Complex Analytic and Differential Geometry 2025, 22 - Cohomology Groups of O(k) and Ampleness

A complex vector space V of dim n + 1 with the exact sequence 0 → O(-1) → V₀ → H → 0 of vector bundles over ℙⁿ = P(V). det V₀ = det H ⊗ O(-1), TP(V) = H ⊗ O(1). Take K_P(V) det T*P(V) = det H* ⊗ O(-n) = det V₀* ⊗ O(-n-1) where det V₀ is a trivial line bundle. Write C^{○, k}(V*) with differential γ so that for o ≤ p ≤ k, C^{p, k}(V*) = Λ^pV* ⊗ S^k-pV* and 0 otherwise, and γ: Λ^pV* ⊗ S^k-pV* → Λ^p-1V* ⊗ S^k-p+1 V*. γ is the linear map obtinaed by contraction with the Euler Identity vector field through the trivial maps. Write α(z) = Σ_|I|=p α_I(z) dz_I. The exterior derivative d maps d: C^p,k(V*) → C^p+1,k(V*). dγ + γd = k Id_{C^{○, k}(V*)}. For k ≠ -, it's exact and there are canonical isomorphisms C^○,k(V*) = Λ^pV* ⊗ S^k-pV* ≅ Z^p,k(V*) ⊕ Z^p-1,k(V*). The groups H^p,0(P(V), O(k)) are given by H^p,0(P(V), O(k)) ≅ Z^p,k(V*) for k ≥ p ≥ 0, and H^p,0(P(V), O(k)) = 0 for k ≤ p and (k, p) ≠ (0, 0). They vanish in several cases, q ≠ 0, n, p; q = 0, k ≤ p, (k, p) ≠ (0 ,0); q = n, k ≥ -n + p, (k, p) ≠ (0, n); q = p ≠ 0, n, k ≠ 0.

Let E → X be a holomorphic vector bundle over an arbitrary complex manifold X. E is to be globally generated if ∀ x ∈ X the eval map H⁰(X, E) → E_x. E is semi-ample if there is an integer k₀ with S^kE is globally generated for k ≥ k₀. For E globally generated, E has a hermitian metric so that E ≥_Grif 0 and E* ≤_Nak 0. If E is a semi-ample line bundle, then E ≥ 0.

E is to very ample if all eval maps H⁰(X, E) → (J¹E)_x, H⁰(X, E) → E_x ⊕ E_y; x, y ∈ X, x≠y surjective. It's ample if ∃ k₀: S^kE is very ample for k ≥ k_{0. If E is an ample line bundle, then E > 0. If E is a very ample vector bundle then it carries a hermitian metric with E* <Nak 0. In this case, also E >Grif 0.}

For a normal bundle of a closed submanifold Y in a complex n-dim manifold X so that codim_XY = s, the normal bundle of Y is a vector bundle over Y: NY = (TX)_↑Y/TY. Its fibers are given by N_yY = T_yX/T_yY at all points y in Y. The projectivized normal bundle are the projective spaces associated to the fibers of NY. A map σ is the blow-up of X with center Y and E = σ^-1(Y) ≅ P(NY) is the exceptional divisor of the origin of σ.