13 problems

Let the orientation constructed by Joyce and Upmeier be the orientation used to define the motivic Donaldson–Thomas invariants in this paper, and let the orientation used by Behren…

Let $\mathbb k$ be a field, and let $C$ and $C'$ be smooth projective curves over $\mathbb k$ that are D-equivalent and whose geometric irreducible components have genus $1$. The a…

Let $c$ be the motivic birational invariant considered in the paper, and let $S$ and $S'$ be D-equivalent K-nef surfaces. The corresponding elements are the differences…

Let $K_0(\operatorname{Var}_k^{\dim})$ be the graded Grothendieck ring of varieties, equipped with the symmetric-power operations $\operatorname{Sym}^m$ and the involution…

For $e\in \mathbb{Z}^n$ and $f\in \mathbb{Z}^m$, let … Here $P_s(t)$ is a polynomial, and $N_s$ is a sequence of integers with $N_s\to\infty$. Stabilization conjecture. The polynom…

Motivic PT generating-series conjecture. The generating series of the fiber-class invariants is given by the displayed infinite-product and plethystic-exponential formula.

Let $X$ be a smooth projective threefold over $\mathbb{R}$, equipped with an isomorphism … for some line bundle $L$ on $X$. For $n\geq 0$, let $\tilde{I}_n$ be the quadratic degree…

Refined Gopakumar–Vafa/Pandharipande–Thomas correspondence. One should have

Let $G$ be a complex algebraic group, and let $\mathrm{K}\operatorname{M}_{G}$ denote the ambient module of motivic data associated with $G$. Define … the submodule generated by th…

Mozgovoy's conjecture. The invariants $\Om_D(r,d)$ are independent of $d$, without requiring $r$ and $d$ to be coprime, and are given by the formula proposed in the cited work. Thi…

The $c_2$ residue conjecture. If $I_{G_1}=I_{G_2}$ for two primitive log-divergent graphs $G_1$ and $G_2$, then

Hausel–Rodriguez-Villegas conjecture. The functions $H_n^{(p)}(t)$ are polynomials in $t$ and, for the moduli space $\lM=\lM(L,n,d)$ of twisted Higgs bundles of coprime rank and de…

Mixed Hodge structure conjecture. The algebraic counting $c_{alg}$ specializes to $c_{top}$, and there are no further algebraic counting invariants specializing to the Hodge--Delig…