45 problems

Let $\mathbb{V}$ be a variation of Hodge structure on a regular Zariski open subset of a projective variety $X$, and let $S(IC_X(\mathbb{V}))$ denote Kollár's coherent S-sheaf asso…

Soit $X\subset \mathbb{P}^r_k$ un sous-schéma fermé du schéma projectif qui soit localement une intersection complète et dont toute composante irréductible soit de codimension…

Let $C_2$ be a double curve, let ${\mathcal E}$ be a coherent quasi-locally free sheaf on $C_2$, and let ${\mathcal F}$ be an order-two deformation of ${\mathcal E}$ corresponding…

Let $G$ be a connected reductive group, let $\check G$ be its Langlands dual group, and let $\Lambda_G$ be the union of conormal bundles to the $G({\mathbf O})$-orbits in the affin…

Let $X$ be the scheme under consideration, and let … be the heart of the $t$-structure on the bounded quasi-coherent derived category. Let … be its coherent subcategory, and let…

Sheaf upper-bound conjecture. Then

Let $\mathbb{X}_{\mathbf{p},\underline{\lambda}}$ be a weighted projective line, let $\mathfrak{Lg}$ be its corresponding loop Kac–Moody algebra, and use the identification…

Lagrangian–sheaf conjecture. Fundamentally, mirror symmetry is a relation between the Lagrangian submanifolds of a threefold and the semistable coherent sheaves on its mirror.

Let $X$ be a separated scheme over $\mathbb{C}$ of finite type and let $A$ be an abelian group. The semi-topological Quillen-Lichtenbaum conjecture for coherent sheaves. The canoni…

Faithfulness conjecture. For any $\alpha\in({\mathbb Z}^2)^+$, the polynomial representation $P^{\frac12}_\alpha$ of $A^{\frac12}_\alpha$ is faithful.

Cohomological boundedness conjecture. The coherent Springer sheaf $\mathcal{S}$ and each specialization $\mathcal{S}_v$ are honest coherent sheaves; equivalently, their cohomology…

Let $G$ be a reductive group, let $\check{G}$ be its Langlands dual group, and let $\mathrm{KP}_{\mathrm{coh}}^{\widehat{G}_{\mathcal{O}}}(\mathcal{R}_{G,\mathfrak{g}})$ be the Kos…

Let $G$ be a reductive group, let $\check{G}$ be its Langlands dual group, and define…

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…

Let $\mathbb{E}$ be either $\overline{\mathbb{Q}}_\ell$ or $\overline{\mathbb{F}}_\ell$, and regard the indicated compactly induced representations as sheaves on…

Let $I$ be the subset indexing the subalgebra $\bPi_I^{\bullet}$, let $j^*$ and $j_!$ be the associated restriction and extension functors, and let $\tau^*$ be the induced functor…

Let $X$ be a two-sided Hecke pattern of rank $r$. Write $H_*(X)$ for its Borel–Moore homology, and let $W_{\star,\star}(S)$ denote the algebra with the appropriate choices of indic…

For a regular semisimple $gamma$, define the graded module … It is a module over the graded algebra $oplus_{d=0}^{cinfty}{}_0\mathcal A_d$ and therefore defines a quasi-coherent sh…

The construction associates to each $gammabin g(cK)$ a {cC}^-equivariant quasi-coherent sheaf $cFgamma$ on $cmathfrak C{ccheck G}$. Coherence conjecture. For any regular semi…

Let $f:Y\to X$ be a proper morphism between smooth projective algebraic varieties. Assume that the degenerate locus of $f$ is contained in a normal crossing divisor $D\subset X$. L…

Let $K$ be a nonarchimedean local field, let $d$ be a positive integer, and let $]_{ll]}$ be the coefficient ring. Let $L]_d(K)$-representations mean representations on $]_{ll]}$-m…

Let $G$ be the reductive group under consideration, let $\widetilde{G}$ denote its Grothendieck resolution, and let ${\mathcal N}$ be the nilpotent cone in its Lie algebra. Write…

Formal K-flatness conjecture. Formal K-flatness is equivalent to K-flatness.

Let $X$ be a ribbon of genus $g$ with reduced curve $X_{\mathrm{red}}$ of genus $\bar{g}\ge 2$, let $\delta=-\deg({\mathscr N})$, where ${\mathscr N}$ is the conormal sheaf, and le…

Let $X$ be a ribbon, let $\delta=-\deg({\mathscr N})$, and let $\bar{g}$ be the genus of $X_{\mathrm{red}}$. The large-nilradical component conjecture. If … then the only irreducib…