25 problems

Let $X$ be a projective algebraic variety with at worst Gorenstein canonical singularities. Assume that the stringy $E$-function $E_{\operatorname{st}}(X;u,v)$ is a polynomial. Bat…

Let $X$ be a normal $3$-dimensional variety and let $Delta_X$ be an effective ${\Bbb Q}$-divisor such that $(X,\Delta_X)$ is Kawamata log terminal. Let … be a log $(K_X+\Delta_X)$-…

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…

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

Let $V$ and $W$ be $n$-dimensional, possibly singular, complex Calabi–Yau varieties. Their stringy invariants are denoted by $E_{\operatorname{st}}(V;u,v)$ and…

Batyrev–Nill polynomiality conjecture. For arbitrary Gorenstein polytopes, the stringy $E$-function is a polynomial in $u$ and $v$. Nill and Schepers subsequently proved this state…

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…

For the moduli spaces $M_G(A)$ considered in the paper, form the generating series in $q$ of their stringy Hodge polynomials and specialize $(x,y)$ to obtain the stringy signature…

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…

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

NCCR summand-count conjecture. The number of non-isomorphic indecomposable summands of $M$ is equal to

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…

Let $n$ be odd, and let $X_W$ and $Y_W$ be the singular Pfaffian-Grassmannian double mirrors constructed from the linear data $W$. Write $Ell(-;y,q)$ for the appropriately defined…

Orbifold form of the non-linear McKay correspondence. With this definition,

The motivic McKay correspondence for linear actions II. We have

The motivic McKay correspondence for linear actions I. We have

Spherical smoothness conjecture. One has

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…