By Demailly J.-P.

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14) Theorem. Let X be a compact complex manifold, E a holomorphic vector bundle and (F, hF ) a hermitian line bundle with a smooth metric h such that ω = i ΘhF (F ) > 0. e. a function ϕ such that i d′ d′′ ϕ ≥ −Cω for some constant C > 0. Then a) There exists an integer m0 such that H q (X, E ⊗ F ⊗m ⊗ I(ϕ)) = 0 for q ≥ 1 and m ≥ m0 . b) The restriction map H 0 (X, E ⊗ F ⊗m ⊗ I(ϕ)) −→ H 0 (X, E ⊗ F ⊗m ⊗ OX /I(ϕ)) is surjective for m ≥ m0 . b) The vector bundle E ⊗ F ⊗m generates its sections (or jets of any order s) for m ≥ m0 (s) large enough.

E. mj ≥ ⌊αj ⌋ (integer part). Hence I(ϕ) = O(−⌊D⌋) = O(− ⌊αj ⌋Dj ) where ⌊D⌋ denotes the integral part of the Q-divisor D = αj Dj . Now, consider the general case of analytic singularities and suppose that ϕ ∼ α 2 2 near the poles. 10, we may 2 log |f1 | +· · ·+|fN | assume that the (fj ) are generators of the integrally closed ideal sheaf J = J (ϕ/α), defined as the sheaf of holomorphic functions h such that |h| ≤ C exp(ϕ/α). In this case, the computation is made as follows (see also L. Bonavero’s work [Bon93], where similar ideas are used in connection with “singular” holomorphic Morse inequalities).

Amp ◦ amp nef ◦ ) ⊂ (KNS = (KNS ) , a) KNS psef nef KNS . ⊂ ENS amp amp nef ◦ = = (KNS b) If moreover X is projective algebraic, we have KNS ) (therefore KNS psef nef eff KNS ), and ENS = ENS . -P. Demailly, Complex analytic techniques in algebraic geometry amp ⇔ ∃ε > 0, ∃h smooth such that i Θh (L) ≥ εω. c) c1 (L) ∈ KNS nef d) c1 (L) ∈ KNS ⇔ ∀ε > 0, ∃h smooth such that i Θh (L) ≥ −εω. psef ⇔ ∃h possibly singular such that i Θh (L) ≥ 0. e) c1 (L) ∈ ENS f) If moreover X is projective algebraic, then eff ◦ c1 (L) ∈ (ENS ) ⇔ κ(L) = dim X ⇔ ∃ε > 0, ∃h possibly singular such that i Θh (L) ≥ εω.

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Applications of the theory of L2 estimates and positive by Demailly J.-P.
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