Relative analytic reciprocity laws

Abstract

We study reciprocity laws involving complex line bundles on fibrations in oriented circles. In particularly, we prove the following reciprocity law. Let $B$ be a complex manifold and $\pi_i : M_i \to B$ be a fibration in oriented circles, where $i$ runs through a finite set. Let $L_i$ and $N_i$ be complex line bundles on every $M_i$. The reciprocity law states that the sum of all $(\pi_i)_* \left(c_1(L_i) \cup c_1(N_i) \right)$, where $(\pi_i)_*$ is the Gysin map and $c_1$ is the first Chern class, equals zero in $H^3(B, {\mathbb Z})$ when the disjoint union of all $M_i$ is embedded into a holomorphic family of compact Riemann surfaces over the base $B$ such that in every fiber of this family the disjoint union of the embedded circles is the boundary of an embedded compact Riemann surface with boundary, and all $L_i$ and all $N_i$ are restrictions of holomorphic line bundles on this family.