18 problems

Hypertoric Fukaya-category equivalence conjecture. There is a natural equivalence between

Let $\mathcal{A}$ be a unimodular arrangement, let $\mathfrak{M}(\mathcal{A})$ be the associated hypertoric variety, and let $R(\mathcal{A})$ be the ring generated by the elements…

Let $\mathcal{A}$ be a unimodular arrangement and let $\mathfrak{M}(\mathcal{A})$ be the associated hypertoric variety over $\mathbb{C}$. Let…

Let $\mathcal{A}$ be a unimodular arrangement, let $\mathfrak{M}(\mathcal{A})_{\mathbb{C}}$ be its hypertoric variety, and let $R_0(\mathcal{A})$ be the combinatorially defined rin…

Let $V(\Delta)$ be a boundary hypertoric vertex operator superalgebra, with character $\operatorname{ch}_{V(\Delta)}(q)$. Quasimodularity conjecture. The character is (quasi)modula…

Gammage–McBreen–Webster conjecture. There exists an additive variety $\mathfrak{M}$, a divisor $\mathfrak{D} \in \mathfrak{M}$, and an isomorphism

Let $\mathfrak{M}_\mathbb{V}$ be the hypertoric variety associated to the arrangement $\mathbb{V}$, let $M(\mathbb{V})$ be the corresponding Weinstein manifold, and let…

The hypertoric intersection-cohomology conjecture. For all $n$, there exists an isomorphism of graded $\mathfrak{S}_n$-representations

Let $X$ be a holomorphic symplectic quotient with vertex function $V(q,z,a)$, where $q$ is a complex number with $|q|<1$, $z$ denotes the Kähler parameters, and $a$ denotes the equ…

Lau–Zaslow isomorphism conjecture. The variety $\mathfrak{M}\setminus \divisor$ is isomorphic to a multiplicative hypertoric variety.

Additive realization conjecture. Every multiplicative hypertoric variety admits an additive realization: there exists an additive hypertoric variety $\mathfrak{M}$, a divisor…

A multiplicative hypertoric variety is a variety obtained as a multiplicative hyper-Kähler quotient; an additive hypertoric variety is denoted by $\mathfrak{M}_{u,\lambda}$, and an…

Generalization conjecture. The results proved for hypertoric varieties should hold for more general holomorphic symplectic quotients of symplectic affine spaces by a reductive grou…

Extended-core zonotopal tiling conjecture. There exists an integral zonotopal tiling $\mathcal{T}$ in $N_\mathbb{R}$ whose vertex set is

Affine quasihypertoric classification and morphism conjecture. Every affine $T$-quasihypertoric variety is of the form $Y(Z)$ for a single zonotope $Z$, and every morphism between…

Arbo–Proudfoot classification conjecture. Every $T$-hypertoric variety arises via this construction: for every $T$-hypertoric variety $Y$, there exists an integral zonotopal tiling…

Let $K$ be a subtorus of a torus $T$, let $\boldsymbol{\chi}=(\chi_i)$ be the associated characters, and let $\alpha$ be as in the construction of the hypertoric variety. For each…

Let ${\mathcal H}$ be a unimodular central arrangement, let $\mathfrak{M}$ be its hypertoric variety, let $V_0$ be the relevant vector space, and let $R({\mathcal H})$ be the assoc…