22 problems

Let $X$ be a $d$-dimensional algebraic variety with at worst Gorenstein canonical singularities. The bounded denominator conjecture asserts that the denominator of the rational num…

Let $\Delta$ be a Gorenstein polytope of CY-dimension $n$, so that its stringy E-function $E_{\rm st}(\Delta;u,v)$ and CY-dimension are defined as in the preceding discussion. Stri…

Let $X$ be a $d$-dimensional normal irreducible projective variety with at worst log-terminal singularities, and let $\phi$ be a family of bounded motivic invariants of type…

Let $X$ be a smooth Calabi–Yau manifold with an action of a finite group $H$ preserving the holomorphic volume form, let $Y=X/H$, and let $h_{st}^{p,q}(Y)$ denote its stringy Hodge…

Stringy flip conjecture. One has

Physicists' Euler number conjecture. In appropriate circumstances,

Stringy Euler-characteristic conjecture. The stringy Euler characteristic of $X$ can be computed as the Euler characteristic of an NCCR, computed via periodic cyclic homology.

Stringy E-function polynomiality conjecture. The stringy $E$-function $E_{st}(X,u,v)$ is a polynomial.

Topological mirror symmetry in degree zero. For every $\kappa\in\hat{\Gamma}$, with $\gamma=w(\kappa)$,

Special-stabilizer lifting conjecture. If all stabilizers of $\mathcal{X}$ are special groups, then $\operatorname{sep}_\pi^{-1}(0) \subset \mathscr{L}(X)$ is measurable and

Let $(X,\Delta)$ be a Kawamata log-terminal pair, and let $e^{\rm str}_{\rm alg}(X,\Delta)$ denote its algebraic stringy Euler number. Monotonicity conjecture. The algebraic string…

Let $g\colon X\dashrightarrow X^+$ be a birational morphism between $[200~\mathbb{Q}$-Gorenstein varieties, together with birational morphisms $f\colon X\to Z$ and…

Let $X$ be a projective algebraic variety with at worst $\mathbb{Q}$-Gorenstein log-terminal singularities. Positivity conjecture. The algebraic stringy Euler number satisfies … Th…

The motivic McKay correspondence for linear actions II. We have

The motivic McKay correspondence for linear actions I. We have

Crepant invariance conjecture. If $f$ is proper, birational and crepant, then

Spherical smoothness conjecture. One has

Let $P$ be a Gorenstein polytope of Calabi–Yau dimension $n$, so that $E_{st}(P;u,v)$ is its stringy $E$-function. Batyrev–Nill conjecture. The following assertions hold: 1.…

Let $P$ be a Gorenstein polytope of index $r$ and dimension $d$, and let $E_{st}(P;u,v)$ denote its stringy $E$-function. Batyrev–Nill polynomiality conjecture. The function…

Suppose that a finite group $G$ acts linearly on a lattice $M$ of rank $d$ via a homomorphism … Let $P$ and $P^*$ be polar, $G$-invariant, reflexive polytopes, and let $X$ and…

Let $\mathcal{X}$ be a Gorenstein proper Deligne–Mumford stack with projective coarse moduli space $X$, such that $\mathcal{X}\to X$ is proper and birational and $K_\mathcal{X}$ is…

Nonnegativity conjecture. All stringy Hodge numbers $h^{p,q}_{\rm st}(X)$ are nonnegative.