Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. (English) Zbl 0919.32014

Kollár, János (ed.) et al., Algebraic geometry. Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9–29, 1995. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62(pt.2), 285-360 (1997).
This are notes of a series of lectures delivered at the Santa Cruz AMS Summer School on Algebraic Geometry. They are mainly devoted to the study of complex varieties through a few geometric questions related to hyperbolicity in the sense of Kobayashi. A convenient framework for this is the category of “directed manifolds”, that is, the category of pairs \((X,V)\) where \(X\) is a complex manifold and \(V\) a holomorphic subbundle of \(T_X\). If \(X\) is compact, the pair \((X,V)\) is hyperbolic if and only if there are no nonconstant entire holomorphic curves \(f:\mathbb{C}\to X\) tangent to \(V\) (Brody’s criterion). The author describes a construction of projectivized \(k\)-jet bundles \(P_kV\), which generalizes a construction made by Semple in 1954 and allows to analyze hyperbolicity in terms of negativity properties of the curvature.
An overview information on the lecture notes is given by their contents.
1. Hyperbolicity concepts and directed manifolds
2. Hyperbolicity and bounds for the genus of curves
3. The Ahlfors-Schwarz lemma for metrics of negative curvature
4. Projectivization of a directed manifold
5. Jets of curves and semple jet bundles
6. Jet differentials
7. \(k\)-Jet metrics with negative curvature
8. Algebraic criterion for the negativity of jet curvature
9. Proof of the Bloch theorem
10. Logarithmic jet bundles and a conjecture of Lang
11. Projective meromorphic connections and Wronskians
12. Decomposition of jets in irreducible representations
13. Riemann-Roch calculations and study of the base locus
14. Appendix: A vanishing theorem for holomorphic tensor fields.
For the entire collection see [Zbl 0882.00033].


32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
14J40 \(n\)-folds (\(n>4\))
53C55 Global differential geometry of Hermitian and Kählerian manifolds