A Family of Hyperbolic Hypersurfaces in the Complex Projective Space

摘要

In 1970, S. Kobayashi asked whether generic hypersufaces in P~n(C) of large degree are hyperbolic or not([6]). This problem is still open, but some papers were devoted to giving various examples of hyperbolic hypersurfaces. In 1977, R. Brody-M. Green gave an example of hyperbolic hypersurfaces in P~3(C) of degree d = 2p for p ≥ 25. Afterwards, A. Nadel, J. Goul, J. P. Demailly and Y. T. Siu-S. K. Yeung obtained hyperbolic hypersurfaces in P~3(C) of degree d = 6p + 3 for p ≥ 3 in 1989([9]), of arbitrary degree ≥ 14 in 1996([5]) and of arbitrary degree ≥ 11 in 1997([3]) and independently the same result in the same year([l1]), respectively. Recently, M. Shirosaki constructed a hyperbolic hypersurface in P~3 (C) of degree 10([10]). On the other hand, in 1996, K. Masuda and J. Noguchi proved that there exists a smooth hyperbolic hypersurface of every degree d ≥ d(n) for a positive integer d(n) depending only on n and some concrete exmaples of hyperbolic hypersurfaces in P~n(C) for n ? 5([7]). Moreover, in 1999, M. McQuillan showed that a generic hypersurface in P~3 (C) of sufficiently large degree is hyperbolic([8]). The purpose of this talk is to explain some new examples of hyperbolic hypersurfaces in P ~n (C), which are obtained in [4].