Arithmetic properties of equivariant birational types

Symbols and equivariant birational geometry in small dimensions

We discuss the equivariant Burnside group and related new invariants in equivariant birational geometry, with a special emphasis on applications in low dimensions.

等变双有理类型与导出范畴

Equivariant birational types and derived categories

本文从导出范畴的视角研究了有理曲面和三维簇的等变双有理几何。

Witt环、Pfister型与等变双有理几何

Witt rings, Pfister forms, and equivariant birational geometry

该研究针对带有限群作用的二次型的等变双有理几何，发展了Witt群与Pfister理论的等变类似物，拓展了双有理几何的研究范畴。

三次四维流形的等变双有理几何与导出范畴

Equivariant birational geometry of cubic fourfolds and derived categories

本文从导出范畴角度研究等变双有理性，构造了有限群在光滑三次四维流形上非线性化但稳定线性化的作用实例。

等变双有理几何中的线性化概念

Linearizability notions in equivariant birational geometry

本文讨论了有限群作用在代数簇上的双有理性质、线性化、torsors和versality。

Equivariant birational geometry of quintic del Pezzo surface

In this paper we prove that there are exactly two $G$-minimal surfaces which are $G$-birational to the quintic del Pezzo surface, where $G \cong C_5 \rtimes C_4$. These surfaces are the quintic del Pezzo surface itself and the surface $\mathbb{P}^1 \times \mathbb{P}^1$.

Burnside groups and orbifold invariants of birational maps

We construct new invariants of equivariant birational isomorphisms taking values in equivariant Burnside groups.

$\mathbb{P}^3$ 的一个 $1$-上同调双曲双有理映射，具有超越算术度

A $1$-cohomologically hyperbolic birational map of $\mathbb{P}^3$, with a transcendental arithmetic degree

构造 $\mathbb{P}^3$ 上 $1$-上同调双曲双有理映射，其第一动力次数与在代数闭包点上的算术度均为超越数。

Birational smooth minimal models have equal Hodge numbers in all dimensions

This is a resume of the author's talk at the Worhshop on Arithmetic, Geometry and Physics around Calabi-Yau Varieties and Mirror Symmetry (July 23-29, 2001), the Fields Institute. The aim of this note is to prove that birational smooth minimal models over C have equal Hodge numbers in all dimensions by an arithmetic method. Our method is a refinement of the method of V. Batyrev and C.-L. Wang on Betti numbers who used p-adic integration and the Weil conjecture. Our ingredient is to use further arithmetic results such as the Chebotarev density theorem and p-adic Hodge theory.

Equivariant pliability of the projective space

We classify finite subgroups $G\subset\mathrm{PGL}_4(\mathbb{C})$ such that $\mathbb{P}^3$ is not $G$-birational to conic bundles and del Pezzo fibrations, and explicitly describe all $G$-Mori fibre spaces that are $G$-birational to $\mathbb{P}^3$ for these subgroups.

Birational geometry for number theorists

Awfully idiosyncratic lecture notes from CMI summer school in arithmetic geometry July 31-August 4, 2006. Does not include: rationality problems, techniques of the minimal model problem and much of the rest. Includes: Lecture 0: geometry and arithmetic of curves Lecture 1: Kodaira dimension and properties, rational connectendess, Lang's and Campana's conjectures. Lecture 2: Campana's program; Campana constellations framed in terms of b-divisors, to allow for a definition of Kodaira dimension directly on the base. A speculative notion of firmaments which may deserve further investigation, espec…

On the birational invariance of the arithmetic genus and Euler characteristic

The aim of this note is to use elementary methods to give a large class of examples of projective varieties $ Y \subseteq \mathbb{P}^d_k$ over a field $k$ with the property that $Y$ is not isomorphic to a hypersurface $H\subseteq \mathbb{P}^N_k$ in projective space $\mathbb{P}^N_k$ with $N:=dim(Y)+1$. We apply this construction to the study of the arithmetic genus $p_a(Y)$ of $Y$ and the problem of determining if $p_a(Y)$ is a birational invariant of $Y$ in general. We give an infinite number of examples of pairs of projective varieties $(Y, Y')$ in any dimension $dim(Y)=dim(Y')\geq 4$ where $…

Toric Fano varieties and birational morphisms

In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have $0\leqρ_X-ρ_D\leq 3$, for the difference of the Picard numbers of X and D. Moreover, if $ρ_X-ρ_D>0$ (with some additional hypotheses if $ρ_X-ρ_D=1$), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have $ρ_X\leq 9$. In the second part of the paper, we study equivariant birational morphisms f whose source is Fano. We give some general results, and in dimension …

Equivariant Burnside groups and toric varieties

We study $G$-equivariant birational geometry of toric varieties, where $G$ is a finite group.

环面轨形的双有理分类

Birational classification of toric orbifolds

本文借助相伴堆叠扇，对平凡一般稳定子的光滑真环面 Deligne–Mumford 堆叠的环面等变双有理等价类给出完整分类。

Birational maps and Nori motives

In this note, we sketch an approach to the problems of equivariant birational geometry developed by M. Kontsevich and Yu. Tschinkel, where Burnside invariants were introduced. We are making explicit the role of Nori constructions in this environment.

一种研究$G$-等变双有理不变量的Gromov-Witten方法

A Gromov-Witten approach to $G$-equivariant birational invariants

该研究依托21世纪初的几何论证建立Chen-Ruan上同调与若干$G$-双有理不变量的关联并给出应用，结合原子理论将推动等变原子理论构建，是双有理与光滑不变量关联研究计划的延伸。

Birational transformations of weighted graphs

We introduce the notion of a standard weighted graph and show that every weighted graph has an essentially unique standard model. Moreover we classify birational transformations between such models. Our central result shows that these are composed of elementary transformations. The latter ones are defined similarly to the well known elementary transformations of ruled surfaces. In a forthcoming paper, we apply these results in the geometric setup to obtain standard equivariant completions of affine surfaces with an action of certain algebraic groups. We show that these completions are unique u…

Epsilon constants and equivariant Arakelov Euler characteristics

We study equivariant Arakelov-Euler characteristics of hermitian sheaves on arithmetic varieties which support a tame action by a finite group G. The tameness of the group action allows us to produce an equivariant Arakelov-Euler characteristic in a particularly fine "projective" arithmetic class group. We then show that the equivariant Arakelov-Euler characteristics of various complexes of differentials determine the epsilon constants of the L-functions of the motives obtained from the arithmetic variety using symplectic representations of the group G. Our results may be viewed firstly as a h…

等价极小模型程序的进展及其在复动力学和算术动力学中的应用

Advances in the equivariant minimal model program and their applications in complex and arithmetic dynamics

本文探讨了非同构可缩映射的等价极小模型程序进展及其在复动力学和算术动力学中的应用。

On the birational geometry of toric Fano 4-folds

We announce a factorization result for equivariant birational morphisms between toric 4-folds whose source is Fano: such a morphism is always a composite of blow-ups along smooth invariant centers. Moreover, we show with a counterexample that, differently from the 3-dimensional case, even if both source and target are Fano, the intermediate varieties can not be chosen Fano.

Combinatorial Burnside groups

We study the structure of combinatorial Burnside groups, which receive equivariant birational invariants of actions of finite groups on algebraic varieties.

Applications of an Equivariant Etale Cohomology to Arithmetic Topology

A. Sikora has shown results which confirm the analogy between number fields and 3-manifolds. However, he has given proofs of his results which are very different in the arithmetic and in the topological case. In this paper, we show how to provide a unified approach to the results in the two cases. For this we introduce an equivariant cohomology which satisfies a localization theorem. In particular, we obtain a satisfactory explanation for the coincidences between Sikora's formulas which leads us to clarify and to extend the dictionary of arithmetic topology.

Arithmetic dynamics on smooth cubic surfaces

We study dynamical systems induced by birational automorphisms on smooth cubic surfaces defined over a number field $K$. In particular we are interested in the product of non-commuting birational Geiser involutions of the cubic surface. We present results describing the sets of $K$ and $\bar{K}$-periodic points of the system, and give a necessary and sufficient condition for a dynamical local-global property called strong residual periodicity. Finally, we give a dynamical result relating to the Mordell--Weil problem on cubic surfaces.

过滤强等变 $\mathcal{D}^{\dagger}$-模的算术柏原正则性与轨道分类

Arithmetic Kashiwara Regularity and Orbit Classification for Filtered Strongly Equivariant $\mathcal{D}^{\dagger}$-Modules

本文证明形式旗流形上过滤强等变 Berthelot 算术 $\mathscr D^\dagger$-模的柏原正则性的算术类比，并通过轨道和等变 $F$-等晶体分类简单 Frobenius 过滤强等变相干算术 $\mathscr D^\dagger$-模。

例外类型中nilpotent轨道覆盖的birational诱导

Birational induction of nilpotent orbit covers in exceptional types

本文研究了例外类型中nilpotent轨道覆盖的birational诱导问题，确定了唯一birationally刚性的诱导数据。

orbifold上全纯挠的渐近性与等变算术Hilbert-Samuel定理

Asymptotics of holomorphic torsion on orbifolds and an equivariant arithmetic Hilbert-Samuel theorem

本文计算orbifold与等变挠的正线丛p次幂全纯挠渐近展开的log p部分，基于此推导Arakelov几何中的等变算术Hilbert-Samuel定理。

An arithmetic Lefschetz-Riemann-Roch theorem. With an appendix by Xiaonan Ma

In this article, we consider regular projective arithmetic schemes in the context of Arakelov geometry, any of which is endowed with an action of the diagonalisable group scheme associated to a finite cyclic group and with an equivariant very ample invertible sheaf. For any equivariant morphism between such arithmetic schemes, which is smooth over the generic fibre, we define a direct image map between corresponding higher equivariant arithmetic K-groups and we discuss its transitivity property. Then we use the localization sequence of higher arithmetic K-groups and the higher arithmetic conce…

Pfaffians, the G-Signature Theorem and Galois de Rham discriminants

We study equivariant de Rham discriminants associated to arithmetic varieties which support a tame action by a finite group; we form these discriminants by endowing the de Rham cohomology with pairings arising from duality theory. Such equivariant discriminants are shown to break up naturally into a metric part and a signature part. In a previous paper we described the equivariant Arakelov discriminants, obtained by endowing the equivariant determinant of de Rham cohomology with various metrics. In this paper we study the associated equivariant signature information; in particular, we show tha…

Localization theorem for higher arithmetic K-theory

Quillen's localization theorem is well known as a fundamental theorem in the study of algebraic K-theory. In this paper, we present its arithmetic analogue for the equivariant K-theory of arithmetic schemes, which are endowed with an action of certain diagonalisable group scheme. This equivariant arithmetic K-theory is defined by means of a natural extension of Burgos-Wang's simplicial description of Beilinson's regulator map to the equivariant case. As a byproduct of this work, we give an analytic refinement of the Riemann-Roch theorem for higher equivariant algebraic K-theory. And as an appl…

两个二次曲面交线上线性空间的算术与双有理性质

Arithmetic and birational properties of linear spaces on intersections of two quadrics

该文研究非闭域上两二次曲面光滑交线的线性空间Fano概形有理性，通过二次曲面束双曲约化证明其与对称幂双有理等价，给出不同维数下的有理性结论及实数域判别准则，推广了前人结果。

非常一般对称Verra四重态等变无理性的研究

Equivariant irrationality of very general symmetric Verra fourfolds

利用Katzarkov-Kontsevich-Pantev-Yu引入的原子理论及其等变版本，证明非常一般的复对称Verra四重态不与P^4在Z/2-双有理等价。

某些有理自映射的周期点与算术度

Periodic points and arithmetic degrees of certain rational self-maps

本文研究代数数域上的上同调双曲双有理自映射，给出其周期点高度的有界性结果，还将二维有理自映射的Kawaguchi--Silverman猜想归约为动力Mordell--Lang猜想，验证了若干新情形并证明对应Zariski稠密轨道存在。

中间雅可比和伯恩赛德不变量

Intermediate Jacobians and Burnside invariants

本文提出结合等变中间雅可比和伯恩赛德形式化方法的新不变量，用于研究具有有限群作用的光滑有理连通三维流形。

从算术拓扑角度看降阶理论与等变范畴

An Arithmetic Topology viewpoint on Descent theory and Equivariant Categories

本文通过算术拓扑视角，统一了降阶理论与等变范畴，建立了算术与拓扑不变量的类比，并展示了其在范畴层面的应用。

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…

Kummer quartic double solids

We study equivariant birational geometry of (rational) quartic double solids ramified over (singular) Kummer surfaces.

Equivariant derived equivalence and rational points on K3 surfaces

We study arithmetic properties of derived equivalent K3 surfaces over the field of Laurent power series, using the equivariant geometry of K3 surfaces with cyclic groups actions.

Hyperbolicity of varieties of log general type

These notes provide an overview of various notions of hyperbolicity for varieties of log general type from the viewpoint of both arithmetic and birational geometry. The main results are based on our paper entitled "Hyperbolicity and uniformity of varieties of log general type." They are expanded notes from a minicourse the authors gave as part of the Geometry and arithmetic of orbifolds workshop at UQÁM.

Cohomology of finite subgroups of the plane Cremona group

An equivariant stable birational invariant of an action of a finite group on a smooth projective variety is the first cohomology group of the Picard module. Bogomolov-Prokhorov and Shinder computed this for actions of cyclic groups on rational surfaces, with maximal stabilizers, in terms of the geometry of the fixed point locus. Using the Brauer group of the quotient stack, we extend the computation to more general actions and relate it to the equivariant Burnside group formalism.

On the periods of motives with complex multiplication and a conjecture of Gross-Deligne

We prove that the existence of an automorphism of finite order on a (defined over a number field) variety X implies the existence of algebraic linear relations between the logarithm of certain periods of X and the logarithm of special values of the Gamma-function. This implies that a slight variation of results by Anderson, Colmez and Gross on the periods of CM abelian varieties is valid for a larger class of CM motives. In particular, we prove a weak form of the period conjecture of Gross-Deligne. Our proof relies on the arithmetic fixed point formula (equivariant arithmetic Riemann-Roch theo…

The moduli of curves of genus 6 and K3 surfaces

We prove that the coarse moduli space of curves of genus 6 is birational to an arithmetic quotient of a bounded symmetric domain of type IV by giving a period map to the moduli space of some lattice-polarized K3 surfaces.

D^\dagger-affinity of formal models of flag varieties

Let G be the group of L-rational points of a connected split reductive group over a finite extension L of Q_p. We show that formal models of the algebraic flag variety X of G are D-affine for certain sheaves of arithmetic differential operators. We then introduce the category of coadmissible G-equivariant arithmetic D-modules on the system of formal models of X and prove that it is anti-equivalent to the category of admissible locally L-analytic G-representations with trivial infinitesimal character. We compute the equivariant arithmetic D-modules of certain classes of representations.

Divisorial instability and Vojta's Main Conjecture for $\mathbb{Q}$-Fano varieties

We study Diophantine arithmetic properties of birational divisors in conjunction with concepts that surround $\mathrm{K}$-stability for Fano varieties. There is also an interpretation in terms of the barycentres of Newton-Okounkov bodies. Our main results show how the notion of divisorial instability, in the sense of K. Fujita, implies instances of Vojta's Main Conjecture for Fano varieties. A main tool in the proof of these results is an arithmetic form of Cartan's Second Main Theorem that has been obtained by M. Ru and P. Vojta.

链的摇摆模空间、等变多重性和$\mathrm{U}(n_0,n_1)$-Higgs丛

Wobbly moduli of chains, equivariant multiplicities and $\mathrm{U}(n_0,n_1)$-Higgs bundles

本文通过引入$\U(n_0,n_1)$-wobbliness概念，研究Higgs丛固定点区域的摇摆性，并提出可计算的判据。

A fixed point formula of Lefschetz type in Arakelov geometry II: a residue formula

This is the second of a series of papers dealing with an analog in Arakelov geometry of the holomorphic Lefschetz fixed point formula. We use the main result of the first paper to prove a residue formula "`a la Bott" for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; recent results of Bismut-Goette on the equivariant (Ray-Singer) analytic torsion play a key role in the proof.