Complex Analytic and Differential Geometry 2025, 16 - Line Bundles

Define a trivial bundle V₀ P(V) × V with V a complex vector space with dim at least 2, and P(V) its projective space. O(-1) = {([x], ξ) ∈ P(V) × V; ξ ∈ ℂ ⋅ x } ⊂ V₀ is the tautological line subbundle. It's holomorphic. For all natural k, O(k) is

O(1) = O(-1)*, O(0) = P(V) × ℂ

O(k) = O(1)^⊗k = O(1) ⊗ ... ⊗ O(1) ∀ k ≥ 1

O(-k) = O(-1)^⊗k ∀ k ≥ 1

The quotient vector bundle V₀/O(-1) of rank n. The canonical exact sequences of vector bundles over P(V):

0 → O(-1) → V₀ → H → 0, 0 → H* → V*₀ → O(1) → 0

The holomorphic map μ: O(-1) → V

μ: O(1) → V₀ = P(V) × V →^pr₂ V

sends the zero section to the origin and induces a bi-holomorphism O(-1) \ (P(V) × {0}) to V \ {0}. H⁰(P(V), O(k)) = 0 for k < 0 and H⁰(P(V), O(k)) ≅ S^kV* ∀ k ≥ 0. There is a canonical isomorphism of bundles TP(V) ≅ H ⊗ O(1).

The cohomology algebra H^○(ℙⁿ, ℤ) ≅ ℤ[h]/(hⁿ⁺¹) with h = c₁(O(1)) in H²(ℙⁿ, ℤ). Write the quotient line bundle π*E/S as O_E(1). It's the tautological line bundle associated to E.

0 → S → π*E → O_E(1) → 0

For all natural k, the direct sheaf π_*O_E(k) on X vanishes for k < 0 and is isomorphic to O(S^kE) for k ≥ 0.

G_r(V) is the set of all r-codimensional vector subspaces of a complex vector space V. ∀ a ∈ G_r(V), W ⊂ V: V = a ⊕ W, dim_ℂW = r all its subspaces in the open subsets Ω_W = {x ∈ G_r(V); x ⊕ W = V} can be written uniquely as a graph of a linear map u in Hom(a, W). G_r(V) is a compact complex analytic manifold of dim n = r(d - r). TG_r(V) = Hom(S, Q) = S* ⊗ Q