Monodromy of the Hitchin Map over Hyperelliptic Curves

Monodromy of rank 2 twisted Hitchin systems and real character varieties

We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of $L$-twisted $G$-Higgs bundles, for the groups $G = GL(2,\mathbb{C})$, $SL(2,\mathbb{C})$ and $PSL(2,\mathbb{C})$. We also determine the twisted Chern class of the regular locus, which obstructs the existence of a section of the moduli space of $L$-twisted Higgs bundles of rank $2$ and degree $deg(L)+1$. By counting orbits of the monodromy action with $\mathbb{Z}_2$-coefficients, we obtain in a unified manner the number of components of the chara…

Chart description for hyperelliptic Lefschetz fibrations and their stabilization

Chart descriptions are a graphic method to describe monodromy representations of various topological objects. Here we introduce a chart description for hyperelliptic Lefschetz fibrations, and show that any hyperelliptic Lefschetz fibration can be stabilized by fiber-sum with certain basic Lefschetz fibrations.

Topology of representation spaces of surface groups in PSL(2,R) with assigned boundary monodromy and nonzero Euler number

In this paper we complete the topological description of the space of representations of the fundamental group of a punctured surface in SL(2,R) with prescribed behavior at the punctures and nonzero Euler number, following the strategy employed by Hitchin in the unpunctured case and exploiting Hitchin-Simpson correspondence between flat bundles and Higgs bundles in the parabolic case. This extends previous results by Boden-Yokogawa and Nasatyr-Steer. A relevant portion of the paper is intended to give an overview of the subject.

平移曲面分层中的单值核

Monodromy kernels for strata of translation surfaces

本文证明平移曲面分层中非超椭圆分量$\mathcal{H}(3,1)$和$\mathcal{H}^{nh}(4)$的单值映射核包含秩2非阿贝尔自由群，从而未能验证与映射类群的可公度性猜想。

Counting Dirac braid relators and hyperelliptic Lefschetz fibrations

We define a new invariant $w$ for hyperelliptic Lefschetz fibrations over closed oriented surfaces, which counts the number of Dirac braids included intrinsically in the monodromy, by using chart description introduced by the second author. As an application, we prove that two hyperelliptic Lefschetz fibrations of genus $g$ over a given base space are stably isomorphic if and only if they have the same numbers of singular fibers of each type and they have the same value of $w$ if $g$ is odd. We also give examples of pair of hyperelliptic Lefschetz fibrations with the same numbers of singular f…

拟希钦表示的熵与支配关系

Entropy and domination for quasi-Hitchin representations

针对亏格≥2的闭定向曲面，研究拟希钦表示的熵与支配关系，证明弯曲所得拟希钦表示的希尔伯特熵严格大于原希钦表示的熵，推广了此前有限层叠的相关结果。