Chang C.-HCheng J.-HI-HSUN TSAI2022-11-152022-11-15202210506926https://www.scopus.com/inward/record.uri?eid=2-s2.0-85122954169&doi=10.1007%2fs12220-021-00774-2&partnerID=40&md5=704c79c8b5fe17d6f32b4abf9d6eaa01https://scholars.lib.ntu.edu.tw/handle/123456789/625096Let M be a complex torus, Lμ^→ M be positive line bundles parametrized by μ^ ∈ Pic (M) , and E→ Pic (M) be a vector bundle with E| μ^≅ H(M, Lμ^). We endow the total family {Lμ^}μ^ with a Hermitian metric that induces the L2-metric on H(M, Lμ^) hence on E. Using theta functions {θm}m on M× M as a family of functions on the first factor M with parameters in the second factor M, our computation of the full curvature tensor Θ E of E with respect to this L2-metric shows that Θ E is essentially an identity matrix multiplied by a constant 2-form, which yields in particular the adiabatic curvature c1(E). After a natural base change M→ M^ so that E× M^M: = E′, we obtain that E′ splits holomorphically into a direct sum of line bundles each of which is isomorphic to Lμ^=0∗. © 2021, Mathematica Josephina, Inc.Connection; Curvature; Poincaré line bundle; Theta functionsjournal article10.1007/s12220-021-00774-22-s2.0-85122954169