Hyperbolicity of surfaces modulo rational and elliptic curves (Q1097377)

Let \(X\) be a smooth compact complex surface of general type with irregularity \(q\) and \(D\) the union of all rational and elliptic curves in \(X\). It follows from the Bloch conjecture that the image of every nonconstant holomorphic map \(\mathbb C\to X\) is contained in \(D\) if \(q>2\) or, by a strengthened version of the conjecture for surfaces of general type proved by the author [Duke Math. J. 53, 345--358 (1986; Zbl 0612.14036)], if \(q=2\) and the Albanese torus \(A(X)\) is simple. Call \(X\) hyperbolic modulo \(D\) if the Kobayashi pseudo-distance \(d_ X\) has the property \(d_ X(x,y)>0\) unless \(x=y\) or \(x,y\in D\). Using the above fact and Brody's theorem the author proves that \(X\) is hyperbolic modulo \(D\) if \(q\geq 2\) and if \(X\) admits a nonconstant holomorphic map into a complex torus whose image does not contain an elliptic curve. In particular this is the case if \(q(X)\geq 2\) and the Albanese torus of \(X\) is not isogenous to a product of elliptic curves. The weaker statement that \(X\setminus D\) is hyperbolic is verified for some special situations. For a proof of the Bloch conjecture see \textit{T. Ochiai} [Invent. Math. 43, 83-- 96 (1977; Zbl 0374.32006)], \textit{M. Green} and \textit{P. Griffiths} [Differential geometry, Proc. int. Chern Symp., Berkeley 1979, 41--74 (1980; Zbl 0508.32010)], \textit{Y. Kawamata} [Invent. Math. 57, 97--100 (1980; Zbl 0569.32012)], \textit{P.-M. Wong} [Am. J. Math. 102, 493--502 (1980; Zbl 0439.32010)].