By I. R. Shafarevich

This EMS quantity includes components. the 1st half is dedicated to the exposition of the cohomology thought of algebraic types. the second one half offers with algebraic surfaces. The authors have taken pains to provide the cloth conscientiously and coherently. The booklet comprises various examples and insights on numerous themes. This booklet can be immensely beneficial to mathematicians and graduate scholars operating in algebraic geometry, mathematics algebraic geometry, advanced research and comparable fields. The authors are recognized specialists within the box and I.R. Shafarevich is usually recognized for being the writer of quantity eleven of the Encyclopaedia.

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Additional info for Algebraic Geometry II: Cohomology of Algebraic Varieties: Algebraic Surfaces

Example text

The relation 1 d Lim (z) = Lim−1 (z) (m ≥ 2) dz z is obvious and leads by induction to the extension of the domain of definition of Lim to the cut plane C [1, ∞); in particular, the analytic continuation of the dilogarithm is given by z Li2 (z) = − 1 0 log(1 − u) du u for z ∈ C [1, ∞) . This paper is a revised version of a lecture which was given in Bonn on the occasion of F. Hirzebruch’s 60th birthday (October 1987) and has also appeared under the title “The remarkable dilogarithm” in the Journal of Mathematical and Physical Sciences, 22 (1988).

6) = 1 , 120 ζ(−5) = − 1 , 252 ... Not so the dilogarithm. As far as anyone knows, there are exactly eight values of z for which z and Li2 (z) can both be given in closed form: Li2 (0) = 0, π2 , Li2 (1) = 6 π2 Li2 (−1) = − , 12 1 1 π2 Li2 − log2 (2), = 2 12 2 The Dilogarithm Function √ 3− 5 2 √ −1 + 5 Li2 2 √ 1− 5 Li2 2 √ −1 − 5 Li2 2 Li2 = π2 − log2 15 = π2 − log2 10 √ 1+ 5 2 √ 1+ 5 2 π2 1 =− + log2 15 2 =− π2 1 + log2 10 2 7 , , √ 1+ 5 2 √ 1+ 5 2 , . Let me describe a recent experience where these special values figured, and which admirably illustrates what I said about the bizarreness of the occurrences of the dilogarithm in mathematics.

CJ and C and rational flunctions φ1 (z), . . , φJ (z) is a consequence of the five-term equation. ) The proof is given in §2 of Chapter II. D As well as the Bloch-Wigner function treated in this section, there are several other modifications of the “naked” dilogarithm Li2 (z) which have nice properties. These are discussed in §1 of Chapter II. E Now much more information about the actual order of K2 (OF ) is available, thanks to the work of Browkin, Gangl, Belabas and others. Cf. [7], [3] of the bibliography to Chapter II.

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