关于特征p>0的Grothendieck双有理Anabel猜想的Hom形式

代数闭域上任意特征曲线的双有理 Anabelian 重构定理

A Birational Anabelian Reconstruction Theorem for Curves over Algebraically Closed Fields in Arbitrary Characteristic

本文针对1维情形，证明了代数闭域上曲线的函数域可由其绝对Galois群与仅固定基域的更大域自同构群构成的群对恢复，推进了Bogomolov纲领。

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…

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.

正特征仿射Anabel曲线

Affine anabelian curves in positive characteristic

研究了拓扑上一致的态射，推广到了仿射双曲曲线的所有特征，并部分地重新证明了Tamagawa关于仿射双曲曲线的Anabel几何中的Grothendieck猜想的定理.该定理现在处理π^{tame}的曲线在一个π生成的领域和它的影响的集合上的同构。泛同胚是形式反演的。

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…

正特征映射的拓扑刚性及阿贝尔几何

Topological rigidity of maps in positive characteristic and anabelian geometry

研究正特征域上有限型整概形间的非常值映射对，通过添加纯拓扑判据推广玉川关于映射相差Frobenius幂次的等价性结果。

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…

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…

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。

伽罗瓦-蒂希米歇尔理论中的非交换视角

Anabelian perspectives in Galois-Teichmüller theory

本文探讨了非交换几何在伽罗瓦-蒂希米歇尔理论中的应用，通过算法框架重构绝对伽罗瓦群。

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…

Reconstructing function fields from rational quotients of mod-$\ell$ Galois groups

In this paper, we develop the main step in the global theory for the mod-$\ell$ analogue of Bogomolov's program in birational anabelian geometry for higher-dimensional function fields over algebraically closed fields. More precisely, we show how to reconstruct a function field $K$ of transcendence degree $\geq 5$ over an algebraically closed field, up-to inseparable extensions, from the mod-$\ell$ abelian-by-central Galois group of $K$ endowed with the collection of mod-$\ell$ rational quotients.

用Hilbert格式连接一类刚性双有理非同胚Calabi-Yau三重方程

Connecting certain rigid birational non-homeomorphic Calabi--Yau threefolds via Hilbert scheme

我们将给出一对双有理射影Calabi-Yau三重图，它们是刚性的，非同胚的，但在某些连通基方案上通过射影平坦变形连接。

具有完美剩余域的混合特征完备离散赋值域的绝对伽罗瓦群之间同态的几何性质

Geometric properties of homomorphisms between the absolute Galois groups of mixed-characteristic complete discrete valuation fields with perfect residue fields

本文将 Mochizuki 与 Hoshi 关于 $p$-进局部域绝对伽罗瓦群几何同态的判别条件推广到具有完美剩余域的一般混合特征完备离散赋值域，并得到弱 Isom 远阿贝尔结果。

Birational Motivic Homotopy Theories and the Slice Filtration

This paper is part of an endeavor to define an analogue of the slice filtration in the unstable motivic homotopy category. Our approach was inspired by the fact that the triangulated structures do not play a relevant role for the construction of birational homotopy categories as well as by the work of Kahn-Sujatha \cite{K-theory/0596} on birational motives, where the existence of a connection between the layers of the slice filtration and birational invariants is explicitly suggested. Our main result, shows that there is an equivalence of categories between the orthogonal components for the sl…

Fano hypersurfaces with no finite order birational automorphisms

We use the specialization homomorphism for the birational automorphism group to study finite order birational automorphisms. For a family of varieties over a DVR, we prove that a birational automorphism of order coprime to the residue characteristic cannot specialize to the identity. As an application, we show that very general $n$-dimensional hypersurfaces of degree $d \geq 5 \lceil (n+3)/6 \rceil$ have no finite order birational automorphisms.

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…

Birational morphisms and Poisson moduli spaces

We study birational morphisms between smooth projective surfaces that respect a given Poisson structure, with particular attention to induced birational maps between the (Poisson) moduli spaces of sheaves on those surfaces. In particular, to any birational morphism, we associate a corresponding "minimal lift" operation on sheaves of homological dimension <=1, and study its properties. In particular, we show that minimal lift induces a stratification of the moduli space of simple sheaves on the codomain by open subspaces of the moduli space of simple sheaves on the domain, compatibly with the i…

双有理簇的Chern类

Chern classes of birational varieties

Batyrev的一个定理断言，如果两个非奇簇V，W是双有理的，并且它们的典范丛在拉回到它们之间的双有理映射的不确定性的分解后一致，则V和W的贝蒂数重合。证明了在相同的假设条件下，V和W的全同调Chern类在归结的Chow群中是同类的推进类.例如，它遵循一个奇异簇的一个crepant分辨率的全陈类的推进是独立的分辨率。

稳定双有理不变量复形

Complexes of stable birational invariants

本文引入了一个新的稳定双有理不变量，以链复形同伦类型的形式分类化动机体积，并由此构造了回缩有理性的动机障碍及一类普遍的不变量。

New counterexamples to the birational Torelli theorem for Calabi--Yau manifolds

We produce counterexamples to the birational Torelli theorem for Calabi-Yau manifolds in arbitrarily high dimension: this is done by exhibiting a series of non birational pairs of Calabi-Yau $(n^2-1)$-folds which, for $n \geq 2$ even, admit an isometry between their middle cohomologies. These varieties also satisfy an $\mathbb L$-equivalence relation in the Grothendieck ring of varieties, i.e. the difference of their classes annihilates a power of the class of the affine line. We state this last property for a broader class of Calabi-Yau pairs, namely all those which are realized as pushforwar…

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.

MBM loci in families of hyperkahler manifolds and centers of birational contractions

An MBM class on a hyperkahler manifold M is a second cohomology class such that its orthogonal complement in H^2(M) contains a maximal dimensional face of the boundary of the Kahler cone for some hyperkahler deformation of M. An MBM curve is a rational curve in an MBM class and such that its local deformation space has minimal possible dimension 2n-2, where 2n is the complex dimension of M. We study the MBM loci, defined as the subvarieties covered by deformations of an MBM curve within M. When M is projective, MBM loci are centers of birational contractions. For each MBM class z, we consider …

Moduli spaces of fundamental groups of curves in positive characteristic I

In this series of papers, we investigate a new anabelian phenomenon of curves over algebraically closed fields of positive characteristic. Let $\overline M_{g, n}$ be the moduli space of curves of type $(g, n)$ over $\overline {\mathbb{F}}_p$. We introduce a topological space $\overline Π_{g, n}$ which can be determined group-theoretically from admissible fundamental groups of pointed stable curves of type $(g, n)$. By introducing a certain equivalence relation $\sim_{fe}$ on the underlying topological space $|\overline M_{g, n}|$ of $\overline M_{g, n}$, we obtain a topological space $\overli…

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。

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。

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。

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…

从其étale topos重构概形

Reconstruction of schemes from their étale topoi

本文对任意特征的无限域证明了Grothendieck关于将 $k$-概形映射至其étale topos构成完全忠实函子的猜想，实现了概形的重构并推广了Voevodsky的工作。

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 $U$.

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。

On homomorphisms between Cremona groups

We look at algebraic embeddings of the Cremona group in $n$ variables $Cr_n(C)$ to the group of birational transformations $Bir(M)$ of an algebraic variety $M$. First we study geometrical properties of an example of an embedding of $Cr_2(C)$ into $Cr_5(C)$ that is due to Gizatullin. In a second part, we give a full classification of all algebraic embeddings of $Cr_2(C)$ into $Bir(M)$, where $dim(M)=3$, and generalize this result partially to algebraic embeddings of $Cr_n(C)$ into $Bir(M)$, where $dim(M)=n+1$, for arbitrary $n\geq 2$. In particular, this yields a classification of all algebraic…

由projective geometry引起的光滑簇之间的极值压缩

Some extremal contractions between smooth varieties arising from projective geometry

我们构建明确的例子基本极值收缩，双有理和纤维型，从光滑的射影n维品种，n \geq 4，到光滑的射影品种，产生于经典的射影几何和定义在足够小的领域，不一定代数封闭。考虑的例子来自特定的特殊homaloidal和subhomaloidal线性系，这通常是退化的一般现象经典研究的博尔迪加，塞维里，托德，房间，法诺，森普尔和Tyrrell和最近的艾因和谢泼德-巴伦。第一系列的例子与P^m，3 \leq m \leq 5的特定余维2行列式光滑子变种相关联。通过考虑P^5中某些曲面的特殊三次超曲面，或P^7中某些具有一个明显二重点的三重曲面，我们得到了另一系列的例子。最后的例子来自一个有趣的双理性基本极值收缩在6维，研究了森普尔和泰瑞尔和充分描述的最后一节。

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。

A-infinity algebras associated with curves and rational functions on M_{g,g}. I

We consider the natural A-infinity structure on the Ext-algebra $Ext^*(G,G)$ associated with the coherent sheaf $G={\cal O}_C\oplus {\cal O}_{p_1}\oplus...\oplus {\cal O}_{p_n}$ on a smooth projective curve $C$, where $p_1,...,p_n\in C$ are distinct points. We study the homotopy class of the product $m_3$. Assuming that $h^0(p_1+...+p_n)=1$ we prove that $m_3$ is homotopic to zero if and only if $C$ is hyperelliptic and the points $p_i$ are Weierstrass points. In the latter case we show that $m_4$ is not homotopic to zero, provided the genus of $C$ is at least 2. In the case $n=g$ we prove tha…

Rational curves on hyperkahler manifolds

Let $M$ be an irreducible holomorphically symplectic manifold. We show that all faces of the Kahler cone of $M$ are hyperplanes $H_i$ orthogonal to certain homology classes, called monodromy birationally minimal (MBM) classes. Moreover, the Kahler cone is a connected component of a complement of the positive cone to the union of all $H_i$. We provide several characterizations of the MBM-classes. We show the invariance of MBM property by deformations, as long as the class in question stays of type (1,1). For hyperkahler manifolds with Picard group generated by a negative class $z$, we prove tha…

簇的Grothendieck环与特征处的数值不变量

Grothendieck ring of varieties and at-the-characteristic numerical invariants

本文在正特征弱奇点消解假设下，证明Hodge数等特征处数值不变量可由Grothendieck环同态给出，并回答了两个相关问题，附录中无条件证明低维双有理Calabi--Yau簇的Hodge数相同。