38 problems

Top-codimension conjecture. For all $g\geq 1$,

Polynomial-cycle injectivity conjecture. If the cohomology class of $P$ vanishes, then $P$ vanishes in the Chow group:

Let $X$ be any flag variety, and let $\mathcal M_{0,n}(X,\beta)$ be the open part of the moduli space of stable maps, parametrizing maps without boundary degenerations. Let the ind…

Let $X$ be a complex hyper-Kähler variety, and let $A^*(X)=\bigoplus_i A^i(X)$ be its Chow ring with $\mathbb{Q}$-coefficients. A subvariety $Z\subset X$ is a constant cycle subvar…

Let $\mathrm{M}$ be a matroid and let $\mathcal{G}$ be a building set such that $(\mathrm{M},\mathcal{G})$ is a flag or complete built matroid. Koszulity conjecture. The Chow ring…

Let $T$ be a hyperkähler variety. Write $\operatorname{CH}^1(T)$ for the divisor classes, $c_i(T)$ for the Chern classes of $T$, and $\mathrm{cl}$ for the cycle class map. Beauvill…

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a smooth hypersurface, let $h\in A^1(X)$ denote the hyperplane class, and let $i_1,i_2,i_3>0$. Triple-intersection conjecture. … The s…

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a general hypersurface of degree $n+2$, and let $\Gamma\in A^{2n}(X\times X\times X)$ be the cycle defined in the source. Voisin's con…

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a smooth hypersurface. Assume that either the dimension $n$ is odd, or the degree of $X$ is $n+2$, so that $X$ is Calabi–Yau. Intersec…

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a smooth hypersurface, and let $h\in A^1(X)$ denote the hyperplane class. Hartshorne's product conjecture. … This is presented as a co…

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a smooth hypersurface, and let $h\in A^1(X)$ denote the hyperplane class. Hartshorne's conjecture. … Apart from some easy results when…

Totaro's conjecture. Under these hypotheses, this map is an isomorphism. The conjecture compares the Chow ring with Brown–Peterson cohomology after localization. No resolution is s…

Let $G$ be a reductive group over a field $k$, let $P$ be a parabolic subgroup, and let $\Delta$ be the set of simple roots. For each $J\subset\Delta$, let $L_J$ be the correspondi…

Let $M$ be a loopless matroid of rank $r$, let $G$ be a group of automorphisms of $M$, and let … be its complexified matroid Chow ring, with each graded piece carrying the induced…

Let $X$ be a projective hyper-Kähler manifold, and let $C^*$ be the subalgebra of $CH^*(X)$ generated by divisors and Chern classes. Voisin's conjecture. The cycle class map is inj…

Let $\pi:\overline{J}_C\to B$ be a Lagrangian compactified Jacobian fibration as in the motivic Beauville decomposition conjecture, with motivic summands $h_{2i}(\overline{J}_C/B)$…

Young-subgroup positivity conjecture. For $i\leq j\leq k\leq\ell$ with $i+\ell=j+k$, the element

Let $\mathsf{U}_{k,n}$ be the uniform matroid of rank $k$ on $n$ elements, let $\mathrm{A}(\mathsf{U}_{k,n})$ be its Chow ring, and let…

Let $r$ be a positive integer and $d$ a positive odd number. The classes $4c_1,c_24$ generate the polynomial ring describing the presentation, and let…

Let $(g,n)\neq(1,0)$, and let $d\geq 0$ be fixed. For each $m\geq 0$, consider the forgetful morphism … It forgets the last $m$ markings without stabilizing the curve. Pullback inj…

Let $X$ be a hyperkähler variety of dimension $4$ admitting a Lagrangian fibration, and let $A$ be a general fibre. Write $A^i(X):=CH^i(X)_{\mathbb{Q}}$ for the Chow groups with ra…

Injectivity conjecture. There exists $m_0\geq 0$ such that, for every $m\geq m_0$, the pullback

Multiplicativity conjecture. The intersection product preserves the eigenspace grading: products of classes in ${\rm CH}^i(X)_s$ and ${\rm CH}^j(X)_t$ lie in…

Beauville's conjecture. Any polynomial cohomological relation among divisor classes on $M$ that holds in $H^*(M,\mathbb{Q})$ already holds in ${\rm CH}(M)$.

Let $X$ be a K3 surface and ${\mathsf M}$ a moduli space of stable sheaves on $X$ of dimension $2n$. Let $V\subset {\mathsf M}$ be a constant-cycle subvariety, meaning that all poi…