DEFECTS FOR AMPLE DIVISORS OF ABELIAN-VARIETIES, SCHWARZ-LEMMA, AND HYPERBOLIC HYPERSURFACES OF LOW DEGREES

We prove that the defect vanishes for a holomorphic map f from the aff ine complex line to an abelian variety A and for an ample divisor D in A. The proof uses the translational invariance of the Zariski closure of the k-jet space of the image of f and the theorem of Riemann Roch to construct a nonidentically zero meromorphic k-jet differential whos e pole divisor is dominated by a divisor equivalent to pD and which va nishes along the k-jet space of D to order q with p/g smaller than a p rescribed small positive number. Then estimates involving the theta fu nction with divisor D and the logarithmic derivative lemma are used. W e also prove a pointwise Schwarz lemma which gives the vanishing of th e pullback, by a holomorphic map from the affine complex line to a com pact complex manifold, of a holomorphic jet differential vanishing on an ample divisor. This pointwise Schwarz lemma is a slight modificatio n of a statement whose proof Green and Griffiths sketched in their alt ernative treatment of Bloch's theorem on entire curves in abelian vari eties. The log-pole case of the pointwise Schwarz lemma is also given. We construct examples of hyperbolic hypersurface whose degree is only 16 times the square of its dimension.