Complex Analytic and Differential Geometry 2025, 21 - Some Vanishing Theorems

If iΘ(E) is semi-positive and has at least n-s+1 positive eigenvalues at a point in X for some integer s ≤ n, then H^q(X, K_X ⊗ E) = 0 for q ≥ s. Without the integer condition, then H^p,q(X, E) = 0 for p + q ≥ n + s. For a holomorphic vector bundles over compact complex manifold X, a positive integer s and hermitian line bundle E so that iΘ(E) has at least n-s+1 positive eigenvalues at all points in X, then ∃ k₀≥0 with H^q(X, E^k ⊗ F) = 0 for q ≥ s and k ≥ k₀. If A is a hermitian n × n matrix with eigenvalues (λ)_n and corresponding eigenvectors (v)_n, with ψ ∈ C^∞(ℝ, ℝ), define ψ[A] as the hermitian with eigenvalues ψ(λ_j) and vectors v_j, 1 ≤ j ≤ n and the map A → ψ[A] is C^∞ on Herm(ℂⁿ)

E is "Nakano positive" (negative) if θ_E is positive definite (semi-negative) as a hermitian form on TX⊗E. Write <_Nak. E is "Griffiths positive" (semi-negative) if for all ξ∈T_xX, ξ≠0 and s∈E_x, s≠0, then θ_E(ξ⊗s, ξ⊗s) > 0 (≤ 0). Write >_Grif for Griffiths positivity (semi-negativity). Complex vector spaces T, E with dim n, r then for Θ hermitian on T ⊗ E a tensor u ∈ T ⊗ E is of rank m if m is the smallest ≥ 0 integer such that u can be written n = Σ^m_{j = 1}ξ_j⊗s_j, ξ_j∈T, s_j∈E. Θ is m-positive (m-negative) if Θ(u, u) > 0 (≤ 0) for every tensor u ∈ T ⊗ E of rank ≤ m, u ≠ 0. Write Θ >_m 0 (≤_m 0). A bundle E is Griffiths-positive iff E* is Griffiths-negative. Griffiths (semi-)positivity/negativity travels both ways in exact sequences of hermitian vector bundles. Nakano-(semi-)positivity only travels backwards on the sequence.

Assume E >_m 0, then [iΘ(E), Λ] is a hermitian operator and positive definite on Λ^n,qT*X⊗E for q ≥ 1, m ≥ min{n - q + q, r}. If X is a compact connected Kaehler manifold of dim n, E a hermitian vector bundle of rank r, with θ_E ≥_m 0 on X and θ_E >_m 0 in at least one point, then H^n,q(X, E) = H^q(X, K_X⊗E) = 0 for q ≥ 1 and m ≥ min{n-q+1, r}. For the reverse in at least one point, then H^p,0(X, E) = H⁰(X, Ω^p_X ⊗ E) = 0 for p < n and m ≥ min{p + 1, r}. For m = r and E ≥_Nak 0 strictly in in one point, then H^n,q(X, E) = 0 for q ≥ 1. For the opposite, H^p,0(X, E) = 0 for p < n. For any hermitian vector bundle E, E >_Grif 0 ↠ E ⊗ det E >_Nak 0.

A Griffiths (semi-)positive bundle E with rank r at least 2, then for any m ≥ 1 E* ⊗ (det E)^m >_m (≥_m 0). An exact sequence 0 → S → E → Q → 0 of hermitian vector bundles with m ≥ 1 has E >_m 0 ↠ S⊗(det Q)^m >_m 0.