Theta functions and adiabatic curvature on an Abelian variety

TL;DR

This paper investigates theta functions on Abelian varieties, computes the adiabatic curvature of associated vector bundles, and explores their algebraic properties, linking complex geometry with algebraic geometry.

Contribution

It provides an explicit curvature formula for the direct image bundle of theta functions on Abelian varieties, combining differential and algebraic geometry methods.

Findings

Explicit curvature computation of the direct image bundle

Connection between theta functions and geometric properties of line bundles

Insights into algebraic structure of the vector bundle E

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.