Hodge-Theoretic Anabelian Geometry

Grothendieck’s anabelian program asks to what extent an algebraic variety can be reconstructed from its fundamental group together with additional structure. It may be viewed as an arithmetic analogue of rigidity phenomena in geometry: for instance, Mostow rigidity says that finite-volume hyperbolic manifolds of real dimension greater than two are determined by their fundamental groups.

A celebrated result of S. Mochizuki shows that hyperbolic curves over p-adic fields can be recovered from their arithmetic fundamental groups: every isomorphism between such groups comes from a unique isomorphism of the curves. This is surprising from the viewpoint of complex geometry, where the naive analogue fails: compact Riemann surfaces of the same genus have isomorphic topological fundamental groups, but need not be biholomorphic.

Inspired by non-abelian Hodge theory, we propose a Hodge-theoretic analogue of arithmetic fundamental groups for complex Kähler manifolds, and show that an anabelian phenomenon occurs in complex-analytic geometry. In particular, we show that hyperbolic Riemann surfaces are determined by their Hodge-theoretic fundamental groups. We will also discuss some higher-dimensional speculations if time permits.