Algebraic Geometry 1: From Algebraic Varieties to Schemes by Kenji Ueno

By Kenji Ueno

This is often the 1st of 3 volumes on algebraic geometry. the second one quantity, Algebraic Geometry 2: Sheaves and Cohomology, is accessible from the AMS as quantity 197 within the Translations of Mathematical Monographs sequence.

Early within the twentieth century, algebraic geometry underwent an important overhaul, as mathematicians, particularly Zariski, brought a far more suitable emphasis on algebra and rigor into the topic. This used to be by means of one other primary swap within the Sixties with Grothendieck's creation of schemes. at the present time, such a lot algebraic geometers are well-versed within the language of schemes, yet many novices are nonetheless firstly hesitant approximately them. Ueno's booklet presents an inviting creation to the speculation, which may still triumph over such a obstacle to studying this wealthy topic.

The ebook starts with an outline of the normal idea of algebraic kinds. Then, sheaves are brought and studied, utilizing as few necessities as attainable. as soon as sheaf thought has been good understood, the next move is to determine that an affine scheme should be outlined when it comes to a sheaf over the major spectrum of a hoop. by means of learning algebraic kinds over a box, Ueno demonstrates how the proposal of schemes is critical in algebraic geometry.

This first quantity supplies a definition of schemes and describes a few of their hassle-free houses. it really is then attainable, with just a little extra paintings, to find their usefulness. additional houses of schemes might be mentioned within the moment quantity.

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Example text

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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