Let M be a complex torus, Lμ^→M\documentclass[12pt]{minimal}
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\begin{document}$$L_{{\hat{\mu }}}\rightarrow M$$\end{document} be positive line bundles parametrized by μ^∈Pic0(M)\documentclass[12pt]{minimal}
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\begin{document}$${\hat{\mu }}\in \mathrm{Pic}^0(M)$$\end{document}, and E→Pic0(M)\documentclass[12pt]{minimal}
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\begin{document}$$E\rightarrow \mathrm{Pic}^0(M)$$\end{document} be a vector bundle with E|μ^≅H0(M,Lμ^)\documentclass[12pt]{minimal}
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\begin{document}$$E|_{{\hat{\mu }}}\cong H^0(M, L_{{\hat{\mu }}})$$\end{document}. We endow the total family {Lμ^}μ^\documentclass[12pt]{minimal}
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\begin{document}$$\{L_{{\hat{\mu }}}\}_{{\hat{\mu }}}$$\end{document} with a Hermitian metric that induces the L2\documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document}-metric on H0(M,Lμ^)\documentclass[12pt]{minimal}
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\begin{document}$$H^0(M, L_{{\hat{\mu }}})$$\end{document} hence on E. Using theta functions {θm}m\documentclass[12pt]{minimal}
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\begin{document}$$\{\theta _m\}_{m}$$\end{document} on M×M\documentclass[12pt]{minimal}
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\begin{document}$$M\times M$$\end{document} as a family of functions on the first factor M with parameters in the second factor M, our computation of the full curvature tensor ΘE\documentclass[12pt]{minimal}
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\begin{document}$$\Theta _E$$\end{document} of E with respect to this L2\documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document}-metric shows that ΘE\documentclass[12pt]{minimal}
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\begin{document}$$\Theta _E$$\end{document} is essentially an identity matrix multiplied by a constant 2-form, which yields in particular the adiabatic curvature c1(E)\documentclass[12pt]{minimal}
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\begin{document}$$c_1(E)$$\end{document}. After a natural base change M→M^\documentclass[12pt]{minimal}
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\begin{document}$$M\rightarrow {\hat{M}}$$\end{document} so that E×M^M:=E′\documentclass[12pt]{minimal}
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\begin{document}$$E\times _{{\hat{M}}} M:=E'$$\end{document}, we obtain that E′\documentclass[12pt]{minimal}
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\begin{document}$$E'$$\end{document} splits holomorphically into a direct sum of line bundles each of which is isomorphic to Lμ^=0∗\documentclass[12pt]{minimal}
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\begin{document}$$L_{{\hat{\mu }}=0}^*$$\end{document}.
