38 problems

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…

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 $X$ 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…

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 $M$ 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…

Beauville's conjecture. The cycle class map is injective on the subalgebra of $A^{*}(X)$ generated by divisors.

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 $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ 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…

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 $\pi:\overline{J}_C\to B$ 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…

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 ${\rm CH}(M)$.