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Theta functions and adiabatic curvature on an Abelian variety

Ching-Hao Chang, Jih-Hsin Cheng, I-Hsun Tsai

Abstract

For an ample line bundle $L$ on an Abelian variety $M$, we study the theta functions associated with the family of line bundles $L\otimes T$ on $M$ indexed by $T\in \text{Pic}^{0}(M)$. Combined with an appropriate differential geometric setting, this leads to an explicit curvature computation of the direct image bundle $E$ on $\text{Pic}^{0}(M)$, whose fiber $E_{T}$ is the vector space spanned by the theta functions for the line bundle $L\otimes T$ on $M$. Some algebro-geometric properties of $E$ are also remarked.

Theta functions and adiabatic curvature on an Abelian variety

Abstract

For an ample line bundle on an Abelian variety , we study the theta functions associated with the family of line bundles on indexed by . Combined with an appropriate differential geometric setting, this leads to an explicit curvature computation of the direct image bundle on , whose fiber is the vector space spanned by the theta functions for the line bundle on . Some algebro-geometric properties of are also remarked.
Paper Structure (5 sections, 19 theorems, 71 equations)

This paper contains 5 sections, 19 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Holomorphic Line Bundles over $M=V/\Lambda$
  3. A holomorphic line bundle $\widetilde{K}\rightarrow M\times M$
  4. Direct image bundle $K\rightarrow M_{2}$ with an $L^{2}$-metric
  5. Canonical Connection on the Poincaré line bundle and Proof of Theorem \ref{['thm1.2']}

Key Result

Theorem 1.1

(See (E'KL0) )(cf. CCT). Let $E^{\prime }$ and $L$ be as above. Then $E^{\prime }\otimes L$ is a holomorphically trivial vector bundle on $M$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.5
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 24 more