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<p>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.</p>

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Section conjectures over \(\mathbb {C}\) and Kodaira fibrations

  • 期刊论文

  • 发表日期：

    2026年3月21日

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  • Simon Shuofeng Xu,
  • Seraphina Eun Bi Lee,
  • Carlos A. Serván

摘要

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.