Holomorphic curves and integral points off divisors (Q1601785)

The main theorem of the paper is as follows. Let \(M\) be a compact Kähler manifold of dimension \(m\). Let \(\{D_1,\dots,D_\ell\}\) be a collection of hypersurfaces in \(M\) in general position (i.e., any irreducible component of the intersection of any \(r\) of them has codimension \(r\)). Let \(r=\text{rank}_{\mathbb Z}\{c_1(D_i):i=1,\dots,\ell\}\). Let \(f\:\mathbb C\to M\setminus\bigcup_{i=1}^\ell D_i\) be a non-constant holomorphic curve. Let \(W\) be the Zariski closure of the image of \(f\) (i.e., the smallest closed complex subspace containing the image of \(f\)), and let \(q(W)\) denote its irregularity. Then: (i) \(\ell+q(W)\leq\dim W+r\); and (ii) if all of the \(D_i\) are ample, then \((\ell-m)\dim W\leq \max\{m(r-q(W)),0\}\). The proof includes a generalization to the Kähler case of the logarithmic Bloch-Ochiai theorem of \textit{J. Noguchi} [Geometric complex analysis: Proc. Conf. 3rd Internat. Research Inst. Math. Soc. Japan, Hayama, March 19-29(1995), 489-503 (1996; Zbl 0927.32015)]. An analogue of part (ii) of the above theorem is also proved in the context of integral points, where in this case \(M\) and \(D_1,\dots,D_\ell\) are assumed to be algebraic and defined over some given number field. Some corollaries of these two theorems are also given.