Hodge-Theoretic Anabelian Geometry

The anabelian phenomenon may be viewed as an arithmetic analogue of Mostow rigidity: it predicts that certain varieties can be reconstructed from their arithmetic fundamental groups. A celebrated result of S. Mochizuki shows that hyperbolic curves over p-adic fields exhibit this phenomenon.

Inspired by non-abelian Hodge theory, we introduce 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, hyperbolic Riemann surfaces are uniquely determined by their Hodge-theoretic fundamental groups, yielding a complex-analytic analogue of Mochizuki’s result. If time permits, we will discuss higher-dimensional generalizations.