56 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 $T$ be a hyperkähler variety, and let $\operatorname{CH}^1(T)$ denote its divisor classes. The cycle class map sends algebraic cycles to rational cohomology. Beauville's diviso…

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

Ferroni's real-rootedness conjecture. Both $\operatorname{\underline{H}}_M(x)$ and $\operatorname{H}_M(x)$ should be real-rooted.

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 $X$ 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 $G$ be an algebraic group whose classifying space $BG$ has cohomology concentrated in even degrees. Write $CH^*BG$ for its Chow ring and $MU^*BG/MU^{<0}(pt).MU^*BG$ for the ind…

Let $\mathsf{M}$ be a matroid of rank $k$ on a ground set of size $n$, and let $\eta$ be a partition whose Young diagram is contained in the $k\times(n-k)$ rectangle. The Chow clas…

Shen–Vial's conjecture. Then $X$ has a multiplicative Chow–Künneth decomposition. This conjecture seeks to identify a broad class of Fano varieties whose Chow rings admit a multipl…

Totaro's conjecture. This homomorphism is an isomorphism.

Transferred Euler class generation conjecture. The Chow ring $CH^*BG$ is generated by transferred Euler classes.

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 $\mathrm{M}$ 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…

Let $\mathscr{H}_{k,g}$ be the moduli space of simply-branched, degree-$k$, genus-$g$ covers of smooth genus-zero curves, and let $\overline{\mathscr{H}}_{k,g}$ be its admissible c…

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…

Huh–Stevens conjecture. The polynomial $H_M(t)$ has only real zeros.

Ferroni's coefficient formula conjecture. The coefficients of the Chow polynomial $\operatorname{\underline{H}}_{U_{k,n}}(x)$ and augmented Chow polynomial…

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 write … View each graded piece $A^i_{\mathbb{C}}$ as a $\mathbb{C}G$-module, and let…

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