Algebraic Hyperbolicity and Lang Conjecture

Lang猜想指出，各种一般类型的有理曲线和椭圆曲线在代数上是*退化的。我们研究射影空间中一般超曲面的这一猜想。这个问题与复双曲性和小林猜想密切相关。在代数环境中，这相当于*Demailly意义下的代数双曲性。H.Clemens证明了当超曲面的次数足够大时，一般超曲面是*代数双曲的。我们想把他关于超曲面次数的*界推到猜想的最优。我们建议通过构造超曲面族的整体射流空间并研究相应射丛的稳定轨迹来逼近这个*猜想。

Lang conjecture says that rational and elliptic curves on a variety of general type are algebraically***degenerate. We study this conjecture for generic hypersurfaces in projective spaces. This problem is***closely related to complex hyperbolicity and Kobayashi conjecture. In the algebraic setting, this amounts***to algebraic hyperbolicity in the sense of Demailly. H. Clemens proved that a generic hypersurface is***algebraically hyperbolic when the degree of the hypersurface is sufficiently large. We want to push his***bound on the degree of hypersurfaces to the conjectured optimum. We propose to approach this***conjecture by constructing the global jet spaces of versal families of hypersurfaces and studying the stable***loci of the corresponding jet bundles.