A survey on hyperbolicity of projective hypersurfaces (Q2908583)

The paper surveys a number of recent results on hyperbolicity of projective hypersurfaces. These are lecture notes of a course held at IMPA, Rio de Janiero, in September 2010: the purpose was to present recent results on Kobayashi hyperbolicity in complex geometry. NEWLINENEWLINENEWLINENEWLINE In the first chapter the authors state and describe the basic definitions of complex hyperbolic geometry. Then, they state and prove the classical Brody lemma and Picard's theorem. They conclude by giving a brief account of elementary examples and describing the case of Riemann surfaces. NEWLINENEWLINENEWLINENEWLINE Chapter 2 deals with an algebraic analogue of Kobayashi hyperbolicity. The authors explain how Kobayashi hyperbolicity implies restrictions on the ratio between the genus and the degree of algebraic curves contained in a hyperbolic projective algebraic manifold, and take this property as a definition. Then, they discuss some general conjecture and related results, in particular Bogomolov's proof of the finiteness of rational and elliptic curves on surfaces whose Chern classes satisfy a certain inequality. In the second part, an account of known results on algebraic hyperbolicity of generic projective hypersurfaces of high degree is done. NEWLINENEWLINENEWLINENEWLINE Chapter 3 is devoted to the theory of jet spaces and jet differentials. The authors describe the construction of the vector bundle of jet differentials and explain how to build in a functorial way a tower of projective bundles together with the corresponding tautological line bundles on any given manifold which provide a relative compactification of the classical jets spaces. Then, a characterization of jet differentials in terms of direct images of these tautological line bundles is given. NEWLINENEWLINENEWLINENEWLINE In Chapter 4 the authors explain how negativity properties of the curvature of complex manifolds is connected to hyperbolicity. They start with some basic notions of curvature and then prove the classical Ahlfors-Schwarz lemma. Then, they come back to higher order jets, and prove the basic result that every entire curve automatically satisfies every global jet differential with values in an antiample line bundle; as a consequence they deduce Bloch's theorem about entire curves on complex tori. To conclude the chapter they illustrate a general strategy to prove algebraic degeneracy of entire curves. NEWLINENEWLINENEWLINENEWLINE The main topic of this survey is hyperbolicity of generic projective hypersurfaces of high degree. The authors describe in chapter 5 how to prove it in the simpler case of surfaces in projective 3-space. While Kobayashi's conjecture predicts in the case of surfaces a lower bound for the degree equal to 5, nowadays the hyperbolicity is only known for degree greater than or equal to 18, after 36 and 21. NEWLINENEWLINENEWLINENEWLINE The last chapter is devoted to recent results on algebraic degeneracy of entire curves in generic projective hypersurfaces of high degree. The authors prove an algebraic degeneracy result for entire curves in generic projective hypersurfaces of high degree. The first part is concerned with finding jet differentials, then they cite a general result on meromorphic vector fields and discuss some effective aspects of the proof.