21 problems

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…

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…

Let $V$ be a complex projective variety, let $M=V\setminus\operatorname{sing} V$ be its regular part, and equip $M$ with the metric induced by a smooth Kähler metric on projective…

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…

Let $X$ be a proper scheme over $\mathbb C$, let $\pi:X_\bullet\to X$ be a smooth proper hypercovering, and let $H^{2p+i}(X)$ be Deligne's mixed $\mathbb Q$-Hodge structure on…

Let $X$ be a proper integral scheme over $\mathbb C$, and let $\nu^{i,l}$ be the natural map … where $(N^lH^t)^\bullet$ is induced by the coniveau filtration on the components of a…

RCD homeomorphism conjecture. The metric completion of $(X^\circ,\omega,\omega^n)$ is a non-collapsed RCD space homeomorphic to $X$.

Let $X$ be a compact pure-dimensional complex algebraic variety. The Goresky–MacPherson homology $L$-class of $X$ is denoted by $L_*(X)$, and its intersection cohomology Hirzebruch…

Let $X$ be an affine algebraic variety, and let the topological complex be the more general version of the complex considered in Vinogradov's work on invariants produced by derivat…

Constant and intersection-complex $L$-class conjecture. If $X$ is a ${\mathbb Q}$-homology manifold, then

Let $X$ be a singular projective variety, let $\phi:\widetilde{X}\to X$ be any resolution of singularities, and let $E$ be the exceptional divisor. Let $p$ be an integer such that…

The cohomological singular Hodge conjecture. The image of $\mathrm{cl}_p$ coincides with $H^{2p}_{\mathrm{Hdg}}(X)$. This extends the classical Hodge conjecture to singular varieti…

Uniform Sobolev inequality conjecture. The metrics $(Y,\omega_\epsilon)$ have a uniform Sobolev constant for $0<\epsilon\leqslant1$: there exists a constant $C_S$, independent of…

Let $X$ be a compact complex algebraic variety that is a rational homology manifold. Denote by $T_{1*}(X)$ its Hodge-L-class and by $L_*(X)$ its topological Goresky–MacPherson L-cl…

Let $X$ be a $\mathbf{b}{Q}$-Gorenstein variety with a canonical divisor $K_{X}$ big. A closed subvariety $Y\subset X$ is potentially dense if, after a finite extension of the base…

Let $X$ be a complex projective variety defined over $\overline{\mathbb Q}$. A weight $2p$ Hodge cycle is a class in $H^{2p}(X,\mathbb Q(p))$ of Hodge type $(p,p)$. Let … be motivi…

Let $X'\to X$ be a desingularisation of an integral variety $X$, with exceptional fibre $E\hookrightarrow X'$. For a positive integer $n$, let $F^dK_0(X',nE)$ denote the codimensio…

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…

General decomposition conjecture. This description should hold in general for strongly stable varieties of arbitrary dimension.

Let $X$ be a projective complex scheme of dimension $d$. Let $F^dKH^M_0(X)$ be the degree-$d$ part of the Brown filtration on Milnor $KH$-theory, and let … be the Chern-class map.…

Let $X$ be a quasi-projective scheme of dimension $d$ over an algebraically closed field $k$ of characteristic zero. Let $CH^d(X)$ denote the Chow group of zero-cycles and let…