Theta Functions and Adiabatic Curvature on an Abelian Variety

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