DESCRIPTION
Offers an updated, fresh view of hyperbolicity-type results about projective hypersurfaces.
Presents new and classical concepts, like the basics of Kobayashi hyperbolicity and algebraic hyperbolicity.
Is as self-contained as possible, and uses straightforward language.
CONTENT
1 Kobayashi Hyperbolicity: basic theory
1.1 The Kobayashi distance
1.2 Brody’s criteion for hyperbolicity
1.3 Riemann surfaces and uniformization
2 Algebraic hyperbolicity
2.1 Hyperbolicity and genus of curves
2.2 Algebraic hyperbolicity of generic projective hypersurfaces of high degree
2.3 A brief history of the above results
3 Jets spaces
3.1 Projectivization of directed manifolds
3.2 Projectivized jet bundles
3.3 Jet differentials
4 Hyperbolicity and negativity of the curvature
4.1 Curvature and positivity
4.2 The Ahlfors-Schwarz lemma
4.3 A general strategy for algebraic degeneracy
5 Hyperbolicity of generic surfaces in projective 3-space
5.1 General strategy
5.2 Existence of jet differentials
5.3 Global generation of the twisted tangent space of the universal family
5.4 Proof of the hyperbolicity
6 Algebraic degeneracy for generic projective hypersurfaces
6.1 Statement of the result and scheme of the proof
6.2 Existence of the jet differentials
6.3 Proof of the existence of jet differentials in dimension 3
6.4 Proof of the existence of jet differentials in higher dimensions
6.5 Meromorphic vector fields
6.6 Effective aspects
Bibliography
ABOUT THE AUTHORS
Simone Diverio
Erwan Rousseau
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