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A Non-Abelian Approach to Riemann Surfaces Part I: Wronskian Geometry

Mehrzad Ajoodanian

TL;DR

The paper develops a non-Abelian analogue of the Abelian quotient of meromorphic sections for projectively flat holomorphic vector bundles on a Riemann surface. It introduces the Wronskian construction, defines the Wronskian quotient $\frac{A}{B}=W(B)^{-1}W(A)$, and builds the Wronskian line bundle $w(\mathcal{V})$ whose first Chern class encodes Abel’s identity in this non-Abelian setting. Abel’s identity is recast via $w' = p_1 w$ with $p_1=d(\log w)$, and Liouville’s formula connects the Maurer–Cartan data to the determinant bundle, yielding a cocycle interpretation of $c_1(w(\mathcal{V}))$. The authors construct Wronskian vector bundles $W_A(\mathcal{V})$ with determinant $w(\mathcal{V})$, provide explicit examples, and pose an intriguing open question linking $w(\mathcal{V})$ to the determinant and canonical bundles.

Abstract

We study projectively flat holomorphic vector bundles over Riemann surfaces. To each such bundle, we naturally assign a Wronskian line bundle. The main idea is a notion of the division of two meromorphic sections. Abel's identity is interpreted as the first Chern class of the Wronskian line bundle.

A Non-Abelian Approach to Riemann Surfaces Part I: Wronskian Geometry

TL;DR

The paper develops a non-Abelian analogue of the Abelian quotient of meromorphic sections for projectively flat holomorphic vector bundles on a Riemann surface. It introduces the Wronskian construction, defines the Wronskian quotient , and builds the Wronskian line bundle whose first Chern class encodes Abel’s identity in this non-Abelian setting. Abel’s identity is recast via with , and Liouville’s formula connects the Maurer–Cartan data to the determinant bundle, yielding a cocycle interpretation of . The authors construct Wronskian vector bundles with determinant , provide explicit examples, and pose an intriguing open question linking to the determinant and canonical bundles.

Abstract

We study projectively flat holomorphic vector bundles over Riemann surfaces. To each such bundle, we naturally assign a Wronskian line bundle. The main idea is a notion of the division of two meromorphic sections. Abel's identity is interpreted as the first Chern class of the Wronskian line bundle.

Paper Structure

This paper contains 16 sections, 10 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. The Wronskian
  3. Definition and Basic Properties
  4. Change of Coordinates
  5. Change of Coordinates
  6. The Wronskian Quotient
  7. Algebraic Properties of the Quotient
  8. The Rank of the Derivative of the Quotient as an Invariant
  9. Ordinary Differential Equations
  10. Liouville’s Formula, Abel’s Identity, and the First Chern Class
  11. Projectively Flat Vector Bundles on Riemann Surfaces
  12. How to Divide Sections of Projectively Flat Vector Bundles
  13. The Wronskian Line Bundle
  14. The First Chern Class of the Wronskian Line Bundle
  15. The Wronskian Vector Bundles
  16. ...and 1 more sections

Key Result

Lemma 2.1

The map $\frac{A}{B}$ is independent of the choice of basis for $V$ and of the coordinate on $U$.

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: Abel's identity
  • proof
  • ...and 15 more