Relative analytic reciprocity laws

TL;DR

This paper establishes a reciprocity law involving complex line bundles on fibrations in oriented circles, demonstrating a sum of certain pushforward classes vanishes in cohomology under specific geometric conditions.

Contribution

It proves a new reciprocity law relating Chern classes of line bundles on fibrations in circles within holomorphic families of Riemann surfaces.

Findings

Sum of pushforward Chern class cup products equals zero in cohomology.

Reciprocity law holds when the union of fibrations embeds into a holomorphic family with boundary conditions.

The law applies to complex line bundles restricted from a family of Riemann surfaces.

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.