On the completion of the mapping class group of genus two

Mult-Projective Models for Jacobian Varieties of Genus Two Curves

A set of multi-homogeneous equations for the Jacobian of a genus two curve is given. The approach used is to write down affine equations for the Jacobian minus various tranlations of the Theta-divisor by [2]-division points, and then to write down affine glue equations for the overlaps. Taking multi-projective completions for all of these then yields a complete multi-projective two-dimensional variety whose points are in one-to-one correspondence with degree zero divisor classes on the curve (i.e. the Picard group). This multi-projective variety then becomes a complete projective two-dimension…

Completions of mapping class groups and the cycle $C - C^-$

In this paper we study the proalgebraic completion of mapping class relative to their maps to the symplectic group. The main result is that the natural map from the unipotent (a.k.a. Malcev) completion of the Torelli group to the prounipotent radical of the Sp_g completion of the mapping class group is a non trivial central extension with kernel isomorphic to Q, at least when g \ge 8. The theorem is proved by relating the central extension to the line bundle associated to the archemidean height of the cycle C - C- in the Jacobian of the curve C. We also develop some of the basic theory of rela…

Genus 3 Mapping Class Groups are not Kahler

In this paper we prove that finite index subgroups of genus 3 mapping class and Torelli groups that contain the group generated by Dehn twists on bounding simple closed curves are not Kahler. These results are deduced from explicit presentations of the unipotent (aka, Malcev) completion of genus 3 Torelli groups and of the relative completions of genus 3 mapping class groups. The main results follow from the fact that these presentations are not quadratic. To complete the picture, we compute presentations of completed Torelli and mapping class in genera > 3; they are quadratic. We also show th…

The Riemann-Hilbert mapping for $\mathfrak{sl}_2$ -systems over genus two curves

We prove in two different ways that the monodromy map from the space of irreducible $\mathfrak{sl}_2$-differential-systems on genus two Riemann surfaces, towards the character variety of $\mathrm{SL}_2$-representations of the fundamental group, is a local diffeomorphism. This is motivated by a question raised by Étienne Ghys about Margulis' problem: existence of curves of negative Euler characteristic in compact quotients of $\mathrm{SL}_2(\mathbb{C})$.

Mapping Class Groups of Trigonal Loci

In this paper we study the topology of the stack $\mathcal{T}_g$ of smooth trigonal curves of genus g, over the complex field. We make use of a construction by the first named author and Vistoli, that describes $\mathcal{T}_g$ as a quotient stack of the complement of the discriminant. This allows us to use techniques developed by the second named author to give presentations of the orbifold fundamental group of $\mathcal{T}_g$, of its substrata with prescribed Maroni invariant and describe their relation with the mapping class group $\mathcal{M}ap_g$ of Riemann surfaces of genus g.

Fano-Mukai fourfolds of genus $10$ as compactifications of $\mathbb{C}^4$

It is known that the moduli space of smooth Fano-Mukai fourfolds $V_{18}$ of genus $10$ has dimension one. We show that any such fourfold is a completion of $\mathbb{C}^4$ in two different ways. Up to isomorphism, there is a unique fourfold $V_{18}^{\mathrm s}$ acted upon by $\operatorname{SL}_2(\mathbb{C})$. The group $\operatorname{Aut}(V_{18}^{\mathrm s})$ is a semidirect product $\operatorname{GL}_2(\mathbb{C})\rtimes(\mathbb{Z}/2\mathbb{Z})$. Furthermore, $V_{18}^{\mathrm s}$ is a $\operatorname{GL}_2(\mathbb{C})$-equivariant completion of $\mathbb{C}^4$, and as well of $\operatorname{GL}…

Operads of genus zero curves and the Grothendieck-Teichmüller group

We show that the group of homotopy automorphisms of the profinite completion of the genus zero surface operad is isomorphic to the (profinite) Grothendieck-Teichmüller group. Using a result of Drummond-Cole, we deduce that the Grothendieck-Teichmüller group acts nontrivially on $\overline{\mathcal{M}}_{0,\bullet+1}$, the operad of stable curves of genus zero. As a second application, we give an alternative proof that the framed little 2-disks operad is formal.

Relative Weight Filtrations on Completions of Mapping Class Groups

This paper gives an exposition of relative weight filtrations on completions of mapping class groups associated to a stable degeneration of marked genus g curves. These relative weight filtrations have been constructed using Galois theory (with Matsumoto) and Hodge theory (with Pearlstein and Terasoma). It is shown that the level 0 part of the relative weight filtration is an analogue of a parabolic subalgebra of a Kac-Moody Lie algebra. It is shown that all such subalgebras correspond to equivalence classes of pants decompositions of the surface -- two being equivalent if and only if they det…

Congruence topologies on the mapping class group

Let $Γ(S)$ be the pure mapping class group of a connected orientable surface $S$ of negative Euler characteristic. For ${\mathscr C}$ a class of finite groups, let $\hatπ_1(S)^{\mathscr C}$ be the pro-${\mathscr C}$ completion of the fundamental group of $S$. The \emph{${\mathscr C}$-congruence completion $\checkΓ(S)^{\mathscr C}$ of $Γ(S)$} is the profinite completion induced by the embedding $Γ(S)\hookrightarrow{\operatorname{Out}}(\hatπ_1(S)^{\mathscr C})$. In this paper, we begin a systematic study of such completions for different ${\mathscr C}$. We show that the combinatorial structure o…

曲线模空间中相对完备化与分离边界除子

Relative completions and separating boundary divisors in moduli of curves

本文研究曲线模空间基本群的相对完备化，证明特定分离除子可给出相同完备化，并给出所需除子数量的下界。

亏格1纤维化K3曲面的模空间与周期映射

Moduli spaces and period mappings of genus one fibered K3 surfaces

本文构建了多种亏格1纤维化K3曲面的模空间，建立其与局部对称空间的周期映射并计算轨形基本群，相关结果异于极化K3曲面模空间情形，可应用于K3曲面的光滑映射类群研究。

Mapping class groups of surfaces of genus $\geq 3$ do not virtually surject to $\mathbb{Z}$

We prove a well known conjecture of Nikolai Ivanov which states that if $X$ is a surface of genus $\geq 3$ (with any number of punctures and boundary components), $\rm{Mod}(X)$ is the mapping class group of $X$, and $K < \rm{Mod}(X)$ is a finite-index subgroup, then $K$ does not virtually surject to $\mathbb{Z}$. As a corollary of this we get that $H_1(Z; \mathbb{Q}) = 0$ whenever $Z$ is a finite cover of $\mathcal{M}_{g,n}$, the moduli space of complex algebraic curves of genus $g\geq 3$ with $n$ marked points.

Homomorphisms between mapping class groups

Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$ has genus $g\geq 6$ and $Y$ has genus at most $2g-1$; in addition, suppose that $Y$ is not closed if it has genus $2g-1$. Our main result asserts that every non-trivial homomorphism $\Map(X) \to \Map(Y)$ is induced by an {\em embedding}, i.e. a combination of forgetting punctures, deleting boundary components and subsurface embeddings. In particular, if $X$ has no boundary then every non-trivial endomorphism $\Map(X)\to\Map(X)$ is in fact an isomorphism. As an application of our main theorem we obtain that, under the s…

Algebraic isomonodromic deformations and the mapping class group

The germ of the universal isomonodromic deformation of a logarithmic connection on a stable n-pointed genus g curve always exists in the analytic category. The first part of this paper investigates under which conditions it is the analytic germification of an algebraic isomonodromic deformation. Up to some minor technical conditions, this turns out to be the case if and only if the monodromy of the connection has finite orbit under the action of the mapping class group. The second part of this paper studies the dynamics of this action in the particular case of reducible rank 2 representations …

相对完备化的幂单Chabauty-Kim-Kantor方法

The Unipotent Chabauty-Kim-Kantor Method for Relative Completions

该研究基于Kantor的相对完备化思路，开发了新的p进解析方法，验证了相关维数不等式，得到Faltings与Siegel定理的新条件证明，还取得多项关键技术贡献。

关于超椭圆映射类群的注记

Notes on hyperelliptic mapping class groups

本文系统研究超椭圆映射类群，给出Putman-Wieland猜想在亏格2情形的反例，并将同余子群性质推广到一般超椭圆映射类群，确定其profinite完备化中多重扭转和开子群的中心化子。

Torsion Elements in the Mapping Class Group of a Surface

Given a finite set of $r$ points in a closed surface of genus $g$, we consider the torsion elements in the mapping class group of the surface leaving the finite set invariant. We show that the torsion elements generate the mapping class group if and only if $(g, r) \neq (2, 5k+4)$ for some integer $k$.

A Geometric Interpretation of the χ-{y} Genus on Hyper-Kahler Manifolds

The group SL(2) acts on the space of cohomology groups of any hyper-Kahler manifold X. The χ_{y} genus of a hyper-Kahler X is shown to have a geometric interpretation as the super trace of an element of SL(2). As a by product one learns that the generalized Casson invariant for a mapping torus is essentially the χ_{y} genus.

The Hodge de Rham theory of relative Malcev completion

The Hodge de Rham theory of relative Malcev completion is developed in this paper. In the special case where one takes the corresponding reductive group to be trivial, one recovers Chen's de Rham theory of the fundamental group and the corresponding Hodge theory due to Morgan and the author. This work is a principal technical tool in the author's work on the mapping class groups.

同构于拼接裤复形的共轭完成的扩展映射类群的自同构

Automorphisms of the procongruence pants complex

该研究证明了扩展映射类群的共轭完成的自同构均为内自同构，并展示了其与拼接裤复形自同构群的同构关系。

Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces

We study several geometric and group theoretical problems related to Kodaira fibrations, to more general families of Riemann surfaces, and to surface-by-surface groups. First we provide constraints on Kodaira fibrations that fiber in more than two distinct ways, addressing a question by Catanese and Salter about their existence. Then we show that if the fundamental group of a surface bundle over a surface is a ${\rm CAT}(0)$ group, the bundle must have injective monodromy (unless the monodromy has finite image). Finally, given a family of closed Riemann surfaces (of genus $\ge 2$) with injecti…

无理 pencils 与通过基本群的 profinite 完成来刻画与积同构的代数簇

Irrational pencils, and characterization of Varieties isogenous to a product, via the Profinite completion of the Fundamental group

本文通过基本群的 profinite 完成，刻画了与积同构的代数簇，并证明了无理 pencil 与紧 Kahler 流形的对应关系。