arXiv subjects

Xiyan Zhong

Publications and source records attributed to Xiyan Zhong.

5 recordsLinked to original sources

arXiv2026

Abelianization of Symmetric Mapping Class Groups

Let $\widetilde{S}\to S$ be an unbranched regular $p$-fold cyclic cover of a closed orientable surface $S$ of genus $g$. Two natural groups are associated to this cover. The first is the centralizer $\mathrm{Mod}(\widetilde{S},σ)$ of a chosen generator $σ$ of the deck transformation group in $\mathrm{Mod}(\widetilde{S})$. The second is the finite-index subgroup $\mathrm{Mod}(S,[β])$ of $\mathrm{Mod}(S)$ consisting of mapping classes that fix the nonzero class $[β]\in H_1(S;\mathbb Z/p\mathbb Z)$ corresponding to the cover. For $p=2$, Sato computed the abelianizations of these groups. We compute their abelianizations for every odd prime $p$ and show that they exhibit a splitting phenomenon different from the case $p=2$. We also construct explicit abelianization maps using Morita's crossed homomorphism and the Prym representation.

math.GT↗

arXiv2026

Rigidity of the period map up to finite covers

We first give a complete classification of bi-affine representations of mapping class groups of surfaces with finitely many boundary components or punctures. We also show that every linear representation of the mapping class group of a genus-$g$ surface with two boundary components of dimension at most $3g-3$ is bi-affine. We then classify low-dimensional symplectic representations of the mapping class group associated to triple covers. Let $[β]\in H_1(S_g;\mathbb{Z}/3\mathbb{Z})^*$, and let $\widetilde{S}\to S_g$ be the corresponding triple cover with deck transformation $σ$. For $h\le g$, every non-abelian homomorphism from either $\mathrm{Mod}(S_g,[β])$, the stabilizer of $[β]$ in $\mathrm{Mod}(S_g)$, or $\mathrm{Mod}(\widetilde{S},σ)$, the centralizer of $σ$ in $\mathrm{Mod}(\widetilde{S})$, to $\mathrm{Sp}_{2h}(\mathbb{Z})$ is, up to conjugation, the standard symplectic representation on $H_1(S_g;\mathbb{Z})$. As an application, we obtain a rigidity theorem for holomorphic maps from the moduli space $R_g^{(3)}$ of genus-$g$ curves equipped with a $3$-sheeted (unbranched) normal covering to the moduli space $\mathcal{A}_h$ of $h$-dimensional principally polarized abelian varieties. We prove that, for $g\ge 6$ and $h\le g$, the unique nonconstant holomorphic map from $R_g^{(3)}$, equipped with either of its two natural complex-orbifold structures, to $\mathcal{A}_h$ is the period map sending a cover $Y\to X$ to the Jacobian of the base curve $X$.

math.GT↗

arXiv2026

Stable cohomology of universal character varieties

We study the universal PGL_n character variety over M_g whose fiber over a point [C] is the space of PGL_n-local systems on the curve C. We use nonabelian Hodge theory and properties of Saito's mixed Hodge modules to show that the Leray-Serre spectral sequence for the projection to M_g degenerates at E_2. As an application, we prove that the rational cohomology of these varieties stabilizes as g goes to infinity and compute the stable limit. We also deduce similar results for the universal G-character variety over M_{g,1} whose fiber over a punctured curve is the variety of G-local systems with fixed central monodromy around the puncture, for G = GL_n or SL_n. The paper concludes with a computation of the ring structure on the stable cohomology and with an appendix by Anne Larsen and Mirko Mauri proving related results for the intersection cohomology of singular universal character varieties.

math.AG↗

arXiv2026

Linear representations of the mapping class group of dimension at most $3g-3$

We classify representations of the mapping class group of a surface of genus $g$ (with at most one puncture or boundary component) up to dimension $3g-3$. Any such representation is the direct sum of a representation in dimension $2g$ or $2g+1$ (given as the action on the (co)homology of the surface or its unit tangent bundle) with a trivial representation. As a corollary, any linear system on the moduli space of Riemann surfaces of genus $g$ in this range is of algebro-geometric origin.

math.GT↗

arXiv2025

Prym Representations and Twisted Cohomology of the Mapping Class Group with Level Structures

We compute the twisted cohomology of the mapping class group with level structures, with coefficients in the $r$-tensor powers of the Prym representations for any positive integer $r$. When $r\ge 2$, we show that the cohomology exhibits instability for large genus, whereas it remains stable for $r=0$ or $r=1$. As a corollary, we prove that the symplectic Prym representation associated with any finite abelian regular cover of a non-closed finite-type surface is infinitesimally rigid.

math.GT↗