Exotic torsion, Frobenius splitting and the slope spectral sequence

Frobenius-torsion 类群方案的 Iwasawa 理论

Iwasawa theory of Frobenius-torsion class group schemes

本文为特征 p 射影直线上几何 Z_p 塔中曲线雅可比簇的 Frobenius 核建立新 Iwasawa 理论，并证明相关主要猜想。

Frobenius gauges and a new theory of p-torsion sheaves in characteristic p. I

We develop a new cohomology theory in characteristic p>0, the so called F-gauge cohomology, a cohomology with values in the category of so-called F-gauges, which refines the cristalline cohomology. In this first paper we mainly discuss the theory for smooth projective varieties over perfect fields of charatceristic p. We also compare our theory with other existing structures like the F-zips of Moonen and Wedhorn and the displays of Langer and Zink.

The algebra of discrete torsion

We analyze the algebraic structures of G--Frobenius algebras which are the algebras associated to global group quotient objects. Here G is any finite group. These algebras turn out to be modules over the Drinfeld double of the group ring k[G]. We furthermore prove that discrete torsion is a universal group action of H^2(G,k^*) on G--Frobenius algebras by isomorphisms of the underlying linear structure. These morphisms are realized explicitly by taking the tensor product with twisted group rings. This gives an algebraic realization of discrete torsion and allows for a treatment analogous to the…

Frobenius splitting and ordinarity

We examine the relationship between the notion of Frobenius splitting and ordinarity for varieties. We show the following: a) The de Rham-Witt cohomology groups $H^i(X, W({\mathcal O}_X))$ of a smooth projective Frobenius split variety are finitely generated over $W(k)$. b) we provide counterexamples to a question of V. B. Mehta that Frobenius split varieties are ordinary or even Hodge-Witt. c) a Kummer $K3$ surface associated to an Abelian surface is $F$-split (ordinary) if and only if the associated Abelian surface is $F$-split (ordinary). d) for a $K3$-surface defined over a number field, t…

Dormant opers and Gauss maps in positive characteristic

The Gauss map of a given projective variety is the rational map that sends a smooth point to the tangent space at that point, considered as a point of the Grassmann variety. The present paper aims to generalize a result by H. Kaji on Gauss maps in positive characteristic and establish an interaction with the study of dormant opers, as well as Frobenius-projective structures. We first prove a correspondence between dormant opers on a smooth projective variety $X$ and closed immersions from $X$ into a projective space with purely inseparable Gauss map. By using this, we determine the subfields o…

正特征中的拟$F$-分裂本原辛簇

Quasi-$F$-split primitive symplectic varieties in positive characteristic

本文证明正特征中射影超凯勒簇的良好约化是拟$F$-分裂当且仅当Frobenius分裂，并推广到本原辛簇，给出Hodge-良性的开放性及Hilbert概型和广义Kummer簇的形变性质。