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

Simon Shuofeng Xu

TL;DR

This work develops complex-analytic analogues of Grothendieck's section conjecture via two invariant frameworks: the topological fundamental group and the category of graded-polarizable admissible $\mathbb{Z}$-VMHS. It provides injectivity results for both approaches in the setting of Kodaira fibrations and related families of Jacobians, showing that monodromy with no invariants yields injectivity, while surjectivity fails in general (including for abelianized sections). It also introduces a weak topological section conjecture and demonstrates counterexamples of surjectivity arising from Lee–Serván constructions, linking the existence of algebraic sections to splitting properties of topological exact sequences. On the Hodge-theoretic side, injectivity is established for families of curves using the canonical VMHS and period maps, illustrating that distinct algebraic sections are detected by their induced VMHS pullbacks. Together, these results illuminate how Kodaira fibrations mirror arithmetic phenomenon (finite algebraic sections, monodromy constraints) while highlighting the limits of both topological and Hodge-theoretic invariants for fully recapturing the section conjecture in the complex setting.

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 Serván) and families of Jacobians.

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

TL;DR

This work develops complex-analytic analogues of Grothendieck's section conjecture via two invariant frameworks: the topological fundamental group and the category of graded-polarizable admissible -VMHS. It provides injectivity results for both approaches in the setting of Kodaira fibrations and related families of Jacobians, showing that monodromy with no invariants yields injectivity, while surjectivity fails in general (including for abelianized sections). It also introduces a weak topological section conjecture and demonstrates counterexamples of surjectivity arising from Lee–Serván constructions, linking the existence of algebraic sections to splitting properties of topological exact sequences. On the Hodge-theoretic side, injectivity is established for families of curves using the canonical VMHS and period maps, illustrating that distinct algebraic sections are detected by their induced VMHS pullbacks. Together, these results illuminate how Kodaira fibrations mirror arithmetic phenomenon (finite algebraic sections, monodromy constraints) while highlighting the limits of both topological and Hodge-theoretic invariants for fully recapturing the section conjecture in the complex setting.

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 Serván) and families of Jacobians.
Paper Structure (10 sections, 33 theorems, 59 equations)

This paper contains 10 sections, 33 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Some construction of Kodaira fibrations
  3. Algebraic sections to family of Jacobians
  4. Family of Jacobians and injectivity
  5. Family of Jacobians and non-surjectivity
  6. Weak section conjecture and counterexample to surjectivity
  7. Hodge theoretic section question and family of curves
  8. Preliminary and formulation of the question
  9. Injectivity for family of curves
  10. Proof of Proposition \ref{['Prop: surjectivity of top. section conjecture']} part (2)

Key Result

Theorem 1.0.6

If $f:X\to B$ is a Kodaira fibration whose monodromy representation $\rho$ has no invariants, then $\sec_{\text{top}}$ is injective.

Theorems & Definitions (67)

  • Remark 1.0.1
  • Conjecture 1.0.2: Grothendieck's section conjecture
  • Remark 1.0.3
  • Definition 1.0.4: Kodaira fibration
  • Theorem 1.0.6: See Cor. \ref{['injectivity of top. section question']} in Section \ref{['section:family of Jacobians']}
  • Theorem 1.0.7
  • Proposition 1.0.8
  • Conjecture 1.0.9: Weak topological section conjecture
  • Proposition 1.0.10: see Cor. \ref{['Cor: weak=surjectivity for Kodaira fibrations']}
  • Remark 1.0.11
  • ...and 57 more