Complex Analytic and Differential Geometry 2025, 14 - Hermitian Vector Bundles

A hermitian complex vector bundle if a positive definite hermitian form is given on each fiber so that its induced map is smooth. A trivialization with corresponding frame, the inner product is given by a positive definite hermitian matrix with C^∞ coefficients. It follows

C^∞_p(M, E) × C^∞_q(M, E) → C^∞_{p + q}(M, ℂ)

(s, t) → {s, t}

For an orthonormal frame of E_↑Ω, the trivialization θ(s) = σ = (σ_λ), θ(t) = τ = (τ_λ), then {s, t} = {σ, τ} = Σσ_λ∧τ'_λ,

d{s, t} = {dσ, &tau} + (-1)^p{σ, dτ}

A sheaf S of modules of rings R are locally free of rank k if all points in the base has a neighborhood Ω so that S_↑Ω is R-isomorphic to R^k_↑Ω. There is a covering (V_α) of M and sheaf isomorphism θ_α: S_↑Vα → 𝔼^r_↑Vα for 𝔼 the sheaf of germs of C^∞ complex functions on M. The group of isomorphism classes of complex C^∞ line bundles is in 1-to-1 correspondence with H¹(M, 𝔼^☆)

Assume that E is a complex line bundle, D a connection on E and Θ(D) a closed 2-form on M, with D' a connection on E so that D' = D + Γ∧○, Γ∈C₁^∞(M, ℂ). It follows that Θ(D') = Θ(D) + dΓ. The coboundary morphism H¹(M, 𝔼^☆) →^∂ H²(M, ℤ) is isomorphic. The first Chern class of a line bundle E is c₁(E) in H² of the Cech cohomology class of the 1-cocycle (g^αβ): c₁(E) = ∂{(g_αβ)}. It coincides with the De Rahm cohomology class associated to any hermition connection D on E. Take {|Z|} = c₁(E)_ℝ