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On Bloch’s conjecture. (English) Zbl 0569.32012
In this paper everything is defined over $${\mathbb{C}}$$. Let A be an abelian variety, let X be a closed reduced irreducible subset of A and let $$\alpha$$ : $$X\to A$$ be the inclusion. Let T be the space of 1-forms on A. We shall prove the following theorem 1. If X is an algebraic variety of general type of dimension n, then there exists a system $$\{\omega_ 1,...,\omega_{n+1}\}$$ in T such that $$\{\alpha^*(\omega_ 1\wedge...\wedge \omega_{i-1}\wedge \omega_{i+1}\wedge...\wedge \omega_{n+1})\}_{i=1,...,n+1}$$ are linearly independent. - T. Ochiai proved in Invent. Math. 43, 83-96 (1977; Zbl 0374.32006)] that from theorem 1 we can deduce theorem 2 (Bloch’s conjecture). Let X be a projective algebraic manifold of dimension n with irregularity $$>n$$. Then, any holomorhic curve $$f: {\mathbb{C}}\to K$$ is degenerate.
[See also the paper by P.-M. Wong in Am. J. Math. 102, 493-502 (1980; Zbl 0439.32010).]

##### MSC:
 32H99 Holomorphic mappings and correspondences 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 14K20 Analytic theory of abelian varieties; abelian integrals and differentials
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##### References:
 [1] Noguchi, J.: Some surfaces of hyperbolic type and holomorphic curves. Hiroshima Math. J., in press (1980) · Zbl 0435.32017 [2] Noguchi, J.: Holomorphic curves in algebraic varieties. Hiroshima Math. J., in press (1979) · Zbl 0412.32025 [3] Ochiai, T.: On holomorphic curves in algebraic varieties with ample irregularity. Invent. Math.,43, 83-96 (1977) · Zbl 0374.32006
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