146 problems

Let $f$ be a non-constant polynomial in $\mathbb{Z}[x_1,\ldots,x_n]$, for some $n\geq 1$, with $f(\mathbf{0})=0$. For every prime number $p$, let $Z_{f,p}(s)$ be Igusa's $p$-adic z…

Let $f \in \mathbb{C}[x_1,\dots,x_n] \setminus \mathbb{C}$ and assume $f(o)=0$. The local motivic zeta function $Z_{\operatorname{mot},o}(f;s)$ and the local topological zeta funct…

Nagai's conjecture. The monodromy operators satisfy

Let $K$ be the local field, let $X$ be a proper smooth variety over $K$, and let $H^j(X^{\mathrm{ca}},\overline{\mathbb{Q}}_l)$ carry its monodromy filtration $M_\bullet$ and weigh…

Monodromy Conjecture. If $s_0$ is a pole of $Z_{f,W}^{\mathrm{mot}}(s)$, then $(s-s_0)\mid b_f(s)$, and $e^{s_0\cdot 2\pi \sqrt{-1}}$ is an eigenvalue of a local monodromy at some…

Let $W$ be the Weyl group associated with a singularity of type $D_n$ or $E_n$. For some proper smooth surface $X/K$, consider an integral model of $X$ whose special fiber has a si…

Let $\mathfrak{g}$ be a simple Lie algebra, let $V(\lambda_1),\ldots,V(\lambda_n)$ be irreducible representations, write…

Let $H=H^{\mathrm{SL}_n}$ be the maximal torus of $SL_n(\mathbb C)$, let $\vartheta\in\mathfrak h^*/W$, and set $k=c-1$. Let $\nabla^{K}(\vartheta,k)$ be the trigonometric KZ conne…

Let $k$ be the base field, let ${\mathcal {P}}$ be the class of schemes considered in the source, and let $X\in {\mathcal {P}}$. The monodromy operator is … The associated objects…

Vanishing-integral conjecture. (i) If $\gamma(t)$ is totally unbalanced, then

Let $K$ be a discretely valued field with separable closure $K^{\operatorname{sep}}$, let $X$ be a variety over $K$ with semistable model, and let $E_r^{p,q}$ be its weight spectra…

Let $S\subset {\mathbb P}^n$ be a primitive K3 surface, and let $d>0$. Consider the family of rational curves in the linear system $|{\mathcal O}_S(d)|$. Transitive monodromy conje…

Let $\pi:\mathcal X\to\Delta$ be a degeneration of an irreducible symplectic $2n$-fold. For each $m$, let $T_m$ be the associated monodromy operator on…

Let $f$ be a singular function and let a Dynkin diagram encode its algebraic monodromy. Consider the braid subgroup associated to such a Dynkin diagram, allowing conjugation in the…

Let $f$ be a singular function, with braid monodromy group acting by the Artin representation on its algebraic monodromy homomorphism. Stabilizer conjecture. The braid monodromy gr…

Let $X$ be a variety with totally degenerate reduction, and let $N_{1,L}$ and $N'_{1,L}$ be the enriched monodromy operators defined in Section 4.3. Write…

Let $f:({\mathbb C}^{n+1},0)\to({\mathbb C},0)$ be a germ of an analytic function, and let $Z_{top,0}(f,s)\in{\mathbb Q}(s)$ be its local topological zeta function. A pole $s_0$ of…

Assume $Y={\mathbb C}^{n+1}$ and that $f$ is weighted homogeneous. Let $C$ be the matrix of the monodromy operator $h_*$ on $H_n(X_\delta,{\mathbb C})$ with respect to a basis, let…

Let ${\cal F}_v$ be a leaf indexed by a positive vector $v$ in the K3 period description, let $\Gamma_v$ be the corresponding stabilizer group, and let $Aut_{{\bf Z}PL,v}(S^2)$ be…

Loeser's generalized monodromy conjecture. For almost all finite places $\mathfrak P$, if $H$ is a polar hyperplane in $\mathcal P(Z_\phi(F_{\mathfrak P},\chi;s))$, with $\phi$ res…

Let $h$ be a complex polynomial and let $Z_{\mathrm{top},0}(h,s)$ be its local topological zeta function at the origin. If $s_0$ is a pole, write it in the source's notation as…

Let $k$ be a field of characteristic zero, let $X=\mathbb A^d_{\mathbb C}$, let $h\in\mathbb C[x_1,\ldots,x_d]$ satisfy $h(\mathbf 0)=0$, and let $Z_{DL}(h,T)$ be the local Denef–L…

Monodromy restriction conjecture. The restriction homomorphism

Let $(X,\omega)$ be a simply connected symplectic $4$-manifold. For the degree-$N_k$ symplectic branched covering associated with sufficiently large $k$, let $G_k^0$ be the relevan…

Let $X_K$ be a smooth scheme of finite type over $K$. Let $H^{n}_{pst}(X_K)$ be the finite-dimensional $\mathcal{H}^{ur}_1$-module equipped with its increasing weight filtration…