Complex Analytic and Differential Geometry 2025, 25 - Surjective Bundle Morphisms

An integer k with 0 ≤ k ≤ n, r = rkE, q = rkQ, s = rkS = r - q, m = min{n-k, s} = min{n-k, r-q}. Assume that (X, ω) has a complete Kaehler metric, E ≥_m 0, and L→X is a hermitian line bundle with iΘ(L) - (m + ε)iΘ(det Q) ≥ 0, for ε > 0. For all D''-closed forms f of type (n, k) with values in Q ⊗ L and ||f|| < ∞, there is a D''-closed form h of type (n, k) with values in E ⊗ L with f = gh and ||h||² ≤ (1+m/ε) ||f||². ⟨A_k^-1(β*∧f), (β*∧f)⟩ ≤ (m/ε)|f|². For a weakly pseudoconvex Kaehler manifold (X, ω), g: E → Q a surjective bundle morphism with r, q, m, L, E, as above, then g induces a surjective H^k(X, K_X ⊗ E ⊗ L) → H^k(X, K_X ⊗ Q ⊗ L).

A complete Kaehler manifold X with Kaehler metric ω, let E → Q be a surjective morphism of hermitian vector bundles, for all D''-closed forms f of type (n, k) with values in Q⊗L and h of type (n, k) with values in E⊗L with f = gh

The existence statement and resulting estimates remain true for generically surjective morphisms g: E→Q for weakly pseudoconvex X. Let Ω be a Kaehler open subset of ℂⁿ and φ PSH on Ω. With m = min{n, r-1}, Z = g^-1(0), for all holomorphic f,

An open Ω ⊂ ℂⁿ, then Ω is a domain of holomorphy iff Ω is pseudoconvex, and If Ω is it's own open hull of it's closed finish, and it has a complete Kaehler metric, it's pseudoconvex.

The torsion form of J is defined θ∈C^∞_{2, 0}(M, T^0,1M). THe almost complex structure J is integrable if θ = 0. Every integrable almost complex structure J on M is defined by a unique analytic structure. For every point a in M and positive definite integer s there are complex coordinates centered at a, with d''z_j = O(|z|^s), 1 ≤ j ≤ n. For s larger than 3, this defines strictly PSH functions on a small ball.