Algebraic Geometry

Download e-book for iPad: An Introduction to Algebraic Geometry by K. Ueno

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By K. Ueno

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1 Let D be the subgroup of [ ( X i ) B 3 ) ] generated ’ by all elements of type corresponding to C1 @ C2 @ C3 where the Ci are disjoint simple closed curves in X . Then jl : 1 ~ I3 (Xz ) + A3(X$)’= P mups D onto P and 2 1 (twice the iterated integral homomorphism I : (XiB3)’+ R/Z) when restricted to D factors through jl : D 4 P and defines a homomorphism denoted v =21:P + R/Z. We have shown that for any d E D and u any transposition, I(d a(d))= 0. The kernel of j 3 : D + P is the intersection D n K where K is the subgroup of (Xk)B3generated by all “monomials” m = x @ y @ z with two of the factors z, y, z equal.

R n ~ 9 - ~such ( X ) that u ( D ) E - u(K Symg-l) and none of the 3g - 3 points are equal. So we have proved the existence of the “general” divisors D , and the nontriviality of the section I of the family of primitive intermediate Jacobians, taking a point corresponding to X to the Abel-Jacobi image v ( X ) . However this result does not directly exhibit a Riemann surface X where this AbelJacobi image v(X) is non-zero. A little further on in this chapter we will exhibit such an X , namely a well-known algebraic curve (of genus 3) x4 + y4 = 1 defined over Z: to do this we will just calculate some iterated integrals.

We then say that 2 is homologous to 0. Suppose further there is an algebraic (or complex analytic) subset S of V of complex dimension 2 such that the topological 3-chain W lies on S: we then say that 2 is algebraically equivalent t o 0 (in V).

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