By Jean Dieudonne

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V) is a (p − 1, q)-form (resp. (p, q)-form), with Supp u ∩ Supp v ⊂⊂ Ω. The third equality is simply obtained through an integration par parts, and amounts to observe that the formal adjoint of ∂/∂zk is −∂/∂zk . We now prove a useful lemma due to Akizuki and Nakano [AN54]. 18∗ ) ∗ ∗ u → Lu = ω ∧ u, L : C ∞ (X, Λp,q TX ⊗ F ) −→ C ∞ (X, Λp+1,q+1TX ⊗ F ), ∗ ∞ p,q ∗ ∞ p−1,q−1 ∗ Λ = L : C (X, Λ TX ⊗ F ) −→ C (X, Λ TX ⊗ F ). Again, in the flat hermitian complex space (Cn , ω) we find : 4. 19) Lemma. In Cn , we have [d′′⋆ , L] = i d′ .

17 a) i∂∂ψε = 2 ∂ψε = j,k γj,k ∧ γj,k e2ψ + ε2 +i j,k − j,k θj gj,k γj,k (e2ψ ∧ + |gj,k |2 (θj ∂∂θj − ∂θj ∧ ∂θj ) e2ψ + ε2 j,k θj gj,k γj,k 2 ε )2 , . 17 b) 2 2 1 ε2 1 j,k |γj,k (ξ)| j,k θj gj,k γj,k (ξ) − ≥ |γj,k (ξ)|2 . 17 a) is uniformly bounded below by a fixed negative hermitian form −Aω, A ≫ 0, and therefore i∂∂ψε ≥ −Aω. Actually, for every pair of indices (j, j ′ ) we have a bound C −1 ≤ |gj,k (z)|2 / k |gj ′ ,k (z)|2 ≤ C on B j ∩ B j ′ , k since the generators (gj,k ) can be expressed as holomorphic linear combinations of the (gj ′ ,k ) by Cartan’s theorem A (and vice versa).

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. Proof. c) and d) are already known and e) is a definition. amp nef a) The ample cone KNS is always open by definition and contained in KNS , so the amp first inclusion is obvious (KNS is of course empty if X is not projective algebraic). psef nef nef Let us now prove that KNS . Let L be a line bundle with c1 (L) ∈ KNS ⊂ ENS .

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