Theta Functions and Adiabatic Curvature on an Elliptic Curve
Journal
Journal of Geometric Analysis
Journal Volume
32
Journal Issue
3
Date Issued
2022
Author(s)
Abstract
Let 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.
Subjects
Connection; Curvature; Poincaré line bundle; Theta functions
Type
journal article