Section conjectures over $\mathbb{C}$ and Kodaira fibrations
Abstract
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 Serv\'{a}n) and families of Jacobians.
Keywords
Cite
@article{arxiv.2407.03248,
title = {Section conjectures over $\mathbb{C}$ and Kodaira fibrations},
author = {Simon Shuofeng Xu},
journal= {arXiv preprint arXiv:2407.03248},
year = {2025}
}
Comments
25 pages; v3 major revision in some parts of the paper (especially in section 3); right exactness of sequence of Hodge theoretic fundamental group now moved to arXiv:2503.13307; added a new appendix by Seraphina Lee and Carlos Serv\'{a}n which gives counterexamples to the surjectivity part of the top. section conjecture; the injectivity results remain unchanged