Cyclic p-groups and semi-stable reduction of curves in equal characteristic p>0

塔中形变循环覆盖

Deforming cyclic covers in towers

本文证明了在等特征情形下，特征 p>0 曲线上的循环覆盖可在塔中提升，采用 Hurwitz 树技术和 Obus-Wewers 的工具，肯定回答了 Saïdi 的问题。

On the stability of tangent bundles on cyclic coverings

Let $Y$ be a smooth projective surface defined over an algebraically closed field $k$ with ${\rm Char}\ k\nmid n$, and let $π:X\rightarrow Y$ be a $n$-cyclic covering branched along a smooth divisor $B$. We show that under some conditions $\mathscr{T}_X$ is semi-stable with respect to $π^*\mathcal {H}$ if the tangent bundle $\mathscr{T}_Y$ is semi-stable with respect to some ample line bundle $\mathcal {H}$ on $Y$.

Cyclic Length in the Tame Brauer Group of the Function Field of a p-Adic Curve

Let $F$ be the function field of a smooth curve over the $p$-adic number field $\Q_p$. We show that for each prime-to-$p$ number $n$ the $n$-torsion subgroup $\H^2(F,μ_n)={}_n\Br(F)$ is generated by $\Z/n$-cyclic classes; in fact the $\Z/n$-length is equal to two. It follows that the Brauer dimension of $F$ is two (first proved in \cite{Sa97}), and any $F$-division algebra of period $n$ and index $n^2$ is decomposable.

Mass formulas and stringy point-count for semi-direct products of tame abelian groups and wild symmetric or cyclic groups in characteristic three

In positive characteristic, there exist counterexamples to the statement corresponding to Batyrev's theorem concerning the McKay correspondence. In this paper, we give another computation of the counterexamples by using stringy-point count for some sequences of quotient varieties. There is a proof of the Serre-Bhagava's mass formula, which is important formula in the number theory, from the computation of stringy-point count of a quotient variety associated to a representation of symmetric group. Our computation of the stringy-point count in this paper is regarded as a variation of this proof …

循环群与循环群半直积扩张的上同调核

Cohomological Kernels for Cyclic by Cyclic Semi-Direct Product Extensions

本文针对 Galois 闭包为循环群半直积的域扩张，利用 Positelski 的工具建立六项正合序列以刻画 Galois 上同调核，并涵盖相对 Brauer 群情形。

Derived categories of cyclic covers and their branch divisors

Given a variety $Y$ with a rectangular Lefschetz decomposition of its derived category, we consider a degree $n$ cyclic cover $X \to Y$ ramified over a divisor $Z \subset Y$. We construct semiorthogonal decompositions of $\mathrm{D^b}(X)$ and $\mathrm{D^b}(Z)$ with distinguished components $\mathcal{A}_X$ and $\mathcal{A}_Z$, and prove the equivariant category of $\mathcal{A}_X$ (with respect to an action of the $n$-th roots of unity) admits a semiorthogonal decomposition into $n-1$ copies of $\mathcal{A}_Z$. As examples we consider quartic double solids, Gushel-Mukai varieties, and cyclic cub…

Moment curves and cyclic symmetry for positive Grassmannians

We show that for each k and n, the cyclic shift map on the complex Grassmannian Gr(k,n) has exactly $\binom{n}{k}$ fixed points. There is a unique totally nonnegative fixed point, given by taking n equally spaced points on the trigonometric moment curve (if k is odd) or the symmetric moment curve (if k is even). We introduce a parameter q, and show that the fixed points of a q-deformation of the cyclic shift map are precisely the critical points of the mirror-symmetric superpotential $\mathcal{F}_q$ on Gr(k,n). This follows from results of Rietsch about the quantum cohomology ring of Gr(k,n). …

循环开-闭映射与霍奇结构的变分

The cyclic open--closed map and variations of Hodge structures

本文在半正情形下为体形变的相对Fukaya范畴构造循环开-闭映射，证明其是半无限霍奇结构极化变分的态射，并给出同构判据，从而在Calabi-Yau镜像对中由同调镜像对称推出枚举镜像对称。

乘法群与循环群乘积的分类空间的 Chow-Witt 环

Chow-Witt Rings of Classifying Spaces of Products of Multiplicative and Cyclic Groups

本文基于 di Lorenzo 和 Mantovani 的策略，计算了单位根分类空间及其与乘法群乘积的全 Chow-Witt 环，并进一步确定了射影空间乘积和 BG_m × BG_m 的全 I^j-上同调与 Chow-Witt 环。