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**Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees.**
*(English)*
Zbl 0947.32012

Summary: We prove that the defect vanishes for a holomorphic map \(f\) from the affine 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 whose pole divisor is dominated by a divisor equivalent to \(pD\) and which vanishes along the \(k\)-jet space of \(D\) to order \(q\) with \(p/q\) smaller than a prescribed small positive number. Then estimates involving the theta function with divisor \(D\) and the logarithmic derivative lemma are used.

We also prove a pointwise Schwarz lemma which gives the vanishing of the pullback, by a holomorphic map from the affine complex line to a compact complex manifold, of a holomorphic jet differential vanishing on an ample divisor. This pointwise Schwarz lemma is a slight modification of a statement whose proof Green and Griffiths sketched in their alternative treatment of Bloch’s theorem on entire curves in abelian varieties. 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.

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 whose pole divisor is dominated by a divisor equivalent to \(pD\) and which vanishes along the \(k\)-jet space of \(D\) to order \(q\) with \(p/q\) smaller than a prescribed small positive number. Then estimates involving the theta function with divisor \(D\) and the logarithmic derivative lemma are used.

We also prove a pointwise Schwarz lemma which gives the vanishing of the pullback, by a holomorphic map from the affine complex line to a compact complex manifold, of a holomorphic jet differential vanishing on an ample divisor. This pointwise Schwarz lemma is a slight modification of a statement whose proof Green and Griffiths sketched in their alternative treatment of Bloch’s theorem on entire curves in abelian varieties. 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.

### MSC:

32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |

32H30 | Value distribution theory in higher dimensions |