96 problems

Let $X$ be a complex Shimura variety, let $X^{\text{BB}}$ be its Baily--Borel compactification, and write $IH^k(X^{\text{BB}};\mathbb{C})$ for its intersection cohomology and…

Let $X\subseteq\mathbb{P}^N$ be a projective variety, let $X_{\mathrm{reg}}$ be its regular locus, and equip $X_{\mathrm{reg}}$ with the Fubini–Study metric $ds_{\mathrm{FS}}^2$. H…

Cappell–Shaneson conjecture.

Let $X$ be an equal-rank locally symmetric space, let $X^*$ be a Satake compactification, and let $\widehat{X}$ be Zucker's reductive Borel–Serre compactification. Let … be the quo…

Let $\Delta\subset\mathbb{R}^d$ be a polytope, let $\Phi$ be its dual fan, and let $\overline{M}_\Phi$ be the quotient combinatorial intersection cohomology module defined from the…

Let $\Delta$ be a $d$-dimensional polytope, and let $Gh_k(\Delta)$ denote the components of Stanley's generalized $h$-vector. For a polytope that is integral, these numbers agree w…

Consider a Shimura datum $(G,X)$ with $G=\operatorname{Res}_{F/\mathbb{Q}}H$, where $H$ is a connected reductive group over a totally real field $F$. Let…

Let $G$ be the group, let $X\in G$ be regular, let $w\in W$, and let $\mathcal{Y}_w(X)$ be the corresponding variety. Let $\Omega_w$ be the Schubert variety and let…

Let $X$ and $X^!$ be symplectic dual cones with Poisson brackets of degree two. Let $\operatorname{HP}_0(\mathcal{O}(X))$ denote the zeroth Poisson homology of the coordinate ring…

Intersection-cohomology–quantum-cohomology conjecture. The intersection cohomology of $X$ should be isomorphic to the quantum cohomology of $\tilde{X}$ specialized at $q=1$.

Malkin–Ostrik–Vybornov conjecture. The pairs

Consider the singularity types $a_2$, $ac_2$, $ag_2$, $c_2$, and $cg_2$. Malkin–Ostrik–Vybornov's conjecture. The singularities of types $a_2$, $ac_2$, and $ag_2$, respectively…

Let $V=\bigoplus_i V_i$ be a finite-dimensional vector space, with each $V_i$ equipped with a non-degenerate symmetric bilinear pairing, and let $K\subset V$ be a subspace such tha…

Let $\Delta$ be a complete fan of dimension $n$, and let ${\mathcal A}(B\Delta)$ be its associated graded algebra with elements $L_i\in {\mathcal A}(B\Delta)$ of degree $e_i$, for…

Let $S$ be a noncompact smooth curve with smooth compactification $\overline S$, let $j:S\hookrightarrow\overline S$ be the inclusion, and let $L_{\rho}$ be the local system associ…

Let $M$ be a manifold with boundary fibration … and suppose that the metric on $M$ is quasi-isometric near the boundary to … where $R\in[1,\infty)$, $-1\leq c<0$, and $k$ is a symm…

Let $X^n$ be a pseudomanifold and let $Y\subset X$ be a compact codimension-$1$ submanifold such that … where $Y\subset\subset X^{\mathrm{reg}}$. For a perversity function…

Let $\mathcal{A}$ be a unimodular arrangement, let $\mathfrak{M}(\mathcal{A})$ be the associated hypertoric variety, and let $R(\mathcal{A})$ be the ring generated by the elements…

Let $\mathcal{A}$ be a unimodular arrangement and let $\mathfrak{M}(\mathcal{A})$ be the associated hypertoric variety over $\mathbb{C}$. Let…

Let $\mathcal{A}$ be a unimodular arrangement, let $\mathfrak{M}(\mathcal{A})_{\mathbb{C}}$ be its hypertoric variety, and let $R_0(\mathcal{A})$ be the combinatorially defined rin…

The affine Grassmannian singularities of types $a_2$, $ac_2$, $ag_2$, $c_2$, and $cg_2$ all have codimension $4$. Their intersection cohomology agrees within the pairs of types…

Bressler–Lunts hard Lefschetz conjecture. For every $i<d/2$, this map is a bijection. This conjecture proposes the hard Lefschetz property for intersection cohomology of arbitrary…

Let ${\cal P}$ be a convex polytope, and let $h({\cal P})=(h_0,h_1,\ldots,h_d)$ be its toric $h$-vector. Define the associated $g$-vector by the successive differences of the first…

Geometric interpretation conjecture. In the Borel-subgroup case with trivial vector bundle, $p_{y,w}|_{v=\sqrt q}$ is, up to normalization, the restriction to $Z_y^F$ of the charac…

Fix a cocompact lattice $J\subset Z(\mathbbA)/Z(F)$ and an admissible $n$-tuple $\overline{\omega}$. Let … be the cuspidal intersection cohomology representation, and let…