Genus 2 mapping class groups are not Kahler

Orbifold Kähler Groups related to Mapping Class groups

We construct certain orbifold compactifications of the moduli stack of pointed stable curves over $\mathbb C$ and study their fundamental groups by means of their quantum representations. This enables to construct interesting Kähler groups and to settle most of the candidates for a counter-example to the Shafarevich conjecture on holomorphic convexity proposed in 1998 by Bogomolov and Katzarkov, using TQFT representations of the mapping class groups.

单连通凯勒流形的映射类群

Mapping Class Groups of Simply Connected Kähler Manifolds

本文综述并补充单连通紧凯勒流形映射类群与Torelli群的结果，重点分析P^4中超曲面情形，证明模空间基本群到映射类群的同态具有大核和无限指数像，并提出若干开放问题。

On the completion of the mapping class group of genus two

In this paper, we will study the Lie algebra of the prounipotent radical of the relative completion of the mapping class group of genus two. In particular, we will partially determine a minimal presentation of the Lie algebra by determining the generators and bounding the degree of the relations.

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。

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…

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 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.

Finiteness and infiniteness results for Torelli groups of (hyper-)Kähler manifolds

The Torelli group $\mathcal T(X)$ of a closed smooth manifold $X$ is the subgroup of the mapping class group $π_0(\mathrm{Diff}^+(X))$ consisting of elements which act trivially on the integral cohomology of $X$. In this note we give counterexamples to Theorem 3.4 of Verbitsky's paper "Mapping class group and a global Torelli theorem for hyperkähler manifolds" (Duke Math.~J.~162 (2013), no.~15, 2929-2986) which states that the Torelli group of simply connected Kähler manifolds of complex dimension $\ge 3$ is finite. This is done by constructing under some mild conditions homomorphisms $J: \mat…

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 …

具有非负Ricci曲率的完备Kähler流形上的Nevanlinna理论

Nevanlinna theory on complete Kähler manifolds with non-negative Ricci curvature

该研究在非负Ricci曲率的完备Kähler流形框架下发展亚纯映射的Nevanlinna等分布理论，得到带精细误差的第二基本定理、精确亏量关系，还推广了五值定理等多个唯一性结果。

Relative Pro-$\ell$ Completions of Mapping Class Groups

Fix a prime number ell. In this paper we develop the theory of relative pro-ell completion of discrete and profinite groups -- a natural generalization of the classical notion of pro-ell completion -- and show that the pro-ell completion of the Torelli group does not inject into the relative pro-ell completion of the corresponding mapping class group when the genus is at least 3. As an application, we prove that when g > 2, the action of the pro-ell completion of the Torelli group T_{g,1} on the pro-ell fundamental group of a pointed genus g surface is not faithful. The choice of a first-order…

Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges

We show that the stringy Kähler moduli space of a generic genus one curve of degree $N$, for $N\le 5$, is the $Γ_1(N)$ modular curve $X_1(N)$. This implies a correspondence between the cusps of the modular curves and certain large volume limits in the stringy Kähler moduli spaces of genus one fibered Calabi-Yau manifolds with $N$-sections. Using Higgs transitions in M-theory and F-theory as well as modular properties of the topological string partition function, we identify these large volume limits with elements of the Tate-Shafarevich group of the genus one fibration. Singular elements appea…

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.

Deformations of harmonic mappings and variation of the energy

We study the deformations of twisted harmonic maps $f$ with respect to the representation $ρ$. After constructing a continuous "universal" twisted harmonic map, we give a construction of every first order deformation of $f$ in terms of Hodge theory; we apply this result to the moduli space of reductive representations of a Kähler group, to show that the critical points of the energy functional $E$ coincide with the monodromy representations of polarized complex variations of Hodge structure. We then proceed to second order deformations, where obstructions arise; we investigate the existence of…