Conjecture de Bloch et nombres de Milnor

非交换相交理论与幂幺 Deligne–Milnor 公式

Non-commutative intersection theory and unipotent Deligne-Milnor formula

本文利用导出与非交换代数几何范畴化 Bloch 相交数，并结合非交换 Chern 特征，在新情形下推广 Bloch 导子猜想，建立幂幺 Deligne–Milnor 公式。

Rationality problems and conjectures of Milnor and Bloch-Kato

We show how the techniques of Voevodsky's proof of the Milnor conjecture and the Voevodsky- Rost proof of its generalization the Bloch-Kato conjecture can be used to study counterexamples to the classical Lüroth problem. By generalizing a method due to Peyre, we produce for any prime number l and any integer n >= 2, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree n unramified étale cohomology class with l-torsion coefficients. When l = 2, the varieties that are constructed are furthermore unirational and non-rationality cannot be detec…

On extending Soulé's variant of Bloch-Quillen identification

Based on Balmer's tensor triangular Chow group [2], we propose (Milnor)K-theoretic Chow groups of derived categories of schemes. These Milnor K-theoretic Chow groups recover the classical ones [6] for smooth projective varieties and can detect nilpotent, while the classical ones can't do. As an application, we extend Soulé's variant of Bloch-Quilled identification from smooth projective varieties to their trivial infinitesimal thickenings. This answers affirmatively a question by Green-Griffiths for trivial deformations, see Question 1.1 below.

Deligne-Milnor猜想的证明

Proof of the Deligne-Milnor conjecture

本文运用导出与非交换代数几何方法研究算术概形上的分歧现象，证明了Deligne-Milnor猜想，并在纯特征情形下推广了Bloch导体猜想。

Infinitesimal part of Weak Lefschetz using Milnor $K-$ theory

Let $X$ be a smooth projective variety over a field of characteristic $0$ and $L$ an ample divisor. In this paper we study the Weak Lefschetz conjecture for Chow groups using the technique employed by Grothendieck in his study of the problem for Picard groups, and using Bloch's formula to interpret Chow groups in terms of Milnor $K$- theory we prove the infinitesimal part of the conjecture.

Pro-representability of $K^M$-cohomology in weight 3 generalizing a result of Bloch

We generalize a result, on the pro-representability of Milnor $K$-cohomology groups at the identity, that's due to Bloch. In particular, we prove, for $X$ a smooth, proper, and geometrically connected variety defined over an algebraic field extension $k/\mathbb{Q}$, that the functor \[\mathscr{T}_{X}^{i,3}(A)=\ker\left(H^i(X_A,\mathcal{K}_{3,X_A}^M)\rightarrow H^i(X,\mathcal{K}_{3,X}^M)\right),\] defined on Artin local $k$-algebras $(A,\mathfrak{m}_A)$ with $A/\mathfrak{m}_A\cong k$, is pro-representable provided that certain Hodge numbers of $X$ vanish.