双有理Anabel猜想的$p$ Hom-形式

Galois theory and projective geometry

We explore connections between birational anabelian geometry and abstract projective geometry. One of the applications is a proof of a version of the birational section conjecture.

关于Anabel几何中的“截面猜想”

On the "Section Conjecture" in anabelian geometry

设X是域K上亏格>1的光滑射影曲线，域K是在有理数上生成的。格罗滕迪克的阿那贝尔几何中的截面猜想说，从X的算术基本群到K的绝对伽罗瓦群的正则投影的截面（直到共轭）与X的K-有理点一一对应。双有理变体在函数域K（X）的绝对伽罗瓦群代替基本群的情况下，得到类似的对应。本文证明了当K被p-adic数域所取代时，对所有X的双有理截口猜想。它否定了实或p-adic代数数域的两个命题。并给出了几乎任意域上X的K-有理点在双有理背景下诱导的截面的一个纯群论刻画。作为一个biproduct，我们得到伽罗瓦理论的标准，在一个以上的变量的多项式方程的根可解性，并为一个领域是PAC，是大的，或希尔伯特。

On some anabelian properties of arithmetic curves

In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicate to describe the position of decomposition groups of points at the boundary of the scheme $\Spec \caO_{K,S}$, where $K$ is a number field and $S$ a set of primes of $K$, intrinsically in terms of the fundamental group. We prove that it is equivalent to give the following pieces of information additionally with the fundamental group $π_1(\Spec \caO_{K,S})$: the location of decomposition groups of boundary p…

On the birational section conjecture with local conditions

A birationally liftable Galois section s of a hyperbolic curve X/k over a number field k yields an adelic point x(s) in the smooth completion of X. We show that x(s) is X-integral outside a set of places of Dirichlet density 0, or s is cuspidal. The proof relies on $GL_2(F_\ell)$-quotients of $π_1(U)$ for some open U of X. If k is totally real or imaginary quadratic, we prove that all birationally adelic, non-cuspidal Galois sections come from rational points as predicted by the section conjecture of anabelian geometry. As an aside we also obtain a strong approximation result for rational poin…

Some implications between Grothendieck's anabelian conjectures

Grothendieck gave two forms of his "main conjecture of anabelian geometry", i.e. the section conjecture and the hom conjecture. He stated that these two forms are equivalent and that if they hold for hyperbolic curves then they hold for elementary anabelian varieties too. We state a stronger form of Grothendieck's conjecture (equivalent in the case of curves) and prove that Grothendieck's statements hold for our form of the conjecture. We work with DM stacks, rather than schemes. If $X$ is a DM stack over a field $k\subseteq\mathbb{C}$ finitely generated over $\mathbb{Q}$, we prove that whethe…

Maximums of generalized Hasse-Witt invariants and their applications to anabelian geometry

Let $(X, D_{X})$ be an arbitrary pointed stable curve of topological type $(g_{X}, n_{X})$ over an algebraically closed field of characteristic $p>0$. We prove that the generalized Hasse-Witt invariants of prime-to-$p$ cyclic admissible coverings of $(X, D_{X})$ attain maximum. As applications, we obtain an anabelian formula for $(g_{X}, n_{X})$, and prove that the field structures associated to inertia subgroups of marked points can be reconstructed group-theoretically from open continuous homomorphisms of admissible fundamental groups. Moreover, the formula for maximum generalized Hasse-Witt…

关于Hodge理论的anabelian几何注记

A Note on Hodge theoretic anabelian geometry

本文提出一个基于Hodge理论的anabelian猜想，通过非阿贝尔Hodge理论中的ℂ×作用替代Galois作用，并证明了双曲曲线和复双曲流形的类比定理。

有限域上曲线的平展下降障碍与阿纳贝尔几何

Etale descent obstruction and anabelian geometry of curves over finite fields

本文通过建立平展基本群之间的态射与全局函数域上曲线的局部常值阿代尔点之间的双射，将Grothendieck的阿纳贝尔猜想与Sutherland和第二个作者的最新猜想联系起来，为阿纳贝尔猜想提供了进一步证据。

The motivic anabelian geometry of local heights on abelian varieties

We study the problem of describing local components of height functions on abelian varieties over characteristic $0$ local fields as functions on spaces of torsors under various realisations of a $2$-step unipotent motivic fundamental group naturally associated to the defining line bundle. To this end, we present three main theorems giving such a description in terms of the $\mathbb Q_\ell$- and $\mathbb Q_p$-pro-unipotent étale realisations when the base field is $p$-adic, and in terms of the $\mathbb R$-pro-unipotent Betti--de Rham realisation when the base field is archimedean. In the cours…

Universal birational invariants and $\mathbb{A}^1$-homology

Let $k$ be a field admitting a resolution of singularities. In this paper, we prove that the functor of zeroth $\mathbb{A}^1$-homology $\mathbf{H}^{\mathbb{A}^1}_0$ is universal as a functorial birational invariant of smooth proper $k$-varieties taking values in a category enriched by abelian groups. For a smooth proper $k$-variety $X$, we also prove that the dimension of $\mathbf{H}^{\mathbb{A}^1}_0(X;\mathbb{Q})(\mathrm{Spec} k)$ coincides with the number of $R$-equivalence classes of $X(k)$. We deduce these results as consequences of the structure theorem that for a smooth proper $k$-variet…

基本群概形的双有理不变性

The Birational Invariance Of Fundamental Group Schemes

本文建立了Tannakian范畴间自然同态成为同构的统一准则，并证明在完美域上光滑射影簇的双有理映射下，多种基本群概形（如S、N、EN等）均自然同构。

Burnside 环与带对数极点的体积形式

Burnside rings and volume forms with logarithmic poles

本文发展了带有对数体积形式的代数簇的双有理等价理论中的 Burnside 环，引入了留数同态，构造了双有理态射的加性不变量，并定义了特化同态。

双元导几何范畴

Categories derivees et geometrie birationnelle

代数簇上相干层的导出范畴最初是一种技术工具，在过去的二十年里，它已经成为代数簇双有理研究中的一个重要不变量。受到Kontsevich同调镜像对称猜想和Mori的极小模型程序的启发，双有理不变性和导出范畴的极小化问题已经出现。我们提出的主要结构和证明在3维和特定类别的触发器。

Quotients of groups of birational transformations of cubic Del Pezzo fibrations

We prove that the group of birational transformations of a Del Pezzo fibration of degree 3 over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order 2. As a consequence we also obtain that the Cremona group of rank 3 is not generated by birational maps preserving a rational fibration. Besides, its subgroup generated by all connected algebraic subgroups is a proper normal subgroup.

作为双有理对象的 $(\mathbb{P}^2, \Omega)$ 的万有镜像

A universal mirror to $(\mathbb{P}^2, Ω)$ as a birational object

本研究针对双有理几何中的 $(\mathbb{P}^2, \Omega)$ 构造万有镜像对象，证明其同调镜像对称，并揭示自同构与辛同胚的对应关系。

Factorization for stacks and boundary complexes

We prove a weak factorization result on birational maps of Deligne-Mumford stacks, and deduce the following: Let $U \subset X$ be an open embedding of smooth Deligne-Mumford stacks such that $D = X-U$ is a normal crossings divisor, then the the simple homotopy type of the boundary complex $Δ(X,D)$ depends only on $U$.

镜像六次双平面纤维化的三维簇

Threefolds Fibred by Mirror Sextic Double Planes

本文系统研究以镜像六次双平面K3曲面为纤维的三维簇，提出广义函数与同调不变量，构造Weierstrass形式及极小形式，并分析其奇异纤维、典范除子和Betti数。

高阶极点的单值群猜想与留数反导数的范数

The monodromy conjecture for high order poles, and norms of residue antiderivatives

该研究证明了n元复多项式的朴素动机zeta函数高阶极点的强局部单值群猜想，构造了对应本征闭链，其证明结合了拓扑技术与AI发现的双有理几何事实。

所有三角剖分都有一个共同的星形细分

All triangulations have a common stellar subdivision

本文解决了PL拓扑与双有理几何中的两个长期未决问题：证明了Oda强因子分解猜想的加权版本及双有理环面簇的共同迭代爆破，并证明了Alexander于1930年提出的关于PL同胚多面体具有共同星形细分的猜想。

Quadratic solutions of quadratic forms

We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$, defined over a field $K$, where the $x_i$ are themselves homogeneous polynomials of some degree $d$ in $r+1$ variables. Equivalently, we are looking at rational maps from projective $r$-space $P^r$ to a quadric hypersurface $Q$, defined over a field $K$. The space of maps of $P^1$ to a quadric $Q$ is stably birational to $Q$ if $d$ is even and to the orthogonal Grassmannian of lines in $Q$ if $d$ is odd. Most of the paper is devoted to obtaining similar descriptions for the spaces parametrizing maps of $P^2$ to quadr…