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Additional resources for Abelian varieties
If we know already that j:l = v1J = I(Z(J:l)) for all prime ideals j:l , this follows from Z (J:l) c Z (I). 6). As the formulation of this theorem is already quite involved and its proof is rather lengthy, we will not attempt to present it here. We will state the following theorem without proof. It should however be noted that we actually present only two aspects of the full result. Theorem 1 . 30 Let X c e n be an irreducible analytic germ defined by a prime ideal p C Ocn , o . Then one can find a coordinate system such that the projection (z 1 , .
First note that for any local biholomorphic map f the pull-back f*w is the associated fundamental form to f*g.
7 ( a-Poincare lemma in one variable) on Be satisfies a = ag. Proof. Note first, that for = x+iy one has dwl\dw = (dx+idy)l\ (dx-idy) = -2idx 1\ dy. The existence of g as well as the assertion a = ag will be shown by splitting g into two parts. This splitting will depend on a chosen point w zo E Be or rather on a neighbourhood of such a point. Let z0 E B := Be and let '1/J : B JR. 3 zo Differential Forms h := ( 1 =h+ (w) dw 1\ dw, � = 1 , 2. 1 -fi 9i (z) := -. 1 21f� B w - z 45 h := of E V C B.