13 problems

Let $G$ be a finite simple group, let $V$ be a faithful representation of $G$ over $\text{\bf C}$, and let $V/G$ denote the quotient variety. Bogomolov's stable rationality conject…

Let $G$ be a reductive group acting on a vector space, let $T$ be a maximal torus with Weyl group $W$, and write $Y=V//T$ and $X=V//G$. Let $L_r$ be the bundle on $Y$ associated to…

Let $G$ be a semisimple group, let ${\mathsf V}$ be a finite-dimensional $G$-module, and let … be the quotient morphism. It is Popov's equidimensional quotient conjecture. If $\pi$…

Let $X$ be a quasi-projective variety over an algebraically closed field of characteristic zero, and let a finite group $G$ act effectively on $X$. For $g\in G$, write $X^g$ for th…

Let $G$ be a finite group acting effectively on a smooth variety $X$. For each conjugacy class $[\lambda]\in G/\sim$, let $X^\lambda$ be the fixed locus of $\lambda$, let…

Unwinding conjecture. In general, one can "unwind" the problem of taking intersections in $X^n/G$ into taking intersections in $X$.

Wang's conjecture. The author's use of the main theorem to restore Cheah's result about the Hodge numbers of $X^n/G$ can vastly generalize.

Let $V_{32}$ be the double cover considered in the paper, and let $\alpha_1$ and $\alpha_2$ be the two involutions acting on it. Write $[V_{32}/\alpha_i]_p$ for the number of point…

Let $X$ be a smooth variety over an algebraically closed field of characteristic zero, and let a finite group $G$ act effectively on $X$. For $\lambda\in G/{\sim}$, let $X_\lambda$…

The quotient modified diagonal conjecture. If

Let $G$ be a reductive group, let $w$ be a suitable elliptic element of its Weyl group, and let $G\bsl\tfB_w$, $G\bsl\fB_w$, $G^F\bsl\tX_w$, and $G^F\bsl X_w$ denote the orbit vari…

Quotient conjecture. For any hyperplane $L\subset\mathbb{P}^{4}$, $L\cap I_4$ is the quotient of $X_L$ by the Heisenberg group, and

Let $X$ be a smooth projective variety over a field $k$ of characteristic $0$, and let $G$ be a finite group acting on $X$. Suppose that … for all $i$. Write $(X,\pi)$ for the moti…