Smooth blowup square for motives with modulus

带模的动机，II：真模对的带转移模层

Motives with modulus, II: Modulus sheaves with transfers for proper modulus pairs

本研究构建了真模对范畴上的层与上同调理论，补充了此前[KMSY21]提出的非真模对范畴的相关理论，完善了带模动机的理论体系。

Nisnevich topology with modulus

In Voevodsky's theory of motives, the Nisnevich topology on smooth schemes is used as an important building block. In this paper, we introduce a Grothendieck topology on proper modulus pairs, which will be used to construct a non-homotopy invariant generalization of motives. We also prove that the topology satisfies similar properties to the Nisnevich topology.

Nori motives of curves with modulus and Laumon 1-motives

Let $k$ be a number field. We describe the category of Laumon 1-isomotives over $k$ as the universal category in the sense of Nori associated with a quiver representation built out of smooth proper $k$-curves with two disjoint effective divisors and a notion of $H^1_\dR$ for such "curves with modulus". This result extends and relies on the theorem of J. Ayoub and L. Barbieri-Viale that describes Deligne's category of 1-isomotives in terms of Nori's Abelian category of motives.

带模动机的固体实现

Solid realization of motives with modulus

本文构造了带模动机的固体实现函子，并利用pro-固体比较定理计算了特定开浸入锥的固体实现，通过形式完备化给出显式描述。

Motives with modulus

We construct and study a triangulated category of motives with modulus $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ over a field $k$ that extends Voevodsky's category $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ in such a way as to encompass non-homotopy invariant phenomena. In a similar way as $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ is constructed out of smooth $k$-varieties, $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ is constructed out of \emph{proper modulus pairs}, that is, pairs of a proper $k$-variety $X$ and an effective divisor $D$ on $X$ such that $X \setminus |D|$ is smooth. To a mo…

Ordinarity of Local Galois Representation Arising from Dwork Motives

In this paper, we prove that for suitably chosen Dwork motives, the local Galois representation arising from middle cohomology of fibers over a point with $p$-adic valuation $<0$ on the base is regular and ordinary. The result will be crucial in the forthcoming work Potential Automorphy for $GL_n$. The proof consists of the construction of a semistable blowup and a use of Hyodo-Kato's log crystalline cohomology theory.

带有界极点的Hochschild和循环同调

Hochschild and cyclic Homologies with bounded poles

本文通过为模对构造Hochschild与循环同调，证明了其在带模动机范畴中的可表示性，并建立了模版HKR定理及模循环同调的可表示性。

一般化θ除子上的混合Hodge结构与图动机

Mixed Hodge Structures on Generalized Theta Divisors and Graph Motives

本文研究一般化θ除子上的混合Hodge结构，并探讨其与图动机的可能联系。

Semi-purity for cycles with modulus

In this paper, we prove a form of purity property for the $(\mathbb{P}^1, \infty)$-invariant replacement $h_0^{\overline{\square}}(\mathfrak{X})$ of the Yoneda object $\mathbb{Z}_{\rm tr} (\mathfrak{X})$ for a modulus pair $\mathfrak{X}=(\overline{X}, X_\infty)$ over a field $k$, consisting of a smooth projective $k$-scheme and an effective Cartier divisor on it. As application, we prove the analogue in the modulus setting of Voevodsky's fundamental theorem on the homotopy invariance of the cohomology of homotopy invariant sheaves with transfers, based on a main result of "Purity of reciprocit…

带模的 Hodge-Witt 层的爆破不变性

Blow-up invariance for Hodge-Witt sheaves with modulus

本文证明带模的 Hodge-Witt 层在爆破下保持不变，并据此在奇点消解假设下建立其在带模动机范畴中的可表性。

Albanese varieties with modulus over a perfect field

Let X be a smooth proper variety over a perfect field k of arbitrary characteristic. Let D be an effective divisor on X with multiplicity. We introduce an Albanese variety Alb(X, D) of X of modulus D as a higher dimensional analogon of the generalized Jacobian of Rosenlicht-Serre with modulus for smooth proper curves. Basing on duality of 1-motives with unipotent part (which are introduced here), we obtain explicit and functorial descriptions of these generalized Albanese varieties and their dual functors. We define a relative Chow group of zero cycles w.r.t. the modulus D and show that Alb(X,…

Dynamical Mordell-Lang and Automorphisms of Blow-ups

We show that if $ϕ: X \to X$ is an automorphism of a smooth projective variety and $D \subset X$ is an irreducible divisor for which the set of $d$ in $D$ with $ϕ^n(d)$ in $D$ for some nonzero $n$ is not Zariski dense, then $(X, ϕ)$ admits an equivariant rational fibration to a curve. As a consequence, we show that certain blowups (e.g. blowups in high codimension) do not alter the finiteness of $\textrm{Aut}(X)$, extending results of Bayraktar-Cantat. We also generalize results of Arnol'd on the growth of multiplicities of the intersection of a variety with the iterates of some other variety …