75 problems

Laurent-polynomial conjecture. For every degree $d$, $\Omega_{d[L]}$ is a Laurent polynomial in $q_0$, $q_1$, $q_2$, and $q_3$.

Hebey–Vaugon conjecture. If $(M^n,g)$ is not conformal to the standard metric on $S^n$, or if the action of $G$ has no fixed point, then

Let $G$ be a finite group acting effectively on a smooth quasiprojective variety $X$. For $g\in G$, let $X^g$ denote the invariant subvariety and let $C(g)$ be the centralizer of…

Let $G$ be an algebraic group, let $X$ be a smooth projective variety with a $G$-action, and let $\pi:X\to Y$ be a $G$-contraction. Let $\mathrm{Stab}(X_G)$ denote the space of sta…

Equivariant rationality conjecture. The series $\widetilde Z^{S}_{\beta}(q\mid s_1,s_2)$ is the Laurent expansion of a rational function in $q,s_1,s_2$. This is presented as an equ…

Equivariant torus-fibration mirror-symmetry conjecture. There should be equivalences

Let $G$ be an algebraic group, let $\Psi$ be a semisimple conjugacy class in $G$, choose $h\in\Psi$, and let $Z={\mathcal Z}_G(h)$. For a $G$-space $X$, write…

Let $X$ be a smooth quasi-projective $3$-fold over $\mathbb{C}$ equipped with an action of an algebraic torus $T$, and suppose that the fixed locus $X^T$ is compact. Let…

Let $R_{\sigma\tau}^{(n)}(t,\lambda)$ be the coefficient of $\hbar^n$ in the asymptotic solution matrix $R_{\sigma\tau}(t,\lambda,\hbar)$ of the equivariant quantum differential eq…

Assume that the equivariant quantum cohomology has convergent structure constants around $x=\lambda=0$, so that the canonical coordinates $\mathsf{u}_\sigma(t,\lambda)$ are defined…

Let $S\subset Y$ be a nonsingular, $T$-invariant divisor, let $\beta\in H_2(Y,\mathbb Z)$ satisfy $\int_\beta[S]\geq 0$, and let $\eta$ be a partition of $\int_\beta[S]$ weighted b…

Let $G$ be a finite group, or more generally a linear algebraic group, acting on a variety $X$, and let $X\to Y:=X/G$ be the quotient. A $G$-variety is a variety with a $G$-action,…

Local structure conjecture. (1) There exists a proper continuous map

Millson–Zombro conjecture. There exists a $PGL(2,\mathbb{C})$-equivariant embedding

Let the equivariant Toda lattice be the integrable hierarchy associated with the equivariant Gromov–Witten theory of $\mathbb{CP}^1$, and let $z_k$ denote its constants for positiv…

Let $X$ be a quasiprojective scheme over a field $k$, let $G$ be a finite group acting on $X$, and assume that $\operatorname{char} k$ does not divide $|G|$. Let…

Equivariant derived McKay conjecture. The derived category of $\mathbb C^{\ast}$-equivariant sheaves on $X$ is equivalent to the derived category of graded finitely generated…

Let $M$ be a Kähler manifold with a Hamiltonian $S^1$-action, and let $M_+$ and $M_-$ be the associated symplectic cut spaces, with reduced space $M_0$ and corresponding holomorphi…

Assume Setup, with $X$ a Fano manifold and $\omega \in c_1(X)$. Let $v \colon P \to \mathbb{R}_{>0}$ be a weight, let $\mathscr{V}$ be a canonical extension sheaf with extension cl…

Let Setup be fixed, let $v \colon P \to \mathbb{R}_{>0}$ be a weight, and let $\mathcal{E}$ be a reflexive sheaf of rank $r$ that is $v$-stable. Bogomolov–Gieseker conjecture. The…

Let Setup be fixed. For a weight $v \colon P \to \mathbb{R}_{>0}$, let $C_v>0$ be a constant, and let $\mathcal{E}$ be a $\mathbb{G}$-equivariant torsion-free sheaf. The equivarian…

Let $G_{\mathrm{data}} \subseteq G$ act on $M$, and let $\Gamma$ be a continuous boundary whose symmetry group contains $G_{\mathrm{data}}$. For a target precision $\varepsilon>0$,…

Let $M^{n+1}$ be as in the equivariant setting, with a compact Lie group $G$ acting by isometries and satisfying … A $G$-homology class is represented by a fixed $G$-hypersurface…

Let $M^{n+1}$ be a closed Riemannian manifold equipped with a compact Lie group $G$ acting by isometries, satisfying … A closed embedded minimal hypersurface is $G$-invariant when…

Higher-degree denominator conjecture. For every integer $d>1$,