By Jean-Pierre Demailly

This quantity is a spread 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 strategies invaluable within the research of questions touching on 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 little bit accustomed to the fundamental suggestions of sheaf idea, homological algebra, and complicated differential geometry. within the ultimate chapters, a few very contemporary questions and open difficulties are addressed--such as effects regarding the finiteness of the canonical ring and the abundance conjecture, and effects describing the geometric constitution of Kahler kinds and their confident cones.

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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 . j 1 j N Proof. 11 to obtain the vanishing of the first H 1 group. The asserted surjectivity property follows. 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).

A psh function ϕ is said to have a logarithmic pole of coefficient γ at a point x ∈ X if the Lelong number ν(ϕ, x) := lim inf z→x ϕ(z) log |z − x| is non zero and if ν(ϕ, x) = γ. 6) Lemma (Skoda [Sko72a]). Let ϕ be a psh function on an open set Ω and let x ∈ Ω. (a) If ν(ϕ, x) < 1, then e−2ϕ is integrable in a neighborhood of x, in particular ÇΩ,x . Á(ϕ)x = (b) If ν(ϕ, x) n + s for some integer s 0, then e−2ϕ C|z − x|−2n−2s in a neighs+1 borhood of x and Á(ϕ)x ⊂ mΩ,x , where mΩ,x is the maximal ideal of ÇΩ,x .

NSR (X) ÃNS NS A very important fact is that all four cones braic interpretations. 17) Theorem. Let X be a projective manifold. Then (i) ÃNS is equal to the open cone Amp(X) generated by classes of ample (or very ample) divisors A (Recall that a divisor A is said to be very ample if the linear system H 0 (X, Ç(A)) provides an embedding of X in projective space). (ii) The interior ◦NS is the cone Big(X) generated by classes of big divisors, namely divisors D such that h0 (X, Ç(kD)) c k dim X for k large.