27 problems

Let $F=(f_1,\ldots,f_r)$ be a tuple of polynomials, let $Z_F^{mot}(s_1,\ldots,s_r)$ be its multivariable motivic zeta function, and let $B_F\subset\mathbb{C}[s_1,\ldots,s_r]$ be it…

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…

For a quasi-projective variety $X$ over $K$, let … where $\operatorname{Sym}^n(X)$ is the $n$-fold symmetric power of $X$ and $\operatorname{Sym}^0(X)=\operatorname{Spec}K$. Kapran…

Let $\mathcal{M}$ be a category of finite-dimensional motives over a field $k$, and let $M\in\mathcal{M}$. Write $\det(M)$ for the determinant of $M$, let $\mathbb{L}$ denote the L…

Let $k$ be a field, let $X$ be a variety over $k$, and let ${\mathcal M}_k$ denote the relevant localization of the Grothendieck ring of varieties. Let $Z_{\rm mot}$ be the motivic…

Let $s_0$ be a pole of the motivic zeta function, and let $k$ be its pole order. Stronger Monodromy Conjecture. The number $s_0$ is a root of the Bernstein--Sato polynomial with mu…

Let $f \in \operatorname{\mathbb{C}}[x_1,\ldots,x_n] \setminus \mathbb{C}$ with $f(0)=0$. Write $Z_f^{\operatorname{mot}}(s)$ and $Z_{f,0}^{\operatorname{mot}}(s)$ for the global a…

Let $X$ be an $n$-dimensional smooth complex irreducible variety and let $f:X\to\mathbb A^1$ be a regular function with non-empty zero locus. Let $\alpha$ be a remarkable number of…

Let $f:(\mathbb{C}^d,0)\to(\mathbb{C},0)$ be a complex analytic function germ, and let $Z_{top,0}(f;s)$ be its local topological zeta function. For $a\in\{f=0\}$, let…

Let $X$ be a smooth irreducible complex variety and let $\mathcal I$ be a coherent ideal sheaf of $X$ with a non-empty zero locus. Let $Z_{\mathcal I}^{\mathrm{mot}}(T)$ be its mot…

Let $f\in \mathbb C[x_{1},\dots,x_{n}]\setminus \mathbb C$ be a non-constant polynomial, and let $E(f)$ denote the set of eigenvalues of the cohomology monodromy of the Milnor fibr…

Let $R^{(2,2k)}$ be the germ of the plane curve singularity $y(y-x^k)=0$, let $R^{(2,2k)\oplus N}$ denote the direct sum of $N$ copies of this singularity, and let…

Let $R^{(2,2k)}$ be the germ of the plane curve singularity $y(y-x^k)=0$, and let $\widehat{N\!Z}_{R^{(2,2k)}}(t)$ denote its motivic Cohen–Lenstra zeta function. Here $k$ is a pos…

Let $f\in \mathbb C[z_1,\ldots,z_n]\setminus \mathbb C$ and let $Z_f^{\mathrm{mot}}(s)$ be its motivic zeta function. Let $b_f(s)$ denote the Bernstein–Sato polynomial, and let…

Generalized monodromy conjecture. The following assertions hold: (i) the image under $\operatorname{Exp}$ of the polar locus of the motivic zeta function of $F$ is $S_F$; and (ii)…

Let $f: X \longrightarrow \mathbb{C}$ be a non-constant regular function and let $x \in X$. Denote by $Z_{\textnormal{mot}, x}(f; s)$ the local motivic zeta function and by…

The exact motivic formula conjecture.

Let $X$ be a reduced curve over $\mathbb{F}_q$, and let $\widetilde X$ be its normalization. Let $\#_q$ denote the point-count homomorphism from the completed Grothendieck ring to…

Let $f\in\mathbb{C}[x_0,\ldots,x_n]$ be a nonconstant polynomial. Denote by $Z^{mot}_f(s)$ its motivic zeta function and by $b_f(s)$ its $b$-function. The strong monodromy conjectu…

Monodromy conjecture. Any pole of $Z_{\operatorname{top}}(f;s)$ provides an eigenvalue of the local monodromy action at some point of $f^{-1}(0)$.

Following Kapranov, let $X$ be a variety over a field $k$, and define its motivic zeta function by … where $\operatorname{Sym}^n X=X^n/\Sigma_n$. Let $\mathbb{L}:=[\mathbb{A}^1]$,…

Let $C$ be a rational planar projective reduced curve of arithmetic genus $\delta$, and let $C^{[n]}$ denote its Hilbert scheme of length-$n$ subschemes. Write…

Let $f$ define a hypersurface, let $x$ be a point in its domain, and let $Z_{f,x}(T)$ be the local motivic zeta function. Write ${\mathbb L}$ for the Lefschetz motive and let…

Let $k$ be a field of characteristic zero, let $X$ be an irreducible smooth $k$-variety, and let $f:X\to\mathbb{A}^1_k=\operatorname{Spec} k[t]$ be a dominant morphism. Write…

Let $k$ be a field of characteristic $0$, let $X$ be a smooth $k$-variety of pure dimension, and let $D=V(f)$ be the zero divisor of a $k$-morphism $f:X\to\mathbb A^1_k$. Let…