49 problems

Let $k$ be a field and $R$ a coefficient ring. Write $\operatorname{DM}_{\operatorname{gm}}^{\operatorname{eff}}(k;R)$ for geometric effective motives and…

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 $X$ 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…

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 $\pi:\mathcal{X}\to B$ 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 $k$ be the coefficient field, let $d_1,d_2,d_3$ be nonnegative integers, and write $x=(x_1,\ldots,x_{d_1})$, $y=(y_1,\ldots,y_{d_2})$, and $z=(z_1,\ldots,z_{d_3})$. For a tuple…

Let $\mathcal U$ be a geometrically connected and smooth ungraded variety. For $\in\mathbb Z_{\geq 0}^{r}$, let $\mathcal Z_{,n}$ be the locus of ordered $r$-tuples of effective ze…

Let $\mathcal U$ be a geometrically irreducible ungraded variety, and let $\mathcal X_n$ denote its space of effective zero-cycles of degree $n$. Write $\mathbb L$ for the Lefschet…

Let $f$ be a complex analytic function in $d$ variables vanishing at the origin $O$. Let $F_{f,O}$ be its topological Milnor fiber, let $\mathscr S_{f,O}$ be the motivic Milnor fib…

Motivic jet-space integral conjecture. The identity

Let $A_{\mathbb{Q}_p}^{(N)}(\boldsymbol{s})$ be the $p$-adic Koba–Nielsen amplitudes, and let $A_{\mathbb{C((}T\mathbb{))}}^{(N)}(\boldsymbol{L})$ denote the motivic string amplitu…

Let $K$ be a complete discretely valued field with algebraically closed residue field $k$, and let $X\to\operatorname{Spec}K$ be a smooth Calabi–Yau variety with volume form…

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 $\mathcal I=(f_1,\ldots,f_r)$ be an ideal in $\mathbb C[x_0,\ldots,x_n]$ whose associated subscheme $Y=V(\mathcal I)$ in $\mathbb C^{n+1}$ contains the origin. Let…

Let $k$ be a perfect field of characteristic $p>0$, let $G=\alpha_p$, and let $V$ be a finite-dimensional representation of $G$ with quotient $V/G$. Let $H=\mathbb{Z}/p\mathbb{Z}$…

Let $Y$ be a smooth $k$-variety with the trivial ${\mathbb G}_{m,k}$-action, and let ${\mathbb G}_{m,k}$ act on ${\mathbb A}^n_k$ with weights $w_1,\ldots,w_n>0$. Let … be a…

Let $E_1$, $E_2$, and $E_1\oplus E_2$ be semi-Schur objects in the derived category of coherent sheaves over $Y$. Their extension space decomposes as … Here ${\mathcal S}_0(E)$ den…