By V. I. Danilov (auth.), I. R. Shafarevich (eds.)

This EMS quantity includes components. the 1st half is dedicated to the exposition of the cohomology thought of algebraic kinds. the second one half offers with algebraic surfaces. The authors, who're recognized specialists within the box, have taken pains to provide the fabric conscientiously and coherently. The ebook includes quite a few examples and insights on a variety of issues. This booklet should be immensely priceless to mathematicians and graduate scholars operating in algebraic geometry, mathematics algebraic geometry, complicated research and comparable fields.

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**Extra resources for Algebraic Geometry II: Cohomology of Algebraic Varieties. Algebraic Surfaces**

**Sample text**

Now, making a substitution u = 1- e- 2 , we get the differential u-n- 1 (1- u)-m- 1 du. 5. The Riemann-Roch-Grothendieck Theorem. We will now diseuss Grothendieek's generalization of the Hirzebrueh formula. Grothendieek observed that sinee the Euler eharaeteristic and the Chern eharaeter are additive, one may eonsider arbitrary eoherent sheaves F in plaee of loeally free sheaves E. One ean show that an arbitrary eoherent sheaf F on a smooth variety admits a loeally free resolution 0 ---+ Er ---+ .

There are two reasons. First, we get the independence of the choice of a covering. Second, the I. Cohomology of Algebraic Varieties 31 general definition of cohomology relates quasi-coherent sheaves with arbitrary Abelian sheaves, for example, O:X. Example. We will calculate the cohomology of Ox for the simplest nonaffine variety X = A 2 \ { 0}. Let T1 , T 2 be coordinates in A 2 . Then X is covered by two affine charts: Ui = D(Ti), i = 1, 2. Consider the complex of this covering: K [T11T2,T1- 1] ff>K [T1,T2,T2 1] ~K [T1,T2,T1- 1,T2- 1], where d(fl,h) = h- f2.

The complex K' is built of homomorphisms dq : Kq -7 Kq+l of two A-modules, which we may assume tobe free of finite rank. So, dk is a matrix with coefficients in A. The specialization to a point y E Y (i. , the homomorphism A -7 Ajmy = k(y)) gives a matrix d~ over k(y). Clearly the rank of d~ is a lower semicontinuous function of y, while the corank, i. , the dimension of the kernel of d~, is an upper semicontinuous function of y. Since Hq(X,F0k(y)) A = Hq(K 0 k(y)) = KerdUimdr 1 , we deduce the semicontinuity theorem of Sect.