2019/05/16 by Ching-Hao Chang, Chang, Ching-Hao, Jih-Hsin Cheng +3
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1905.06555
openalex publication_date 2019/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a complex torus, Lμ→ M be positive line bundles parametrized by μ∈ \rm Pic0(M), and E→ \rm Pic0(M) be a vector bundle with E|μ≅ H0(M, L μ). We endow the total family \Lμ\μ with a Hermitian metric that induces the L2-metric on H0(M, L μ) hence on E. By 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 ΘE of E with respect to this L2-metric shows that ΘE 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 also obtain that E' splits holomorphically into a direct sum of line bundles each of which is isomorphic to Lμ=0^*. Physically, the spaces H0(M, L μ) correspond to the lowest eigenvalue with respect to certain family of Hamiltonian operators on M parametrized by μ or in physical notation, by wave vectors \bf k.