The T^1-lifting theorem in positive characteristic

正特征下射影簇的拟典范提升

Quasi-canonical lifting of projective varieties in positive characteristic

本文通过证明 Frobenius 可提升性的下降性质，给出了特征 $p>0$ 下允许在 Witt 向量上平坦提升并附带对数结构与 Frobenius 态射的光滑射影簇新类，并建立了 Mehta-Srinivas 典范提升存在性结果的精细形式。

Lifting iterated Frobenius over $W_n(k)$

Let $X$ be a smooth scheme over a perfect field $k$ of positive characteristic. M.V.~Nori and V.~Srinivas studied infinitesimal liftings of Frobenius morphism $F:X\to X$. Namely, given a Frobenius lifting $F_Y: Y\to Y$ over $W_{n-1}(k)$ the obstruction of lifting $F_Y$ over $W_n(k)$ is a class in $H^1(X, T_X\tensor B_1Ω^1_{X/k})$. We give a modified version of their theory and applications, in which we consider lifting $(n-1)$th iterated Frobenius $F^{n-1}:X\to X$ directly over $W_n(k)$.

Arithmetic Moduli and Lifting of Enriques Surfaces

We construct the moduli space of Enriques surfaces in positive characteristic and eventually over the integers, and determine its local and global structure. As an application, we show lifting of Enriques surfaces to characteristic zero. The key observation is that the canonical double cover of an Enriques surface is birational to the complete intersection of three quadrics in IP^5, even in characteristic 2.

Lifting Theorems and Smooth Profinite Groups

This work is motivated by the search for an "explicit" proof of the Bloch-Kato conjecture in Galois cohomology, proved by Voevodsky. Our concern here is to lay the foundation for a theory that, we believe, will lead to such a proof- and to further applications. Let p be a prime number. Let k be a perfect field of characteristic p. Let m be a positive integer. Our first goal is to provide a canonical process for "lifting" a module M, over the ring of Witt vectors $W_m(k)$ (of length m), to a $W_{m+1}(k)$-module, in a way that deeply respects Pontryagin duality. These are our big, medium and sma…

正特征理想到零特征理想的提升：曲面情况

Liftings of ideals in positive characteristic to those in characteristic zero:Surface case

本文引入了正特征到零特征的理想提升概念，通过骨架方法关联正特征与零特征的奇点不变量，并证明了光滑曲面与多理想对的log discrepancy集合离散，以及正特征中最小log discrepancy和log canonical阈值集合包含于零特征中。

Effective obstruction to lifting Tate classes from positive characteristic

We give an algorithm that takes a smooth hypersurface over a number field and computes a $p$-adic approximation of the obstruction map on the Tate classes of a finite reduction. This gives an upper bound on the "middle Picard number" of the hypersurface. The improvement over existing methods is that our method relies only on a single prime reduction and gives the possibility of cutting down on the dimension of Tate classes by two or more. The obstruction map comes from $p$-adic variational Hodge conjecture and we rely on the recent advancement by Bloch-Esnault-Kerz to interpret our bounds.

自由结合代数上的环面作用、提升与 Białynicki-Birula 型定理

Torus actions on free associative algebras, lifting and Białynicki-Birula type theorems

本文在自由结合代数中研究代数环面作用的线性化问题，证明了 Białynicki-Birula 型定理，并给出构造不可线性化作用的框架及其在结合消去猜想中的应用。

具有有限基本群的正特征簇提升的一个必要条件

A necessary condition for liftings of positive characteristic varieties with finite fundamental groups

研究具有有限基本群的正特征簇提升到特征零的必要条件，基于平展同伦理论和沃尔有限性障碍，通过定义模l有限支配性概念，证明簇提升的链复形条件，还扩展了沃尔有限性障碍讨论。

多项式系统的提升降低混合体积

Liftings of polynomial systems decreasing the mixed volume

本文针对多项式方程组混合体积上界未达到时同伦延拓求解器追踪冗余路径的问题，提出通过提升系统为等价系统以严格降低混合体积的改进策略，并给出了相关提升构造。

投影提升定理、复反射群与Fermat簇的拟Galois点

Projective Lifting Theorem, Complex Reflection Groups, and Quasi-Galois Points for Fermat Varieties

本文证明保持点投影的超曲面双有理自同构可唯一提升为射影变换，并利用复反射群理论确定了Fermat簇的Galois点与拟Galois点的精确数量。

Deformation data, Belyi maps, and the local lifting problem

We prove existence and nonexistence results for certain differential forms in positive characteristic, called {\em good deformation data}. Some of these results are obtained by reduction modulo $p$ of Belyi maps. As an application, we solve the local lifting problem for groups with Sylow $p$-subgroup of order $p$.

The local lifting problem for actions of finite groups on curves

Let $k$ be an algebraically closed field of characteristic $p > 0$. We study obstructions to lifting to characteristic 0 the faithful continuous action $ϕ$ of a finite group $G$ on $k[[t]]$. To each such $ϕ$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $ϕ$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic 0 in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H \subset G$. We determine for which $G$ the KGB obstruction of every $ϕ$ vanishes. We also consider anal…

关于热带三切线类的提升分区定理

A lifting partition theorem for tropical tritangent classes to smooth space sextic curves

本文研究了平滑空间六次曲线的热带三切线类的提升问题，证明了在一般情况下，只有六种可能的提升多重性分区源于三切线类的提升，且这些分区由合适的连通子复形的维度和特定组合类型热带切线的存在性决定。

Homotopical Adjoint Lifting Theorem

This paper provides a homotopical version of the adjoint lifting theorem in category theory, allowing for Quillen equivalences to be lifted from monoidal model categories to categories of algebras over colored operads. The generality of our approach allows us to simultaneously answer questions of rectification and of changing the base model category to a Quillen equivalent one. We work in the setting of colored operads, and we do not require them to be $Σ$-cofibrant. Special cases of our main theorem recover many known results regarding rectification and change of model category, as well as nu…

On the Birational Nature of Lifting

Let $X$ and $Y$ be proper birational varieties, say with only rational double points over a perfect field $k$ of positive characteristic. If $X$ lifts to $W_n(k)$, is it true that $Y$ has the same lifting property? This is true for smooth surfaces, but we show by example that this is false for smooth varieties in higher dimension, and for surfaces with canonical singularities. We also answer a stacky analogue of this question: given a canonical surface $X$ with minimal resolution $Y$ and stacky resolution $\mathcal{X}$, we characterize when liftability of $Y$ is equivalent to that of $\mathcal…

Kawamata-Viehweg Vanishing on Rational Surfaces in Positive Characteristic

We prove that the Kawamata-Viehweg vanishing theorem holds on rational surfaces in positive characteristic by means of the lifting property to W_2(k) of certain log pairs on smooth rational surfaces. As a corollary, the Kawamata-Viehweg vanishing theorem holds on log del Pezzo surfaces in positive characteristic.

Lifting Artin-Schreier covers with maximal wild monodromy

Let $k$ be an algebraically closed field of characteristic $p > 0$. We consider the problem of lifting $p$-cyclic covers of $\Proj_k$ as $p$-cyclic covers of the projective line over some DVR under the condition that the wild monodromy is maximal. We answer positively the question for covers birational to $w^ p-w=tR(t)$ for some additive polynomial $R(t)$.

An algorithm for lifting points in a tropical variety

The aim of this paper is to give a constructive proof of one of the basic theorems of tropical geometry: given a point on a tropical variety (defined using initial ideals), there exists a Puiseux-valued ``lift'' of this point in the algebraic variety. This theorem is so fundamental because it justifies why a tropical variety (defined combinatorially using initial ideals) carries information about algebraic varieties: it is the image of an algebraic variety over the Puiseux series under the valuation map. We have implemented the ``lifting algorithm'' using Singular and Gfan if the base field ar…

关于经典性定理及其在某些GL₂(Q_{p²})情况下的自守提升定理和Breuil-Mézard猜想中的应用

On the classicality theorem and its applications to the automorphy lifting theorem and the Breuil-M$\mathrm{\acute{e}}$zard conjecture in some $\mathrm{GL}_2(\mathbb{Q}_{p^2})$ cases

本文研究了某些 GL₂(Q_{p²}) 情况下的经典性定理及其应用，包括自守提升定理和 Breuil-Mézard 猜想。

Lifting in Frattini Covers and A Characterization of Finite Solvable Groups

We prove a lifting theorem for odd Frattini covers of finite groups. Using this, we characterize solvable groups and more generally p-solvable groups in terms of containing a triple of elements of distinct prime power orders with product 1. This is also related to various questions about Fried's modular tower program and properties of Hurwitz spaces of covers of curves.

Lifting zero-dimensional schemes and divided powers

We study divided power structures on finitely generated $k$-algebras, where $k$ is a field of positive characteristic $p$. As an application we show examples of $0$-dimensional Gorenstein $k$-schemes that do not lift to a fixed noetherian local ring of non-equal characteristic. We also show that Frobenius neighbourhoods of a singular point of a general hypersurface of large dimension have no liftings to mildly ramified rings of non-equal characteristic.

A generalization of Puiseux's theorem and lifting curves over invariants

Let $ρ: G \to \operatorname{GL}(V)$ be a rational representation of a reductive linear algebraic group $G$ defined over $\mathbb C$ on a finite dimensional complex vector space $V$. We show that, for any generic smooth (resp. $C^M$) curve $c : \mathbb R \to V // G$ in the categorical quotient $V // G$ (viewed as affine variety in some $\mathbb C^n$) and for any $t_0 \in \mathbb R$, there exists a positive integer $N$ such that $t \mapsto c(t_0 \pm (t-t_0)^N)$ allows a smooth (resp. $mathbb C^M$) lift to the representation space near $t_0$. ($C^M$ denotes the Denjoy--Carleman class associated w…

Oort groups and lifting problems

Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A_4 in characteristic 2. This proves one direction of a strong form of the Oort Conjecture.

正特征下的解析循环同调

Analytic cyclic homology in positive characteristic

本文在正特征情形下，利用向完备离散赋值环的 pro-dagger 代数提升及管形代数完备化，构造了一种解析循环同调理论，并证明其计算灵活性与三项核心形式性质。