A Note on Hodge theoretic anabelian geometry
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866908870029344768 |
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| author | Wang, Qixiang |
| author_facet | Wang, Qixiang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05968 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Note on Hodge theoretic anabelian geometry Wang, Qixiang Algebraic Geometry Differential Geometry Number Theory 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. |
| title | A Note on Hodge theoretic anabelian geometry |
| topic | Algebraic Geometry Differential Geometry Number Theory |
| url | https://arxiv.org/abs/2603.05968 |