22 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…

Let $\widetilde Q$ be a quiver with potential $\widetilde W$, let $G$ be the relevant group, and let $\upzeta$ be a stability condition. Write $\mathtt{JH}$ for the semisimplificat…

Let $f:X\to\mathbf A^1$ be a function on a smooth complex algebraic variety, and let $x$ be a closed point of $f^{-1}(0)$. Suppose the singular locus of $f$ is a curve with local c…

Mixed realization conjecture. The functor $\mathop{\rm real}\nolimits_\Delta$ should factor through a mixed realization functor sending ${\mathbf D}(\Delta)$ to mixed Hodge modules…

Let $\pi:\mathcal X\to\Delta$ be a projective conifold degeneration, and let $\mathcal P$ be the corrected perverse object constructed from nearby and vanishing cycles. Suppose the…

Let $\pi:\mathcal X\to\Delta$ be a projective conifold degeneration, and let $\mathcal P$ be the corrected perverse object constructed from nearby and vanishing cycles. A mixed-Hod…

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 open quantum systems have effective connections with irregular singularities, such as those arising from non-Markovian reservoirs with sharp spectral cutoffs or from Floquet sy…

Vanishing conjecture for Kollár-hyperbolic varieties. Kollár-hyperbolic varieties satisfy a vanishing theorem: in every nonzero degree, the limit of the normalized dimensions of th…

Let $k$ be a field of characteristic $p>0$, and let the mixed Hodge module de Rham and Higgs subcomplexes $K^*_{\mathrm{dR}}$ and $K^*_{\mathrm{Hig}}$ be as in the preceding constr…

BPS complex conjectures. The following assertions should hold:

Let $X$ be an algebraic variety over $\mathbb{Q}$. An object $M$ of the abelian category $\mathcal{M}_\mathrm{univ}(X,\mathbb{Z})$ is torsion-free if multiplication by every nonzer…

Let $X$ be a compact Kähler or complex projective manifold that is aspherical or has a nef cotangent bundle, and let $\mathcal{M}$ be a mixed Hodge module on $X$. For an integer…

Let $r,d,d'\in\mathbf{Z}$ with $r\geqslant1$. For positive multiples of $(r,d)$, write … and let the corresponding algebra object be obtained by the symmetric product with the equi…

Eyssidieux's conjecture. These filtrations endow

Let $X$ be a complex projective manifold, let $\mathbb{M}$ be a Mixed Hodge Module on $X$, and let $H^k_2(\widetilde{X},\mathbb{M})$ denote the associated $L_2$ cohomology, equippe…

Let $X$ be a complex scheme. The compact objects in the homotopy category … form a triangulated category, where $\mathcal{H}$ is the absolute Hodge spectrum and…

Let $cM$ be a regular holonomic $cD_X$-module, and let $W\to X$ be a relative compact open subset. Suppose that, locally on $W$, … for some $n\le l\le n+r$. For a support locus…

Let $\widehat{\mathbb{C}}^{n+1}$ and $\mathbb{C}^{n+1}$ be dual affine spaces, and let…

Let $Q_L^n$ be the Quot schemes associated with the local line $L\subset\mathbb A^3$, and let…

Let $\boldsymbol{f}=(f_1,\dots,f_p)$ be a tuple of functions, and let $\mathscr{M}$ be a mixed Hodge module with underlying $\mathcal{D}$-module $\mathcal{M}$. Assume that the pair…

Let $X$ be a compact complex algebraic variety that is also a rational homology manifold. The class $\widehat{T}_{1*}(X)$ denotes the specialization at $y=1$ of the normalized Hirz…