We show that there are no non-trivial stratified bundles over a smooth simply connected quasi-projective variety over the algebraic closure of a finite field, if the variety admits a normal projective compactification with boundary locus of codimension $\ge 2$. In the appendix, various strong forms of the Lefschetz LEF are proven. Final version, to appear in Compositio.
Convergent isocrystals on simply connected varieties
de Jong证明了：若$X$是特征为$p>0$且具有平凡基本群的代数闭域$k$上的连通光滑射影簇，则$X/W$上的任何等晶都是平凡的.我们在两个额外的假设下证明了这个猜想。版本2：主要的变化是一个增编。证明了若$X$是特征$p>0$具有平凡基本群的代数闭域$k$上的连通光滑投影簇，则$X/W$上的任何无穷小等晶都是平凡的.为了这个目的，我们写了一些一般的事实，这样的无穷小isocrystals超过W这是失踪的文献。
On rational chain connectedness of globally +-regular varieties
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；Michigan State University（密歇根州立大学）；Peking University（北京大学）
In this paper we describe a construction method for numerical Godeaux surfaces based on homological algebra. We show the existence of an 8-dimensional locally complete family of simply connected numerical Godeaux surfaces.
Voisin constructed a series of examples concerning simply connected compact Kähler manifolds of even dimensions, which do not have the rational homotopy type of a complex projective manifold starting from dimension six. In this note, we prove that Voisin's examples of dimension four also does not have the rational homotopy type of a complex projective manifold. Oguiso constructed simply connected compact Kähler manifolds starting from dimension four, which can not deform to a complex projective manifold under a small deformation. We also prove that Oguiso's examples do not have the rational ho…
We use birational geometry to show that the existence of rational points on proper rationally connected varieties over fields of characteristic $0$ is a consequence of the existence of rational points on terminal Fano varieties. We discuss several consequences of this result, especially in relation to the $C_1$-conjecture. We also provide evidence that supports the conjecture in dimension $3$ for $C_1$ fields of characteristic $0$.
We study zero cycles on rationally connected varieties defined over characteristic zero Laurent fields with algebraically closed residue fields. We show that the degree map induces an isomorphism for rationally connected threefolds defined over such fields. In general, the degree map is an isomorphism if rationally connected varieties defined over algebraically closed fields of characteristic zero satisfy the integral Hodge/Tate conjecture for one cycles, or if the Tate conjecture is true for divisor classes on surfaces defined over finite fields. To prove these results, we introduce technique…
We study the prime-to-$p$ Hecke action on the projective limit of the sets of connected components of Shimura varieties with fixed parahoric or Bruhat--Tits level at $p$. In particular, we construct infinitely many Shimura varieties for CM unitary groups in odd variables for which the considering actions are not transitive. We prove this result by giving negative examples on the question of Bruhat--Colliot-Thélène--Sansuc--Tits or its variant, which is related to the weak approximation on tori over $\mathbb{Q}$.
