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Relative analytic reciprocity laws

Denis V. Osipov

TL;DR

The paper extends reciprocity laws to the setting of complex line bundles on fibrations in oriented circles by introducing a relative analytic reciprocity law under a star condition. It shows that when the disjoint union of fibers embeds into a holomorphic family of compact Riemann surfaces with boundary and bundles extend holomorphically, a global vanishing identity holds in $$H^3(B, \mathbb{Z})$$. Holomorphic families are then shown to automatically satisfy the star condition, enabling the reciprocity to hold in that holomorphic context; a corollary yields vanishing results for determinant gerbes (12 times the determinant gerbe) in $$H^3(B, \mathbb{Z})$$, connecting to Deligne pairing and topological Riemann–Roch. Overall, the work links local-to-global data through Gysin pushforwards, Chern classes, and Deligne-type pairings in a geometrically rich setting.

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 $π_i : M_i \to B$ 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 $(π_i)_* \left(c_1(L_i) \cup c_1(N_i) \right)$, where $(π_i)_*$ 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.

Relative analytic reciprocity laws

TL;DR

The paper extends reciprocity laws to the setting of complex line bundles on fibrations in oriented circles by introducing a relative analytic reciprocity law under a star condition. It shows that when the disjoint union of fibers embeds into a holomorphic family of compact Riemann surfaces with boundary and bundles extend holomorphically, a global vanishing identity holds in . Holomorphic families are then shown to automatically satisfy the star condition, enabling the reciprocity to hold in that holomorphic context; a corollary yields vanishing results for determinant gerbes (12 times the determinant gerbe) in , connecting to Deligne pairing and topological Riemann–Roch. Overall, the work links local-to-global data through Gysin pushforwards, Chern classes, and Deligne-type pairings in a geometrically rich setting.

Abstract

We study reciprocity laws involving complex line bundles on fibrations in oriented circles. In particularly, we prove the following reciprocity law. Let be a complex manifold and be a fibration in oriented circles, where runs through a finite set. Let and be complex line bundles on every . The reciprocity law states that the sum of all , where is the Gysin map and is the first Chern class, equals zero in when the disjoint union of all is embedded into a holomorphic family of compact Riemann surfaces over the base 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 and all are restrictions of holomorphic line bundles on this family.

Paper Structure

This paper contains 5 sections, 5 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Deligne reciprocity law
  3. Fibration in oriented circles
  4. Relative analytic reciprocity law
  5. Holomorphic families of compact Riemann surfaces

Key Result

Theorem 1

Let $F$ and $G$ be two ${\mathbb C}^*$-valued $C^{\infty}$-functions on a compact Riemann surface $\Sigma$ with boundary such that they are holomorphic on the interior $\Sigma \setminus \partial \Sigma$ of $\Sigma$. Let the $C^{\infty}$-functions $f$ and $g$ be the restrictions of the functions $F$ where for every $1 \le i \le n$ one restricts $f$ and $g$ to $\gamma_i$ to calculate ${\mathbb T}_

Theorems & Definitions (23)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1: Analytic reciprocity law
  • Remark 4
  • Remark 5
  • Remark 6
  • Definition 1
  • Definition 2
  • Definition 3
  • ...and 13 more