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

TL;DR

This paper explores topological and Hodge theoretic versions of Grothendieck's section conjecture over complex numbers, focusing on families of curves and Jacobians, and establishes results on their injectivity and surjectivity properties.

Contribution

It introduces and analyzes analogues of the section conjecture in complex geometry, providing new insights into their validity for Kodaira fibrations and related families.

Findings

Topological and Hodge-theoretic analogues of injectivity hold for families of curves.

Surjectivity analogue does not hold in general for families of curves or Jacobians.

The paper includes a proof that the surjectivity analogue fails, written by Lee and Serván.

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.