By A. V. Pogorélov

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That L is big. 5 f), there is a singular hermitian metric on L such that the corresponding weight ϕL,0 has algebraic singularities and iΘ0 (L) = 2id′ d′′ ϕL ≥ ε0 ω for some ε0 > 0. On the other hand, since L is nef, there are metrics given by i weights ϕL,ε such that 2π Θε (L) ≥ εω for every ε > 0, ω being a K¨ahler metric. 13. We define a singular metric on F by ϕF = 1 (1 − δ)ϕL,ε + δϕL,0 + ϕD m with ε ≪ δ ≪ 1, δ rational. Then ϕF has algebraic singularities, and by taking δ 1 1 ϕD ) = I( m D).

11) Nadel vanishing theorem ([Nad89], [Dem93b]). Let (X, ω) be a K¨ ahler weakly pseudoconvex manifold, and let F be a holomorphic line bundle over X equipped with a singular hermitian metric h of weight ϕ. Assume that iΘh (F ) ≥ εω for some continuous positive function ε on X. Then H q X, O(KX + F ) ⊗ I(h) = 0 for all q ≥ 1. Proof. Let Lq be the sheaf of germs of (n, q)-forms u with values in F and with measurable coefficients, such that both |u|2 e−2ϕ and |d′′ u|2 e−2ϕ are locally integrable.

Fix a point x0 ∈ X such that singular hermitian on L such that T = 2π the Lelong number of T at x0 is zero, and take a sufficiently positive line bundle A (replacing A by a multiple if necessary), such that A − KX has a singular metric hA−KX of curvature ≥ εω and such that hA−KX is smooth on X {x0 } and has an isolated logarithmic pole of Lelong number ≥ n at x0 . 13 ⊗ hA−KX . 13 implies that H 0 (X, KX + F ) = H 0 (X, mL + A) has a section which does not vanish at x0 . Hence there is an effective divisor Dm such that O(mL+A) = 1 1 1 {Dm } − m c1 (A) = lim m {Dm } is in K eff .