Kobayashi measure hyperbolicity for singular directed varieties of general type (Q309765)

Let \((X, V)\) be a complex directed manifold, i.e., \(X\) is a complex manifold equiped with a holomorphic subbundle \(V \subset TX\), where \(V\) is possible singular.NEWLINENEWLINEThe author proves the non-degeneracy of the Kobayashi-Eisenman volume measure of a singular directed variety \((X, V)\), i.e., the Kobayashi measure hyperbolicity of \((X, V)\), as long as the canonical sheaf \(\mathcal{K}_V\) of \(V\) is big in the sense of Demailly.NEWLINENEWLINEThe result is proved for a smooth directed variety in [\textit{J.-P. Demailly}, Kobayashi pseudo-metrics, entire curves and hyperbolicity of algebraic varieties. Saint-Martin d'Hères: Institute Fourier (2012)].NEWLINENEWLINE\noindent In the absolute case \(V=TX\), the result is proved in [\textit{S. Kobayashi}, et al., J. Math. Soc. Japan 23, 137--148 (1971, Zbl 0203.39101)] and [\textit{P.A. Griffiths}, Ann. Math. (2) 93, 439--458 (1971, Zbl 0214.48601)].