78 problems

For a tuple of polynomials $f=(f_1,\ldots,f_r)$ and a monoid ideal $K\subseteq\mathbb{N}^r$, let $B_f^K\subseteq\mathbb{C}[s_1,\ldots,s_r]$ be the corresponding Bernstein–Sato idea…

Let $X$ be a smooth complex affine variety of dimension $n$, let $F=(f_1,\ldots,f_r)$ be a tuple of nonzero regular functions such that $f=\prod_{i=1}^r f_i$ is not invertible, and…

Let $B_F$ be the Bernstein–Sato ideal associated with the tuple $F$, and let $Z(B_F)\subseteq\mathbb C^r$ denote its zero locus. Budur's weak conjecture. Each irreducible component…

The exactness conjecture. The functor $\mathrm F_{G,\underline{\lambda}}$ is exact.

Let $\rho:\widetilde X\to X$ be a symplectic resolution, with $X$ affine, and let $\operatorname{\mathsf{HP}}_i^{DR}(X)$ denote the Poisson-de Rham homology of $X$. The affine symp…

Let $X$ be a complex manifold, let $\mathcal{D}_X$ be its sheaf of differential operators, and let $\mathcal{M}\in D^b_h(\mathcal{D}_X)$ be a bounded complex with holonomic cohomol…

Let $X$ be a complex variety, let $\mathcal{M}$ be a holonomic $\mathcal{D}$-module on $X$, and let $S(\mathcal{M})$ be the union, over germs of hypersurfaces $Z$ of $X$, of the se…

Let $X$ be a complex manifold of dimension $d$, and let ${\cal M}^\bullet$ be a complex of ${\cal D}_X$-modules with bounded good cohomology. Choose a filtration compatible with th…

Let $A$ be an integer matrix, let $\mathcal{M}_A(\beta)$ be the projectivized $A$-hypergeometric system with parameter $\beta$, and let $Y$ be a coordinate variety. For a filtratio…

Let $A$ be fixed, let $L$ be a filtration, let $\tau$ be the corresponding face or stratum, and let $\mu^{L,\tau}_{A,0}(\beta)$ denote the multiplicity of the associated component…

Let $\pi:{\mathfrak X}\to S[[h]]$ and the quantization sheaf $\operatorname{\cal O}_h$ satisfy the assumptions of Theorem. Consider the global sections functor … from sheaves of fi…

Let $A$ be an integer matrix, let $S_A$ be its semigroup ring, and let $\mathfrak m$ be its homogeneous maximal ideal. For each $i<\dim(S_A)$, let $H^i_{\mathfrak m}(S_A)$ denote t…

Annihilator-generator conjecture. For any central arrangement $Q$,

Regularity conjecture. The module $\mathcal{M}$ is regular holonomic if and only if $Sol^t(\mathcal{M})$ is regular. The text notes that the only-if direction follows from the prec…

Real-constructibility conjecture. The ind-sheaf $Sol^t(\mathcal{M})$ belongs to $D^b_{\mathbb{R}\text{-c}}(\mathrm{I}[\mathbb{C}_X])$. This is presented as a consequence of the pap…

Let $X$ be a complex manifold of dimension $d$, let ${\cal M}^\bullet$ be an object of $\operatorname{D}^b_{good}({\cal D}^{op}_X)$, and let $\Lambda$ be a conic subvariety of…

Consequences conjecture. For generic choices of $u\in\mathbb{C}^{n+1}$:

Integral representation conjecture. For generic choices of $u\in\mathbb{C}^{n+1}$, the morphism $\psi$ is an isomorphism of $D_Z$-modules.

Let $\mathbf{L_H(d)}$ be the Fourier transform associated with the Lagrangian correspondence $\mathbb L_H(d)$, and let $\mathbf{q\text{-}L_H(d)}$ be its quantization to a functor ……

Let $X$ be a smooth scheme. Consider the non-linear $\mathcal{D}$-Fredholm index, its symbol complex, and the cycle defined by the associated graded symbol morphism in the total sp…

Let $X$ be the underlying space, let $\mathcal{M}$ be a regular holonomic $\mathscr{D}_X$-module, and let $\mathcal{N}_0$ be the associated module. For a monoid ideal…

Let $G$ be a connected algebraic group, let $n$ be a rank, and let $q\colon G\twoheadrightarrow G/[G,G]$ be the quotient map. Write $\sim$ for the relation of $D$-module isomorphis…

Let $X$ be the underlying smooth variety, and let $\mathcal{M}$ be a holonomic $\widehat{\mathcal{A}\mathcal{V}}$-module. Here holonomicity is the finiteness condition used in the…

Derived D-module equivalence. There exists a symmetric monoidal equivalence

Let $q$ be a complex number that is not a root of unity, and let $G=\operatorname{GL}_2$ or $G=\operatorname{SL}_2$. A $\mathcal{D}_q(G)$-module is called strongly equivariant when…