13 problems

Affine paving conjecture. There exist affine pavings of ${}^0\operatorname{Hilb}^{(m)}(V)$ and $Z^{(m)}(V)$ by the parts $S_T^{\rm new}$ and $S_{\widehat{T}}$, respectively, such t…

Let $X$ be a complex algebraic variety, let $IT_{y*}(X)$ denote its intersection Hirzebruch class transformation, and let $L_*(X)$ denote the homology $L$-class of Goresky–MacPhers…

Bryant's lifting conjecture. Every such class lifts to the intersection homology of $Y$. The conjecture asks whether algebraic cycles always admit intersection-homology lifts, exte…

Brylinski's conjecture. The associated filtration on $\operatorname{dR}(\mathcal{M})$ induces the desired Hodge structure on $IH_\bullet(X)$.

Cheeger–MacPherson Hodge conjecture. Each class contains a unique harmonic representative, and splitting the harmonic forms into their $(p,q)$-pieces yields a pure Hodge decomposit…

Cheeger–MacPherson $L^2$-duality conjecture. The group $H^i_{(2)}(U)$ is always dual to $IH_i(X)$, and the pairing is given by integration.

Cheeger–MacPherson conjecture. The hard Lefschetz theorem and all the other standard consequences of Hodge theory should hold for intersection homology theory.

Let $X$ be a possibly singular complex projective variety, let $\widetilde{X}$ be its universal cover, and let $\chi_{\mathrm{IH}}(X)$ denote its intersection-homology Euler charac…

Let $G=(G_1,\ldots,G_{n-1}):\mathbb{C}^n\to\mathbb{C}^{n-1}$ be a polynomial mapping with $n\geq 2$ and $K_0(G)=\emptyset$. Assume that there exists a very good projection with res…

Cohomology coincidence conjecture. The $L^2$ cohomology associated to a self-dual mezzoperversity and the cohomology of a self-dual sheaf coincide.

Goresky–MacPherson cellular deformation-retraction conjecture. There exists a $\nabla f$-like vector field $V$ on $X$ such that, with

Let $Z$ be a pure $n$-dimensional compact variety. The class $L_*(Z)\in H_{2*}(Z,\mathbb{Q})$ denotes the Goresky–MacPherson homology $L$-class of the Witt space $Z$. Intersection…

Let $Z$ be a rational homology manifold. The class ${T_1}_*(Z)$ is the specialization at $y=1$ of the Hirzebruch class transformation $T_{y*}$, while the total Goresky–MacPherson h…