Analytic methods in algebraic geometry by Jean-Pierre Demailly

By Jean-Pierre Demailly

This quantity is a diffusion of lectures given by means of the writer on the Park urban arithmetic Institute (Utah) in 2008, and on different events. the aim of this quantity is to explain analytic thoughts valuable within the learn of questions concerning linear sequence, multiplier beliefs, and vanishing theorems for algebraic vector bundles. the writer goals to be concise in his exposition, assuming that the reader is already a bit accustomed to the elemental thoughts of sheaf idea, homological algebra, and intricate differential geometry. within the ultimate chapters, a few very contemporary questions and open difficulties are addressed--such as effects relating to the finiteness of the canonical ring and the abundance conjecture, and effects describing the geometric constitution of Kahler forms and their optimistic cones.

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Assume that X is weakly pseudoconvex. e. H = E −ϕ . Suppose that iΘF,h = id′ d′′ ϕ εω ⋆ for some ε > 0. Then for any form g ∈ L2 (X, Λn,q TX ⊗ F ) satisfying D′′ g = 0, there 2 p,q−1 ⋆ ′′ exists f ∈ L (X, Λ TX ⊗ F ) such that D f = g and 1 |f |2 e−ϕ dVω |g|2 e−ϕ dVω . qε X X Here we denoted somewhat incorrectly the metric by |f |2 e−ϕ , as if the weight ϕ was globally defined on X (of course, this is so only if F is globally trivial). We will use this notation anyway, because it clearly describes the dependence of the L2 norm on the psh weights.

Multiplier ideal sheaves and Nadel vanishing theorem We now introduce the concept of multiplier ideal sheaf, following A. Nadel [Nad89]. The main idea actually goes back to the fundamental works of Bombieri [Bom70] and H. Skoda [Sko72a]. 4) Definition. Let ϕ be a psh function on an open subset Ω ⊂ X ; to ϕ is associated the ideal subsheaf Á(ϕ) ⊂ ÇΩ of germs of holomorphic functions f ∈ ÇΩ,x such that |f |2 e−2ϕ is integrable with respect to the Lebesgue measure in some local coordinates near x.

Then the metric hL = 1/k (hkL+A ⊗ h−1 has curvature A ) iΘL,hL = 1 iΘkL+A − iΘA k 1 − iΘA,hA ; k in this way the negative part can be made smaller than ε ω by taking k large enough. (c) ⇒ (a). Under hypothesis (c), we get L · C = C and every ε > 0, hence L · C 0 and L is nef. i Θ C 2π L,hε ε − 2π C ω for every curve Let now X be an arbitrary compact complex manifold. 10 c) is simply taken as a definition of nefness ([DPS94]): § 6. 11) Definition. A line bundle L on a compact complex manifold X is said to be nef if for every ε > 0, there is a smooth hermitian metric hε on L such that iΘL,hε −εω.

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