Complex Analytic and Differential Geometry 2025, 20 - Positive Vector Bundles

An operator τ of type (1, 0) defined by &tau' = [Λ, d'ω] on C^∞(X, E), then

For a connected C^∞-manifold M, a vector bundle F over M, a second order elliptic differential operator P acting on C^∞(M, F), then any section α∈ker P vanishing on a non-empty open subset of M vanishes identically on M. If X is compact and connected, if [iΘ(E), Λ] + T_ω ∈ Herm(Λ^p,qT*X⊗E) is semi-positive on X, then a positive definite in at least one point in X, and H^p,q(X, E) = 0

A hermitian holomorphic line bundle E on X is positive (negative) if the hermitian matrix c_k of its Chern curvature form iΘ(E) is positive (negative) definite at all points in X. If X has a Kaehler metric ω, and (γ)_k the Eigenvalues of the Chern curvature form wrt. ω at each x ∈ X, iΘ(E)_x the diagonalization, then

⟨[iΘ(E), Λ]u, u⟩ = Σ_J,K(Σ_j∈Jγ_j + Σ_j∈Kγ_j - Σ_1≤j≤nγ_j)|u_J,K|² ≥ (γ₁ + ... + γ_q - γ_p+1 - ... - γ_n)|u|²

A holomorphic line bundle on X E is positive for H^p,q(X, E) = 0 , p + q ≥ n + 1 and negative for H^p,q(X E) = 0, p + q ≤ n - 1. Let (X, ω) be a compact and connected Kaehler manifold and E a line bundle, if the Chern curvature is greater than 0 on X and positive definite in at least one point, then H^q(X, K_K ⊗ E) = 0 for q ≥ 1. If it's less than 0 and negative definite in at least one point, then H¹(X, E) = 0 for q ≤ n - 1.