By Dr. Kenji Ueno (auth.)

ISBN-10: 3540071385

ISBN-13: 9783540071389

ISBN-10: 3540374159

ISBN-13: 9783540374152

**Read Online or Download Classification Theory of Algebraic Varieties and Compact Complex Spaces: Notes written in collaboration with P. Cherenack PDF**

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**Extra resources for Classification Theory of Algebraic Varieties and Compact Complex Spaces: Notes written in collaboration with P. Cherenack**

**Sample text**

Definition 2,11 Let X be an analytic space, subspace (not necessarily reduced) and X. A pair (D, f) X f : X' with center D ~X an analytic the ideal sheaf of consisting of the analytic subspace morphism of analytic spaces formation of J D D D on and a is called the monoidal trans- when the morphism f satisfies the following conditions. I) in ~X' 2) The ideal sheaf generated by the image of is invertible on If g : X" >X X'. is a morphism of analytic spaces having the property I), there exists a unique morphism spaces so that f-I(~)@Ox,-----*Ox, h : X" > X' of analytic g = f oh The existence of monoidal transformations is guaranteed by the 22 results to be found in Hironaka one finds that if the analytic space X' monoidal obtained a sequence 2 12 Let of monoidal satisfies X >X.

Now we shall prove the second part. : ~ ~ WC @N. Let A. We consider again the morphism be an open set defined by X. ' ... N be g l o b a l c o o r d i n a t e s of ~ i = Xi-i ~i+l ' ~i X. ' ~i i We set ~'l = l ~-l(Ai) . i Then the morphism Ai such t h a t Xi+l ~N = X. ' . . ~ i i XN X. l ~ is represented by and 48 ~k = fk(z) fk(z) where k = i, 2, i s holomorphic on ~. 1 projective N Let ~N space w i t h homogeneous c o o r d i n a t e s (We c o n s i d e r i t as t h e d u a l p r o j e c t i v e an open set defined by Y.

D. h. 3), we have only used the fact that the local ring zation domain. ~V,x Hence the mapping x of h V at x is a unique factori- is bijective if the local ring of V at any point Let D be a Cartier divisor on a complex variety be local equations of V. ~V,x D is a unique factorization domain. V and let { ~ with respect to an open covering {Uilie I The complex line bundle [D] associated with the divisor D of has transition functions {gij ) with respect to this covering, where gij = Ti / ~j between on Ov(D) bundle [D].

### Classification Theory of Algebraic Varieties and Compact Complex Spaces: Notes written in collaboration with P. Cherenack by Dr. Kenji Ueno (auth.)

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