Hyperbolicity of General Deformations

This is the content of the talk given at the conference “Effective Aspects of Complex Hyperbolic Varieties”, Aver Wrac’h, France, September 10-14, 07. We present two methods of constructing low degree Kobayashi hyperbolic hypersurfaces in Pn: • The projection method • The deformation method The talk is based on joint works of the speaker with B. Shiffman and C. Ciliberto. 1. DIGEST on KOBAYASHI THEORY 1.1. Kobayashi hyperbolicity. DEFINITION The Kobayashi pseudometric kX on a complex space X satisfies the following axioms : (i) On the unit disc ∆, the Kobayashi pseudometric k∆ coincides with the Poincaré metric; (ii) every holomorphic map φ : ∆ → X is a contraction: φ∗(kX) ≤ k∆; (iii) kX is the maximal pseudometric on X satisfying (i) and (ii). REMARK Every holomorphic map φ : X → Y is a contraction: φ∗(kY ) ≤ kX . DEFINITION X is called Kobayashi hyperbolic if kX is non-degenerate : kX(p, q) = 0 ⇐⇒ p = q. EXAMPLES kCn ≡ 0, kPn ≡ 0, kTn ≡ 0 , where T n = C/Λ is a complex torus, whereas C \ {0, 1} is hyperbolic (the Schottky-Landau Theorem.) 1.2. Classical theorems. According to the above definition and to Royden’s Theorem, X is hyperbolic iff natural analogs of the classical Schottky and Landau Theorems hold for X. Brody-Kiernan-Kobayashi-Kwack THEOREM For a compact complex space X the following conditions are equivalent : 2000 Mathematics Subject Classification: 14J70, 32J25.