arXiv subjects

Simon Shuofeng Xu

Publications and source records attributed to Simon Shuofeng Xu.

2 recordsLinked to original sources

arXiv2025

Section conjectures over $\mathbb{C}$ and Kodaira fibrations

In this paper we propose and study topological and Hodge theoretic analogues of Grothendieck's section conjecture over the complex numbers. We study these questions in the context of family of curves, in particular Kodaira fibrations, and in the context of the family of Jacobians associated to a Kodaira fibration. We showed that in the case of family of curves, both the topological and Hodge-theoretic analogues of the injectivity part of the section conjecture holds, and that the topological analogue of the surjectivity part of the section conjecture does not hold in general for families of curves (proven in the appendix written by Lee and Serván) and families of Jacobians.

math.AG↗

arXiv2025

On exact sequences of Hodge theoretic fundamental groups

The goal of this paper is to first define a Hodge theoretic fundamental group for smooth connected complex algebraic varieties and then prove and study a right exact sequence of Hodge theoretic fundamental groups associated to a smooth projective family of algebraic varieties $f\colon X\to B$. In particular, we study when this right exact sequence is exact, relate this question to some prior results in non-abelian Hodge theory, and give an obstruction to splitting in terms of étale fundamental groups. The main examples we consider in this note is the universal curve $f\colon \mathcal{C}_g\to \mathcal{M}_g$ and the moduli space of degree $1$ line bundles on the universal curves $p \colon \text{Pic}^1_{\mathcal{C}_g/\mathcal{M}_g}\to \mathcal{M}_g$.

math.AG↗