37 problems

Let ${\mathfrak M}$ be the moduli space considered above, equipped with the 2-form $\wedge_{\mathfrak M}$. A 2-form is symplectic if it is closed and nondegenerate. Symplecticity c…

Geometric Langlands conjecture. If $E$ is an absolutely irreducible $n$-dimensional local system on $X$, then there exists a perverse sheaf ${\mathcal S}_E\in P(\Bun_n)$ that is a…

Let $S$ be a K3 surface, and let $M_S^H(c_1,c_2)$ denote the moduli space of sheaves on $S$ with Chern classes $(c_1,c_2)$, for a suitable polarization $H$. Assume that…

Support-space conjecture. The morphism $\Sigma\rightarrow S$ is a disjoint union of locally closed immersions.

Projective fine-moduli-space conjecture. The only fine moduli spaces that are projective are the moduli varieties of stable sheaves, when $r$, $c_1$, and $\chi$ are pairwise relati…

Let $S$ be a base scheme, let ${\cal X}/S$ be the relevant site over $S$, and let $G$ be an abelian affine presentable group sheaf on ${\cal X}/S$. Here, affine means that for ever…

Periodic Kazhdan–Lusztig stalk conjecture. The stalks of simple objects in the category of semiinfinite sheaves are computed by the periodic Kazhdan–Lusztig polynomials $p_{y,w}$.

Choi–van Garrel–Katz–Takahashi conjecture. The polynomial $P_{\beta}(y)$ is divisible by the Poincaré polynomial of $\mathbb P^{w-1}$, and

Let $\mathsf{A}$ be a manifold, and consider the transfinite construction of the Fréchet sheaf $\widetilde{\mathscr{O}}_{\!\mathsf{A}}$ of locally approximable continuous $CR$ func…

Let $M_{d,\chi}$ be the moduli space under consideration, and let $c_k(j)$ denote its tautological generators. The generators in the range $A^{*\leq d-2}(M_{d,\chi})$ are those lis…

Let $X$ be a $2$-dimensional simplicial complex, let ${\mathcal{F}}$ be a sheaf on $X$, and use the notation of the modification process. Set … Assume $M\geq 0$. First-step cup-pro…

Let $X$ be a $2$-dimensional simplicial complex, let ${\mathcal{F}}$ be a sheaf on $X$, and use the notation of the modification process, including the subspaces $E_r$, $E'_r$,…

Let $d,q\in\mathbb{N}$ with $d\geq 2$, and let $f:\mathbb{N}\cup\{0\}\to\mathbb{N}$. Let $X$ be covered by a $q$-thick affine building of dimension $d$, and let ${\mathcal{F}}$ be…

Let $X$ be a simplicial complex and let ${\mathcal{F}}$ be a sheaf on $X$. Consider the linear map … Assume that this map is injective and that…

Let $X$ be a sheaved high-dimensional expander and let the modification process alter its sheaf by successively eliminating excess cohomology classes. Say that the process terminat…

Let $M_{d,\chi}$ be the moduli space under consideration, and let the tautological generators in $A^*(M_{d,\chi})$ be those defined in. A relation among these generators has degree…

Let $(r,c,d)\in\mathbb{Z}\oplus\mathbb{Z}\oplus\tfrac{1}{2}\mathbb{Z}$, and suppose that there are Gieseker-semistable sheaves $E$ on $\mathbb{P}^3$ with … Write $E(r,c,d)$ for the…

Let $G$ be a smooth algebraic group and let $X$ be a left $G$-variety over $\operatorname{Spec} K$. Let $\mathsf{Ner}(\operatorname{\mathbf{SfResl}}_G(X)^{\operatorname{op}})$ be t…

Let $X=D53F^1\timesD53F^1$ and let $M_{H_m}({\bf v})$ be the moduli space associated with a character ${\bf v}=(r,\nu,\Delta)\in K(X)$, where $r\geq 2$ and $\Delta\geq\frac{1}{2}$.…

Let $X$ be a projective complex surface with geometric genus $p_g=0$, and let $\mathcal A=\operatorname{Coh}(X)$. For an ample line bundle on $X$, let $\tau$ denote the resulting G…

Let $v\in K_{\mathrm{num}}(\mathbb{P}^3)$ be a one-dimensional class. For each real number $t$, let $\sigma_t=(\mathscr{A}_t,Z_t)$ denote the Euler stability condition on…

Let $M_{(0,d,\chi)}$ be the moduli space of Gieseker semistable sheaves on $\mathbb P^2$ with Chern character $(0,d,\chi)$, and write its intersection Betti numbers as…

Let $M(r,c,d,e)$ denote the moduli space of Gieseker-semistable sheaves on ${\mathbb{P}}^3$ with Chern character $(r,c,d,e)$. Let $r \in {\mathbb{Z}}_{\geq 0}$,…

Let $P$ be a poset, let $j:P\hookrightarrow \operatorname{Op}(P)^{\operatorname{op}}$ be the inclusion, and let…

Intersection conjecture. One also has