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A Note on Hodge theoretic anabelian geometry

Qixiang Wang

Abstract

Grothendieck's anabelian conjectures predict that certain classes of varieties over number fields are largely determined by their {é}tale fundamental groups. A theorem of Mochizuki shows that for hyperbolic curves over number fields or $p$-adic fields, dominant morphisms bijectively correspond to open homomorphisms between their {é}tale fundamental groups. Motivated by non-abelian Hodge theory, we formulate a Hodge-theoretic version of the anabelian conjecture in which the Galois action is replaced by the natural $\mathbb{C}^\times$-action on the pro-algebraic completion of the fundamental group arising from non-abelian Hodge theory. In particular, we prove a Hodge-theoretic analog of Mochizuki's theorem for smooth projective hyperbolic curves over $\mathbb{C}$. We also obtain a higher-dimensional analogue for complex hyperbolic manifolds of ball quotient type and discuss possible extensions to non-$K(π,1)$ spaces replacing fundamental groups by homotopy types.

A Note on Hodge theoretic anabelian geometry

Abstract

Grothendieck's anabelian conjectures predict that certain classes of varieties over number fields are largely determined by their {é}tale fundamental groups. A theorem of Mochizuki shows that for hyperbolic curves over number fields or -adic fields, dominant morphisms bijectively correspond to open homomorphisms between their {é}tale fundamental groups. Motivated by non-abelian Hodge theory, we formulate a Hodge-theoretic version of the anabelian conjecture in which the Galois action is replaced by the natural -action on the pro-algebraic completion of the fundamental group arising from non-abelian Hodge theory. In particular, we prove a Hodge-theoretic analog of Mochizuki's theorem for smooth projective hyperbolic curves over . We also obtain a higher-dimensional analogue for complex hyperbolic manifolds of ball quotient type and discuss possible extensions to non- spaces replacing fundamental groups by homotopy types.
Paper Structure (8 sections, 12 theorems, 36 equations)

This paper contains 8 sections, 12 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Non-abelian Hodge theory and the $\mathbb{C}^*$-action
  3. Recollection of non-abelian Hodge theory
  4. $\mathbb{C}^*$-action on fundamental groups
  5. A Hodge-theoretic version of anabelian geometry
  6. The case of hyperbolic curves
  7. Higher-dimensional anabelian geometry
  8. Anabelian phenomenon for homotopy type

Key Result

Theorem 1.0.1

Let $X$ and $Y$ be compact hyperbolic manifolds with $\dim X,\dim Y>2$. Then $\pi_1(X)\simeq \pi_1(Y)$ if and only if $X\simeq Y$ as Riemannian manifolds. Moreover,

Theorems & Definitions (24)

  • Theorem 1.0.1: Mostow rigidity
  • Theorem 1.0.2: anabelian
  • Theorem 1.0.3: \ref{['anabelian for curve']},\ref{['ball quotient anabelian']}
  • Theorem 1.0.4
  • Theorem 1.0.5: Schmidt_2016*Theorem 1.2
  • Conjecture 1.0.6
  • Theorem 2.1.1: PMIHES_1992__75__5_0
  • Proposition 2.1.2: PMIHES_1992__75__5_0
  • Example 2.1.3
  • Example 2.1.4: Simpsonyangmill*Proposition 9.1, Proposition 9.8
  • ...and 14 more