By Jean-Pierre Demailly

This quantity is a spread 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 innovations important within the learn of questions referring to 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 slightly familiar with the elemental ideas of sheaf conception, homological algebra, and complicated differential geometry. within the ultimate chapters, a few very contemporary questions and open difficulties are addressed--such as effects concerning 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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The easiest case is the following result of Girbau [Gir76]: let (X, ω) be compact K¨ahler; assume that F is a line bundle and that iΘF,h 0 has at least n − k positive eigenvalues at each point, for some integer k 0; show that H p,q (X, F ) = 0 for p + q n + k + 1. Hint: use the K¨ahler metric ωε = iΘF,h + εω with ε > 0 small. A stronger and more natural “algebraic version” of this result has been obtained by Sommese [Som78]: define F to be k-ample if some multiple mF is such that the canonical map Φ|mF | : X B|mF | → PN−1 has at most k-dimensional fibers and dim B|mF | k.

Since then, a number of other proofs have been given, one based on connections with logarithmic singularities [EV86], another on Hodge theory for twisted coefficient systems [Kol85], a third one on the Bochner technique [Dem89] (see also [EV92] for a general survey). Since the result is best expressed in terms of multiplier ideal sheaves (avoiding then any unnecessary desingularization in the statement), we feel that the direct approach via Nadel’s vanishing theorem is extremely natural. If D = αj Dj 0 is an effective Q-divisor, we define the multiplier ideal sheaf Á(D) to be equal to Á(ϕ) where ϕ = αj |gj | is the corresponding psh function defined by generators gj of Ç(−Dj ).

25. e. that L is big. 16 ii), there is a singular hermitian metric h0 on L such that the corresponding weight ϕ0 has algebraic singularities and iΘL,h0 = 2id′ d′′ ϕ0 ε0 ω for some ε0 > 0. On the other hand, since L is nef, there are metrics given by weights i ϕε such that 2π ΘL,hε −εω for every ε > 0, ω being a K¨ahler metric. Let ϕD = αj log |gj | be the weight of the singular metric on Ç(D). We define a singular metric on F by 1 ϕF = (1 − δ)ϕL,ε + δϕL,0 + ϕD m with ε ≪ δ ≪ 1, δ rational. Then ϕF has algebraic singularities, and by taking δ small 1 1 ϕD ) = Á( m D).