Algebraic Geometry, Hirzebruch 70: Proceedings of an by Mica Szurek, Jarosaw Wisniewski, Piotr Pragacz

By Mica Szurek, Jarosaw Wisniewski, Piotr Pragacz

This booklet provides the court cases from the convention on algebraic geometry in honor of Professor Friedrich Hirzebruch's seventieth Birthday. the development was once held on the Stefan Banach overseas Mathematical heart in Warsaw (Poland). the subjects coated within the e-book contain intersection concept, singularities, low-dimensional manifolds, moduli areas, quantity conception, and interactions among mathematical physics and geometry. additionally incorporated are articles from notes of 2 designated lectures. the 1st, by way of Professor M. Atiyah, describes the $64000 contributions to the sphere of geometry through Professor Hirzebruch. the second one article comprises notes from the debate introduced on the convention by way of Professor Hirzebruch. members to the amount are best researchers within the box.

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The dilogarithm function in geometry and number theory1 The dilogarithm function is the function defined by the power series Li2 (z) = ∞ n=1 zn n2 for |z| < 1 . The definition and the name, of course, come from the analogy with the Taylor series of the ordinary logarithm around 1, − log(1 − z) = ∞ n=1 zn n for |z| < 1 , which leads similarly to the definition of the polylogarithm Lim (z) = ∞ n=1 zn nm for |z| < 1, m = 1, 2, . . 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, ∞) .

The same values reappear in connection with Nahm’s conjecture in the case of rank 1 (see §3 of Chapter II). C Wojtkowiak proved the general theorem that any functional equation of J the form j=1 cj Li2 (φj (z)) = C with constants c1 , . . , 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.

The corresponding analogue £1 (x) = p−1 n 1 n=1 x /n of the 1-logarithm was first proposed (under the name “The 1 2 logarithm”) by M. Kontsevich in a note in the informal Festschrift prepared on the occasion of F. Hirzebruch’s retirement as director of the Max Planck Institute for Mathematics in Bonn [23]. Kontsevich showed that this function, as a function from Fp to Fp , satisfies the 4-term functional equation £1 (x + y) = £1 (y) + (1 − y)£1 x 1−y + y £1 − x y (16) (the mod p analogue of the “fundamental equation of information theory” satisfied by the classical entropy function −x log x − (1 − x) log(1 − x)), midway between the 3-term functional equation of log(xy) − log(x) − log(y) = 0 of the classical logarithm function and the 5-term functional equation of the classical dilogarithm, and also that £1 is characterized uniquely by this functional equation together with the two one-variable functional equations £1 (x) = £1 (1 − x) and x£1 (1/x) = −£1 (x).

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