Height pairing and nearby cycles

Abstract

We prove that, as was conjectured by Spencer Bloch, the Hodge period of some limit Hodge structures equals the height pairing of algebraic cycles on the resolution of singularities of the singular fiber.

The proof makes use of the compatibility of the Hodge realization of motivic sheaves with the 6 functors formalism. It was not fully verified in the existent literature, so the proof is conditional subject to that verification. I am grateful to the referee for this comment