A Note on Hodge theoretic anabelian geometry

Grothendieck's anabelian conjectures predict that certain classes of varieties over number fields are largely determined by their {\'e}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 {\'e}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-$K(\pi,1)$ spaces replacing fundamental groups by homotopy types.