By A. V. Pogorélov

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Low Dimensional Topology

During this quantity, that is devoted to H. Seifert, are papers in line with talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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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 .