18 problems

Monodromy conjecture. The filtration $M_{\bullet}$ is the monodromy filtration. This conjecture was known in the equicharacteristic case by Deligne and in a surface case by Rapopor…

Log $l$-adic relative monodromy-weight conjecture. If $\overset{\circ}{X}$ is projective over $\overset{\circ}{S}$, then $P$ is the monodromy filtration of $N$ relative to…

Log $l$-adic relative monodromy-weight conjecture. If $\overset{\circ}{X}$ is projective over $\kappa$, then the relative monodromy filtration on the cohomology relative to…

Kato's log $l$-adic monodromy-weight conjecture. If $\overset{\circ}{X}$ is projective over $\kappa$, then, for every $e\in{\mathbb Z}_{\geq 1}$, $N$ induces an isomorphism

The paper considers the category of smooth log smooth projective $S$-schemes and seeks a description of the category of resolvable motives using only these schemes. The proposed co…

Let $\Psi$ be the nearby-cycle object with filtrations $F$ and $G$, let $\mathbb{D}$ denote duality, and let $b$ be the isomorphism relating the dual nearby cycles to…

Let $f:X\to X'$ be a morphism between the strictly semi-stable schemes in the paper, with induced morphism $f_\eta$ on generic fibres, and let $D_I\subset X$ and…

Let $X/S$ be as in the paper's setup, with $overset{\circ}{X}\longrightarrow\overset{\circ}{S}_1$ projective, and let $P$ denote the filtration introduced in the explanation for…

Let $C$ be a complex integral projective curve whose normalization has genus zero and whose unique planar singularity is locally given by $f(x,y)=0$. Let…

Monodromy–Frobenius weight conjecture. For every such $s$ and every $m\in\mathbb{Z}$, the monodromy filtration on the stalk of $H^m(X)_{\ell}$ coincides with the Frobenius weight f…

Let $L_{10}=P_{10,425}$ and let $\mathcal{W}_{12}$ denote the weight-12 component. $L_{10}$ weight decomposition conjecture. The period $L_{10}$ mixes weights $12$ through $14$, an…

For $n\geq3$, let $L_{2n}$ denote the ladder graph period. Ladder weight-mixing conjecture. … This extrapolates from the cases $L_6$ and $L_8$. It is a prediction about the weight…

Let $G$ be a primitive log-divergent $\phi^4$ graph with loop number $h_G$ and $c_2(G)=0$. Suppose $P(G)$ is mixed Tate. Refined weight-mixing conjecture. Either $P(G)$ has pure we…

Weight bound conjecture. The amplitude integrand satisfies

Let $X$ be a smooth quasi-projective variety over $\overline{\mathbb Q}$, and let $j$ and $k$ be integers. Let $H^k_{\rm dRB}(X,\mathbb Q(j))_{\rm Gr}$ denote the Grothendieck clas…

Let $X$ be regular, let $W_0$ be the sum of generalized $\phi$-eigenspaces whose eigenvalues are roots of unity, and let $\operatorname{sp}$ be the étale specialization map. Weight…

Let $X$ be regular, and let $W_{i-1}$ denote the indicated weight filtration on the $p$-adic étale cohomology groups of the special and geometric generic fibres. Let…

Let $DM^{eff}_{gm}$ be Voevodsky's category of effective geometric motives, and let $MM$ denote the category of mixed motives, assumed to be the heart of a motivic $t$-structure on…