83 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…

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

Denef and Loeser's motivic monodromy conjecture. There exists a finite subset $\mathcal{S}\subset\mathbb{Z}\times\mathbb{Z}_{>0}$ such that

Let $f$ be a polynomial and let $X_s=f^{-1}(0)$ be its complex zero locus. The naïve motivic zeta function $Z_{mot}(f,s)$ is the motivic analogue of the $p$-adic and topological ze…

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…

Motivic fundamental lemma. For every such pair and every $r\in\mathbb{Q}$, one has the identity in $\mathbb{K}$

Let $n$, $k$, and $r$ be the parameters indexing the relevant classical groups and equal-valuation strips. Let $\Theta^{G,+}_{n,k,r}$ and $\Theta^{G,-}_{n,k,r}$ be the Chow motives…

Let $k$ be a field of characteristic $p$, and let $K_0({\mathcal V}_k)$ be the Grothendieck ring of $k$-varieties: the free abelian group on isomorphism classes of geometrically re…

Monodromy conjecture. The poles of $Z_f^{mot}(s)$ are roots of $b_f(s)$.

Let $k$ be a field of characteristic zero, and let $X$ be a $k$-variety. Let $d_1,d_2$ be nonnegative integers, and let $\mathbb{G}_{m,k}$ act on…

Let $X$ be a smooth irreducible complex projective variety, let $(E_i)_{i\in S}$ be a finite set of mutually distinct prime divisors on $X$, and let $\bm a=(a_i)_{i\in S}$ with…

Let $X$ be a smooth projective complex variety and let $\omega^{1/q}$ be a multivalued rational $n$-form on $X$ with no logarithmic poles and whose divisor has normal-crossings sup…

Relative integral identity. The equality

Let $k$ be a field of characteristic not $2$. Let $K_0(\operatorname{Var}_k)$ be the Grothendieck ring of $k$-varieties, let $\operatorname{GW}(k)$ be the Grothendieck–Witt ring, a…

Let $Q$ be a quiver and let $\mathbf{d}\in\mathbb{Z}_{\geq0}^{Q_{0}}\setminus\{0\}$ such that $(Q,\mathbf{d})$ has property (P). Let $\mu_{Q,\mathbf{d}}$ be the corresponding momen…

Let $e$ be a positive integer, and let $Z$ be a locally complete intersection variety of pure dimension over a number field $K$. Define … Here…

Let $J$, $(r_i)_{i\in J}$, the polynomials $f_{ij}$, the ideal $\mathcal{I}$, $r$, and $\sigma_0((f_{ij}))$ be as in the degree-sensitive averaged exponential-sum conjecture. Unifo…

Let $J\subset\mathbb{N}_{\geq1}$ be finite and non-empty. For each $i\in J$, let $r_i$ be positive, let $f_{ij}\in\mathcal{O}_K[x_1,\dots,x_n]$ be non-constant with…

Let $K$ be a number field, let $\mathcal{I}\subset\mathcal{O}_K[x_1,\dots,x_n]$ be a non-zero ideal with $\mathcal{I}_K\neq(1)$, and let $Z\subset\mathbb{A}^n_{\mathcal{O}_K}$ be a…

Let $K$ be a number field, let $f\in\mathcal{O}_K[x_1,\dots,x_n]$ be non-constant, and let $Z\subset\mathbb{A}^n_{\mathcal{O}_K}$ be a subscheme. For a local field $L$ and an addit…

Let $J\subset\mathbb{N}_{\geq1}$ be finite and non-empty, let $r_i$ be positive integers for $i\in J$, and let $f_{ij}\in\mathbb{Z}[x_1,\dots,x_n]$ be non-constant with…

Let $\mathcal{I}\subset\mathbb{Z}[x_1,\dots,x_n]$ be a non-zero ideal with $\mathcal{I}\otimes\mathbb{Q}\neq(1)$, and suppose $\mathcal{I}\otimes\mathbb{Q}$ is generated by…

Let $f$ be a polynomial having $0$ as its only critical value. Let $\operatorname{moi}_{\mathbb{Q}}(f)$ be its motivic oscillation index, and let $b_f(s)$ be its Bernstein–Sato pol…

Let $\pi:\mathcal{X}\to B$ be a Fano fibration and let $M$ range over the Manin components of $\operatorname{Sec}(\mathcal{X}/B)$, with degree tending to infinity. Stability conjec…

Let $f\in\Bbbk[x_1,\ldots,x_n]$ be a regular function over a field $\Bbbk$ of characteristic $0$, with $0\in X_f$, and let $Z_\mathrm{mot}(T)$ be its local motivic zeta function. A…