By A.N. Parshin, I.R. Shafarevich, I. Rivin, V.S. Kulikov, P.F. Kurchanov, V.V. Shokurov

The 1st contribution of this EMS quantity on complicated algebraic geometry touches upon the various crucial difficulties during this sizeable and intensely lively sector of present learn. whereas it's a lot too brief to supply entire insurance of this topic, it offers a succinct precis of the components it covers, whereas supplying in-depth insurance of convinced extremely important fields.The moment half offers a quick and lucid creation to the hot paintings at the interactions among the classical sector of the geometry of advanced algebraic curves and their Jacobian types, and partial differential equations of mathematical physics. The paper discusses the paintings of Mumford, Novikov, Krichever, and Shiota, and will be an outstanding significant other to the older classics at the topic.

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**Extra info for Algebraic geometry III. Complex algebraic varieties. Algebraic curves and their Jacobians**

**Sample text**

The resulting space, if one blows up further the irrelevant points (0, ∞), (∞, 0) is also the Deligne-Mumford compactification for the moduli space of curves of genus 0 with 5 marked points. One gets the picture in Fig. 1 xy=1 y=1 Li=Li(1,1) x= 0 x=1 Li=Li(y) y=0 Li= 0 Fig. 1 Analytic continuation to (1, 1) We now try to compute Li(a,b) (1, 1). First we should interpret it as the value of Li(a,b) on a tangent vector (x, y¯) = (1, 1) at the point (1, 1). The first step in the computation of this value is to restrict to the divisor y = 0.

The equations are therefore going to become ⎧ ⎪ ⎪ d ⎨Li(a−1,b) (x, 0) a > 1 x Li(a,b) (x, 0) = ⎪ ⎪ ⎩0 dx a=1. Since the boundary conditions on these functions are always set so that the constant term at 0 is 0 it follows immediately that the function Li(a,b) (x, 0) is identically 0. This is not surprising based on its expansion and the identity principle. We now repeat the same considerations but this time restricting to the divisor x = 1. In the same way as before the diﬀerential equations are going to be ⎧ ⎪ ⎪ d ⎨Li(a,b−1) (1, y) b > 1 y Li(a,b) (1, y) = ⎪ y ⎪ ⎩ 1−y dy Lia (y) b=1 1 Heidelberg Lectures on Coleman Integration 39 which are of course the same diﬀerential equations satisfied by the single variable Li(a,b) .

Namely, one easily checks that if D = div( f ), then D ω is the global index of Fω and log( f ), and is thus the cup product in H1dR (X/K) of the projections of ω and d f / f , but the projection of d f / f is 0. This method extends to the bad reduction case [Bes05] but it then needs Coleman integration in the bad reduction case, which either uses the methods of Colmez and Zarhin or that of Vologodsky which is far more complicated. (2) Again in the good reduction case it follows from [Bes00a] that the integral factors via the syntomic regulator on the Chow group of zero cycles on X, see below, thus killing principal divisors.