Hyperbolicity of General Deformations: Proofs

We modify the deformation method from [9] in order to construct further examples of Kobayashi hyperbolic surfaces in P of any even degree d ≥ 8. Given a hypersurface Xd = f ∗ d (0) in P n of degree d, we say that a (very) general small deformation of Xd is hyperbolic if for any (very) general degree d hypersurface X∞ = g ∗ d(0) and for all sufficiently small ε ∈ C \ {0} (depending on X∞) the hypersurface Xd,ε = (fd + εgd) (0) is Kobayashi hyperbolic. With this definition let us formulate the following version of the Kobayashi Conjecture. Weak Kobayashi Conjecture. For every hypersurface Xd in P n of degree d ≥ 2n− 1, a (very) general small deformation of Xd is Kobayashi hyperbolic. The original Kobayashi Conjecture claims, in particular, that a (very) general surface Xd of degree d ≥ 5 in P is Kobayashi hyperbolic. This is known to hold indeed for a very general surface of degree at least 21 (see McQuillan [7] and Demailly-El Goul [2]). By Brody’s Theorem, a compact complex space X is hyperbolic if and only if any holomorphic map C → X is constant. Hence the proof of hyperbolicity reduces to a certain degeneration principle for entire curves in X. The Green-Griffiths’ proof of Bloch’s Conjecture [6] provides a kind of such degeneration principle. According to this principle, every entire curve φ : C → X in a very general surface X ⊆ P of degree d ≥ 21 satisfies an algebraic differential equation [2, 7]. See also [8, 12] for recent advances in higher dimensions. The deformation method showed to be quite effective to construct examples of low degree hyperbolic surfaces in P. A nice construction due to J. Duval [3] of a hyperbolic sextic Xε ⊆ P 3 uses this method iteratively in 5 steps, so that ε = (ε1, . . . , ε5) has 5 subsequently small enough components. Hence Xε belongs to a 5-dimensional linear system; however the deformation of X0 to Xε neither is linear nor very generic. In [9] we exhibited examples of some special surfaces Xd in P 3 of any given degree d ≥ 8 such that a general small deformation of Xd is Kobayashi hyperbolic. In these examples Xd = X ′ d ∪X ′′ d , where d = d ′ + d, is a union of two cones in P with distinct vertices over plane hyperbolic curves in general position. Let us indicate briefly the deformation method used in [9] (see also the references in [9, 10]). Given two hypersurfaces Xd,0 and Xd,∞ in P n of the same degree d, we consider the pencil of hypersurfaces {Xd,ε}ε∈C generated by Xd,0 and Xd,∞. Assuming that for a sequence εn → 0, the hypersurfaces Xd,εn are not hyperbolic, there exists a sequence of Brody entire curves φn : C → Xd,εn which converges to a (non-constant) Brody curve φ : C → Xd,0. Suppose in addition that the hypersurface Xd,0 admits a rational map to a hyperbolic variety 2000 Mathematics Subject Classification: 14J70, 32J25.