Complex Analytic and Differential Geometry 2025, 23 - Ampleness for Line Bundles

Let L → X a positive line bundle and L^k the k-th tensor power of L. All N-tuples of distinct points in X are associated with a constant C > 0 so the evaluation maps H⁰(X, L^k) → (J^mL^k)_x1⊕...⊕(J^mL^k)_xN is subjective for all m ≥ 0, k ≥ C(m+1). A blow-up σ of X with a finite set Y at its center and a line bundle O(E) with exceptional divisor E, then H¹(X, O(-mE)⊗σ*L^k) = 0 for m ≥ 1, k ≥ Cm and C ≥ 0 large enough. For a holomorphic line bundle L → X, if L is ample, then L > 0 and vice versa. A compact complex manifold X, dim_ℂX = n, then X carrying a positive line bundle is equivalent to X being projective algebraic, and also to X carrying a Hodge metric. All Kaehler manifold (X, ω) with H²(X, O) is projective. A complex torus X = ℂⁿ/Γ with Γ a lattice of ℂⁿ is an abelian variety iff ∃ h positive definite hermition form on ℂⁿ with Im(h(γ₁, γ₂)) ∈ ℤ for all γ_i∈Γ