By Jean-Pierre Demailly

This quantity is a diffusion of lectures given via the writer on the Park urban arithmetic Institute (Utah) in 2008, and on different events. the aim of this quantity is to explain analytic options beneficial within the research 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 accustomed to the elemental strategies of sheaf thought, 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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Let ψ be a smooth psh exhaustion function on X. 3 globally on X, with the original metric of F multiplied by the factor e−χ◦ψ , where χ is a convex increasing function of arbitrary fast growth at infinity. This factor can be used to ensure the convergence of integrals at infinity. 3, we conclude that H q Γ(X, Ä• ) = 0 for q 1. The theorem follows. 12) Corollary. 11 and let x1 , . . , xN be isolated points in the zero variety V (Á(ϕ)). Then there is a surjective map Ç(KX + L)x H 0 (X, KX + F ) −→ −→ j ⊗ ÇX /Á(ϕ) x .

13) Corollary. 11 and suppose that the weight function ϕ is such that ν(ϕ, x) n + s at some point x ∈ X which is an isolated point of E1 (ϕ). Then H 0 (X, KX + F ) generates all s-jets at x. Proof. The assumption is that ν(ϕ, y) < 1 for y near x, y = x. 6 a). 12. The philosophy of these results (which can be seen as generalizations of the H¨ ormander-Bombieri-Skoda theorem [Bom70], [Sko72a, 75]) is that the problem of constructing holomorphic sections of KX + F can be solved by constructing suitable hermitian metrics on F such that the weight ϕ has isolated poles at given points xj .

It is important for various applications to obtain vanishing theorems which are also valid in the case of semi-positive line bundles. 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.

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