geq 2$, and let $\mathcal{C}_{\varepsilon^\mathbb{Z}}$ be the category whose Grothendieck ring is under consideration. Write $\mathcal G_l$ for a generalized cluster algebra…