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Tau function and moduli of meromorphic forms on algebraic curves

Dmitry Korotkin, Peter Zograf

Abstract

We study the moduli space of meromorphic 1-forms on complex algebraic curves having at most simple poles with fixed nonzero residues. We interpret the Bergman tau function on this moduli space as a section of a line bundle and study its asymptotic behavior near the boundary and the locus of forms with non-simple zeros. As an application, we decompose the projection of this locus to the moduli space of curves into a linear combination of standard generators of the rational Picard group with explicit coefficients that depend on the residues.

Tau function and moduli of meromorphic forms on algebraic curves

Abstract

We study the moduli space of meromorphic 1-forms on complex algebraic curves having at most simple poles with fixed nonzero residues. We interpret the Bergman tau function on this moduli space as a section of a line bundle and study its asymptotic behavior near the boundary and the locus of forms with non-simple zeros. As an application, we decompose the projection of this locus to the moduli space of curves into a linear combination of standard generators of the rational Picard group with explicit coefficients that depend on the residues.
Paper Structure (1 section, 9 theorems, 71 equations)

This paper contains 1 section, 9 theorems, 71 equations.

Table of Contents

  1. Transformation of tau function under a change of canonical basis of cycles

Key Result

Lemma 1

The tau function defined by (tauab3) is independent of $x\in C$.

Theorems & Definitions (20)

  • Definition 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 10 more