By Jean-Pierre Demailly
This quantity is a variety of lectures given by way 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 suggestions valuable within the examine 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 familiar with the fundamental thoughts of sheaf concept, homological algebra, and intricate differential geometry. within the ultimate chapters, a few very fresh questions and open difficulties are addressed--such as effects on the topic of the finiteness of the canonical ring and the abundance conjecture, and effects describing the geometric constitution of Kahler types and their confident cones.
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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 .
In the projective algebraic case, we proceed by induction on n = dim X. If n = 1 the result is clear, as well as if q = 0. Now let A be a nonsingular ample divisor such that E ⊗ Ç(A−KX ) is Nakano positive. Then the Nakano vanishing theorem applied to the vector bundle F = E ⊗ Ç(kL + A − KX ) shows that H q (X, Ç(E) ⊗ Ç(kL + A)) = 0 for all q 1. The exact sequence 0 → Ç(kL) → Ç(kL + A) → Ç(kL + A)↾A → 0 twisted by E implies H q (X, Ç(E) ⊗ Ç(kL)) ≃ H q−1 (A, Ç(E↾A ⊗ Ç(kL + A)↾A ), and we easily conclude by induction since dim A = n − 1.
17 ii), there exists a singular metric h1 on L such that 1 i i ΘL,h1 = ΘA,hA + [D] 2π k 2π 1 ω, k ω= i ΘA,hA . 2π i Now, for every ε > 0, there is a smooth metric hε on L such that 2π ΘL,hε −εω. The 1−kε convex combination of metrics h′ε = hkε h is a singular metric with poles along D 1 ε which satisfies i ΘL,h′ε ε(ω + [D]) − (1 − kε)εω kε2 ω. 2π Its Lelong numbers are εν(D, x) and they can be made smaller than δ by choosing ε > 0 small. We still need a few elementary facts about the numerical dimension of nef line bundles.