Hyperbolicity of Generic High-Degree Hypersurfaces in Complex Projective Space

We use two ingredients to prove the hyperbolicity of generic hypersurfaces of sufficiently high degree and of their complements in the complex projective space. One is the pullbacks of appropriate low pole order meromorphic jet differentials on the complex projective space to a hypersurface. The other is slanted vector fields of low vertical pole order on the vertical jet space of the universal hypersurface. We also present a number of related results, obtained by the same methods, such as: (i) a Big-Picard-Theorem type statement concerning extendibility, across the puncture, of holomorphic maps from a punctured disk to a generic hypersurface of high degree, (ii) nonexistence of nontrivial sets of entire functions satisfying certain polynomial equations with slowly varying coefficients, and (iii) Second Main Theorems for jet differentials and slowly moving targets.


Introduction
In this paper we are going to present the proofs of the following two theorems on the hyperbolicity of generic hypersurfaces of sufficiently high degree and of their complements, together with a number of related results, obtained by the same methods, such as: (i) a Big-Picard-Theorem type statement concerning extendibility, across the puncture, of holomorphic maps from a punctured disk to a generic hypersurface of high degree, (ii) entire holomorphic functions satisfying polynomial equations with slowly varying coefficients, and (iii) Second Main Theorems for jet differentials and slowly moving targets.
Theorem 0.1. For any integer n ≥ 3 there exists a positive integer δ n (which is explicitly expressible as a function of n) with the following property. For any generic hypersurface X in P n of degree δ ≥ δ n there is no nonconstant holomorphic map from C to X. 1 Partially supported by grant DMS-1001416 from the National Science Foundation.
Theorem 0.2. For any integer n ≥ 2 there exists a positive integer δ * n (which is explicitly expressible as a function of n) with the following property. For any generic hypersurface X in P n of degree δ ≥ δ * n there is no nonconstant holomorphic map from C to P n − X.
Theorem 0.1 was presented with a sketch of its proof in [Si02] and [Si04]. The methods used, though rather tedious in some of their details, consist essentially just of some skillful manipulations in linear algebra and the chain rule of differentiation. The underlying ideas in these methods can be traced to the techniques which Bloch developed in his 1926 paper [B26]. To explain this link to Bloch's paper [B26], we first very briefly describe Bloch's techniques with explanations about how they foreshadow to a certain extent our techniques in this paper.

Bloch's Technique of Construction of Jet Differential.
In his 1926 paper [B26] Bloch proved the nonexistence of nonconstant holomorphic maps from C to a submanifold Y of an abelian variety A, which does not contain a translate of a positive-dimensional abelian subvariety of A, by producing sufficiently independent holomorphic jet differentials ω on Y vanishing on some ample divisor of Y and using the fact that the pullbacks of such jet differentials by holomorphic maps from C to Y vanish identically.
He produced such holomorphic jet differentials ω on Y , not by applying to Y the theorem of Riemann-Roch (which was not yet readily available at the time of Bloch's paper for the case needed for its application to Y ), but explicitly by pulling back to Y constant-coefficient polynomials P (with homogeneous weight) of differentials of the coordinates (including higher-order differentials) of the universal coverÃ of the abelian variety A. When the constant-coefficient polynomials P of differentials of coordinates ofÃ are pulled back to Y , the condition of Y not containing a translate of a positive-dimensional abelian subvariety of A causes new vanishing of the pullbacks on Y . Moreover, the new vanishing is on some ample divisor of Y when the constant-coefficient polynomials P of differentials of coordinates ofÃ are appropriately chosen. The reason why it is possible to choose P so that its pullback ω to Y vanishes on an ample divisor of Y is that the condition of not containing a translate of a positive-dimensional abelian subvariety of A guarantees that the dimension of the C-vector space of the pullbacks to Y of all possible such polynomials P is so high that at least one Clinear combination ω of such pullbacks vanishes on some ample divisor of Y .
Bloch's construction is related to the classical construction of a Cbasis of holomorphic 1-forms for a regular plane curve C defined by an equation R(x, y) = 0 of degree δ ≥ 3 in the inhomogeneous coordinates x, y of P 2 , which are constructed by pulling back to C meromorphic 1forms P (x, y) dx R y (x, y) = P (x, y) −dy R x (x, y) of "low pole order" on P 2 , where R x (x, y) and R y (x, y) are the firstorder partial derivatives of R(x, y) and P (x, y) is a polynomial of degree ≤ δ − 3. The adjunction formula for the plane curve C causes new vanishing to cancel the "low pole order" of the meromorphic 1-forms on P 2 to yield holomorphic 1-forms on C when the meromorphic 1forms on P 2 are pulled back to the plane curve C.
In this paper, the construction of holomorphic jet differentials on a generic hypersurface X of sufficiently high degree δ in P n combines Bloch's method and the classical construction of holomorphic 1-forms on plane curves of high degree. We take meromorphic jet differentials of low pole orders (of magnitude δ 1−ε for some appropriate 0 < ε < 1) on P n and pull them back to X. The high degree δ of X will guarantee (according to Lemma 3.4 concerning the injectivity of the pullback map for certain jet differentials) that the dimension of the C-vector space of such pullbacks is so high that some C-linear combination of such pullbacks will be a non identically zero holomorphic jet differential on X vanishing on some ample divisor of X (see Proposition 3.8 below).
One key point in this argument is that, because the dimension of P n is higher than that of X, there are more degrees of freedom in constructing meromorphic (n − 1)-jet differentials of low pole order on P n and, if the pullback map to X of such meromorphic (n − 1)-jet differentials is injective, there are sufficient independent pullbacks to X to form a non identically zero C-linear combination which vanishes on an ample divisor of X.

Key Technique of Slanted Vector Fields.
After so many decades of impasse, the real key which opens the way to the proof of the hyperbolicity of generic hypersurface of sufficiently high degree (in the sense stated in Theorem 0.1) is the introduction in [Si02] [Si04] of the technique of slanted vector fields in the subspace J (vert) n−1 (X ) of vertical (n − 1)-jets in the (n − 1)-jet space J n−1 (X ) of the universal hypersurface X of degree δ in P n ×P N (where N = δ+n n −1). For a complex manifold Y the space J k (Y ) of k-jets of Y consists of all k-jets of Y (each of which is represented by a parametrized complex curve germ). The universal hypersurface X of degree δ in P n × P N (with N = δ+n n − 1) is defined by (0.4.1) ν 0 +···+νn=δ α ν 0 ,··· ,νn z ν 0 0 · · · z ν 0 0 = 0, where α = [α ν 0 ,··· ,νn ] ν 0 +···+νn=δ is the homogeneous coordinate of P N and [z 0 , · · · , z n ] is the homogeneous coordinate of P n . For α ∈ P N let X (α) be the hypersurface of degree δ defined by (0.4.1) when α is fixed as constant. A vertical k-jet of X is a k-jet in X representable by some (parametrized) complex curve germ lying completely in some fiber X (α) of X . We denote by J (vert) k (X ) the space of all vertical k-jets on X . There is a projection map π k,vert : J (vert) k (X ) → P N such that an element P 0 of J (vert) k (X ) is represented by a (parametrized) complex curve germ in X (α) with α = π k,vert (P 0 ).
A slanted vector field ξ on J (vert) k (X ) means a vector field of J (vert) k (X ) which at a generic point P 0 of J (vert) k (X ) is not tangential to the space J k X (α) of k-jets of the fiber X (α) at the point P 0 of J k X (α) with α = π k,vert (P 0 ). When a local k-jet differential form on X (α) defined for α in some open subset U of P N is regarded as a local function on J (vert) k (X ) and is differentiated with respect to ξ, the result is a local function on J (vert) k (X ) which is represented by a local k-jet differential on X (α) for α ∈ U. In the case of k = n − 1, meromorphic slanted vector fields ξ of low vertical pole-order (of the magnitude O Pn (n 2 ) on the vertical fiber), whose existence is given in Proposition 2.17 below, play the following indispensable role in generating sufficiently independent holomorphic (n − 1)-jet differentials on X (α) vanishing on an ample divisor for a generic α and for δ sufficiently large.
On a regular hypersurface X (α) of high degree δ, there cannot be any nonzero meromorphic vector fields on X (α) of low pole order. However, the universal hypersurface X of degree δ in P n × P N has bidegree (δ, 1) with respect to the two hyperplane section line bundles O Pn (1) and O P N (1). Because of the second component 1 in the bidegree (δ, 1) of X , when slanted vector fields are used, it is possible to get meromorphic slanted vector fields of J For a genericα ∈ P n the holomorphic (n − 1)-jet differential ω (α) on X (α) vanishing on an appropriate ample divisor (constructed by pulling back an appropriate meromorphic (n − 1)-jet differential of low pole order on P n according to Proposition 3.8 below) can be extended to a holomorphic family ω (α) on X (α) for α in some open neighborhood U ofα in P N so that successive application of different finite sets of meromorphic slanted vector fields ξ 1 , · · · , ξ ℓ of low vertical pole order (as constructed in in Proposition 2.17 below) would yield sufficiently independent holomorphic jet differentials vanishing on ample divisor on X (α) so that the application of the Schwarz lemma of the vanishing of pullbacks, to C by a holomorphic map C → X (α) , of holomorphic jet differentials vanishing on ample divisor of X (α) would force every holomorphic map from C to X (α) to be constant (see Lemma 4.1, Proposition 4.3, and the proof of Theorem 0.1 given in 4.4 below).
We would like to remark that the Schwarz lemma of the vanishing of pullbacks, to C by a holomorphic map C → X (α) , of holomorphic jet differentials vanishing on ample divisor of X (α) also has its origin in Bloch's 1926 paper [B26], though its formulation and its proof there are in a form very different from our current way of mathematical presentation. Bloch's proof of applying Nevanlinna's logarithmic derivative lemma (p.51 of [Ne25]) to local coordinates which are the logarithms of global meromorphic functions still is the best proof of the Schwarz lemma. It is recast in the current language of mathematical presentation on pp.1162-1164 of [SY97]. The technique of slanted vector fields in a way also finds some remote ancestry in the 1926 paper of Bloch [B26], though the connection is not so transparent. We point this out here in order to dispel the wrong perception that the technique of slanted vector fields is applicable to generic hypersurfaces of high degree because of the variation of complex structure of hypersurfaces as the coefficients of their defining functions vary.
In his proof of the hyperbolicity of a submanifold Y in an abelian variety A which contains no translate of positive-dimensional abelian subvariety of A, Bloch used the vector fields from translations in A to do the differentiation of jet differentials. That is the reason why when such differentiations cannot yield enough independent holomorphic jet differentials to give hyperbolicity of Y , Y must contain a translate of some positive-dimensional abelian subvariety of A. This link of the use of slanted vector fields to Bloch's technique of using vector fields of maps by translation is obscured by the fact that in Bloch's techqniuqe the result of differentiation with respect to vector fields of maps by translation is just the same as the use of a different constant-coefficient polynomial in differentials of the coordinates of the universal coverÃ of A.
Let us return to our situation at hand of using slanted vector fields ξ on J k X (α) of low vertical pole order for k = n − 1. Though the complex structure of the hypersurface X (α) changes as α varies in P N , the slanted vector fields ξ in general do not respect the fibers in the sense that for two distinct points P 0 and P ′ 0 on J k X (α) the two projections π k,vert (ξ P 0 ) and π k,vert ξ P ′ 0 are in general different vectors in the tangent space of P N at α.
For our situation at hand, the geometric picture is not the pulling back of k-jet differential from a neighboring fiber by the slanted vector field ξ, even in the infinitesimal setting. What is relevant is the existence of slanted vector fields pointing in sufficiently many different directions on J (vert) k (X ) at the prescribed point in question. The realization of the irrelevancy of the variation of the complex structure X (α) as α varies in P N , as well as the interpretation of Bloch's technique of differentiation with respect to vector fields of maps by translation, points to the promise of the applicability of our method even to the case of some rigid complex manifolds Z inside some P m as a submanifold of possibly high codimension. In certain cases, though Z may be rigid as a compact complex manifold, yet there is a possibility that appropriate meromorphic vector fields on P n applied to pullbacks to Z of low pole-order meromorphic jet differentials on P n may yield sufficiently independent holomorphic jet differentials on Z vanishing on an ample divisor. 0.6. Necessity of Use of Vertical Jet Space.
The reason why the more complicated space J (vert) n−1 (X ) of vertical (n − 1)-jets of X has to be used instead of the simpler (n − 1)-jet space J n−1 (X ) of X is that, while it is possible to extend a holomorphic (n − 1)-jet differential ω (α) on a hypersurface X (α) for a genericα ∈ P N to a holomorphic family of ω (α) on X (α) for α in an open neighborhood U ofα in P N , it is in general impossible to find a holomorphic (n − 1)jet differential on the part of J n−1 (X ) above some open neighborhood U ofα in P N whose pullback to X (α) is equal to ω (α) .
Difficulty of the latter kind of extension can be illustrated easily in the case of a holomorphic family of plane curves C a given by R(x, y, a) = 0 with a in the open unit disk of C as a holomorphic parameter. A holomorphic 1-form on a single plane curve C a can be constructed as dx R y (x, y, a) = − dy R x (x, y, a) from the vanishing of the differential dR = R x dx + R y dy on C a when a is considered as a constant, but in the total space a∈∆ C a of the family of plane curves it is not easy to carry out a similar construction, because when a is regarded as a variable, the differential dR becomes R x dx + R y dy + R a da and the same method cannot be applied.
Furthermore, it is for this kind of difficulty of constructing (n − 1)jet differentials on the universal hypersurface X that the additional condition (5.9.1) j for 1 ≤ j ≤ n − 1 is introduced into Theorem 5.11 on entire function solutions of polynomial equations with slowing varying coefficients, so that families of vertical (n − 1)-jet differentials on the fibers can be used instead.

Algebraic Geometric Counterpart of Slanted Vector Fields.
In his 1986 paper [Cl86] Clemens introduced a technique (later generalized by Ein [Ei88], and Voisin [Voi96]) to prove the nonexistence of rational and elliptic curves in generic hypersurfaces of high degree by showing that the normal bundle of one such curve in the family of such curves is globally generated by sections with vertical pole order 1. His technique can be considered the counterpart of our method of slanted vector fields and as a matter of fact serves as motivation for our method.
On its face value Clemens's technique of using normal bundle to estimate the genus of a curve is algebraic in nature and cannot possibly have anything to do with the problem of hyperbolicity of transcendental in nature. Its relevancy was realized for the first time in [Si02] and [Si04] partly because of our seemingly completely unrelated earlier work on the deformational invariance of the plurigenera [Si98] [Si00].
Like jet differentials, pluricanonical sections can be naturally pulled back by a map and, as a result, their Lie differentiation can be naturally defined without specifying any special connection. In a hitherto unsuccessful attempt to study deformational invariance of sections of other bundles associated with the tangent bundle (besides pluricanonical sections), we investigated the obstruction of moving jet differentials out of a fiber in a family of compact complex manifolds and considered their Lie derivatives with respect to slanted vector fields. Such investigations, though unsuccessful so far as its original goal is concerned, serendipitously led to the use of slanted vector fields in the study of hyperbolicity problems and to the realization that Clemens's technique is relevant to, and can serve as motivation for, the differentiation of jet differentials by slanted vector fields to produce new ones. 0.8. Linear Algebra versus Theorem of Riemann-Roch.
As already pointed out in the paragraph straddling p.445 and p.446 in [Si02], a non identically zero holomorphic (n − 1)-jet differential on X (α) vanishing on an ample divisor can be constructed from the theorem of Riemann-Roch by using the sufficient positivity of the canonical line bundle of X (α) and the lower bound of the negativity of jet differential bundles of X (α) . Such a jet differential can also be directly obtained by using the linear algebra method of solving a system of linear equations with more unknowns than independent linear equations, which is the method used here in Proposition 3.8 below, as sketched on p.446 of [Si02]. This direct method of construction by linear algebra has the important advantage of better control over the form of the resulting jet differential so that the application of slanted vector fields can produce sufficiently independent jet differentials vanishing on an ample divisor of X (α) for a generic point α of P N (see Lemma 4.1, Proposition 4.3, and the proof of Theorem 0.1 given in 4.4 below).
Of course, the use of the theorem of Riemann-Roch also uses the linear algebra technique of counting the dimension of sections modules and the dimension of obstructional higher cohomology groups, but the process of going through a labyrinth of exact sequences so obscures the eventual form of the resulting jet differential that not enough control can be retained to get beyond the weaker conclusion that holomorphic maps from C to X (α) is contained in some proper subvariety of X (α) .
Recently Diverio, Merker, and Rousseau in [DMR10] used the theorem of Riemann-Roch to construct a holomorphic jet differential on X (α) vanishing on an ample divisor and then used Merker's work [Me09] involving our method of slanted vector fields to arrive at the conclusion that holomorphic maps from C to X (α) is contained in some proper subvariety of X (α) . 0.9. Simplified Treatment in Going from Non Zariski Density of Entire Curves to Hyperbolicity.
In the original sketch of the proof of Theorem 0.1 in [Si02], for the last step discussed on p.447 of [Si02] of going from the non Zariski density of entire curves to hyperbolicity, the method of construction of jet differentials is applied to a hypersurfaceX in Pn constructed from the hypersurface X (α) in P n with a largern so that more jet differentials on X (α) can be obtained from jet differentials onX. The idea is that the zero-set of the jet differentials constructed on X (α) from linear algebra would be defined by the vanishing of polynomials of low degree in loworder partial derivatives of the polynomial f (α) defining X (α) and, in order to take care of such zero-set, the low-order partial derivatives of f are introduced as additional new variables. The genericity condition of f enters in a certain form of independence of the low-order partial derivatives of f .
In this paper we use a simplified treatment of this step and directly use the genericity condition of f as a certain form of independence of the low-order partial derivatives of f . The key point of the argument, given in Lemma 4.1 and Proposition 4.3, is that the pullback, to a generic X (α) , of a meromorphic (n − 1)-jet differential on P n defined by a low-degree polynomials of the inhomogeneous coordinates of P n and their differentials have only low vanishing order at every point of X (α) . 0.10. Techniques Parallel to Those in Gelfond-Schneider-Lang-Bombieri Theory.
Paul Vojta presented in [Voj87] a formal parallelism between the results in diophantine approximation and those in value distribution theory. Along this line, the techniques presented here for the proof of the hyperbolicity of generic hypersurfaces of sufficiently high degree are, in certain ways, quite parallel to the techniques used for the theory of Gelfond-Schneider-Lang-Bombieri ([Sc34], [Ge34], [La62], [La65], [La66], [Bo70], and [BL70]).
(i) The construction of holomorphic jet differentials in Proposition 3.8 by solving a system of linear equations with more unknowns than equations is parallel to the use of Siegel's lemma in the theory of Gelfond-Schneider-Lang-Bombieri to construct a polynomial with estimates on its degree and the heights of its coefficients.
(ii) The requirement that the constructed jet differential vanishing on an ample divisor of high degree in Proposition 3.8 is parallel to the requirement of the vanishing of the constructed polynomial in the theory of Gelfond-Schneider-Lang-Bombieri to high order at certain points.
(iii) Lemma 3.4 concerning the injectivity of the pullback map for certain jet differentials is parallel to the constructed polynomial in the theory of Gelfond-Schneider-Lang-Bombieri being not identically zero due to the assumption of the degree of transcendence of the given functions.
(iv) The use of Nevanlinna's logarithmic derivative lemma and the use of logarithms of global meromorphic functions as local coordinates in the Schwarz lemma to estimate the contribution from the differentials to be of lower order is parallel to the use of the differential equations in the theory of Gelfond-Schneider-Lang-Bombieri.
Such a parallelism between the techniques used in this paper and those in theory of Gelfond-Schneider-Lang-Bombieri lends support to the preferability of the approach used in this paper for the hyperbolicity problem of hypersurfaces. 0.11. Notations and Terminology.
For r > 0 we use ∆ r to denote the open unit disk in C of radius r centered at the origin. When r = 1, we simply use ∆ to denote ∆ 1 when there is no confusion.
For a real number λ denote by ⌊λ⌋ the round-down of λ which means the largest integer ≤ λ and denote by ⌈λ⌉ be the round-up of λ which means the smallest integer ≥ λ.
We use [z 0 , · · · , z n ] to denote the homogeneous coordinates of P n and we use (x 1 , · · · , x n ) to denote the inhomogeneous coordinates of P n with x j = z j z 0 for 1 ≤ j ≤ n. Sometimes we also go to the inhomogeneous coordinates by fixing z 0 ≡ 1 in the homogeneous coordinates when notationally it is more advantageous to do so.
When we present the main ideas of an argument and refer to high vanishing order without explicitly giving a precise number, we mean a quantity of the order of δ. In such a situation, when we refer to low pole order without explicitly giving a precise number, we mean a quantity of the order of δ 1−ε for some appropriate 0 < ε < 1.
The notation at the end of the inequality means that there exist r 0 > 0 and a subset E of R ∩ {r > r 0 } with finite Lebesgue measure such that the inequality holds for r > r 0 and not in E. This is the condition needed for the logarithmic derivative lemma of Nevanlinna as given at the bottom of p.51 of [Ne25].
For a meromorphic function F on C and c ∈ C ∪ {∞} with F (0) = c the counting function is where n(ρ, F, c) is the number of roots of F = c in ∆ ρ with multiplicities counted. The characteristic function is under the assumption that 0 is not a pole of F , where log + means the maximum of log and 0.
For a complex manifold Y with a (1, 1)-form η and for a holomorphic map ϕ : C → Y , the characteristic function of ϕ with respect to η is 0.12. Twice Integration of Laplacian in Nevanlinna Theory.
The technique of twice integrating the Laplacian of a function introduced by Nevanlinnna for his theory of value distribution will be used a number of times in this paper. We put it down here for reference later. For any smooth function g(ζ), from the divergence theorem |ζ|<r ∆g = 2π θ=0 ∂ ∂r g re iθ rdθ and ∆ = 4∂ ζ ∂ ζ it follows that 4 r ρ=r 1 |ζ|<ρ 0.13. Function Associated to Pullback of Jet Differential to Part of Complex Line.
Let ω be a holomorphic k-jet differential on a complex manifold Y of complex dimension n and ϕ be a holomorphic map from an open subset U of C (with coordinate ζ) to Y . The map ϕ induces a map J k,ϕ from the space J k (U) of k-jets on U to the space J k (Y ) of k-jets on Y , which sends a k-jet η on U at ζ 0 represented by a parametrized complex curve germ γ : ∆ → U with γ(0) = ζ 0 to the k-jet represented by the parametrized complex curve germ ϕ • γ : ∆ → Y at ϕ(ζ 0 ). The pullback ϕ * ω of ω by ϕ means the holomorphic k-jet differential on U whose value at a k-jet η of U at ζ 0 is the value of ω at the k-jet J k,ϕ (η) of J k (Y ) at ϕ(ζ 0 ).
The crucial tool in the study of the hyperbolicity problem is the result, usually referred to as the Schwarz lemma, of the vanishing of the pullback of a holomorphic jet differential on a compact complex manifold vanishing on an ample divisor by a holomorphic map from C. Its proof, by Bloch's technique of using the logarithmic derivative lemma of Nevanlinna (p.51 of [Ne25]) with logarithms of global meromorphic functions as local coordinates, first shows the vanishing of a function associated to the pullback of the jet differential and then obtains the vanishing of the pullback of the jet differential by composing the map from C with appropriate holomorphic maps C → C.
In the later part of this article, when the analogues of the Big Picard Theorem are introduced for generic hypersurfaces X of high degree to extend holomorphic maps from C − ∆ r 0 → X to holomorphic maps from C ∪ {∞} − ∆ r 0 → X, appropriate holomorphic maps C − ∆ r 0 to itself are unavailable for proof the full Schwarz lemma. Instead only the vanishing of the function associated to the pullback of the jet differential can be obtained. We now introduce a notation for this function. The function on U, denoted by eval id C (ϕ * ω), at the point ζ 0 is the value of the k-jet ϕ * ω evaluated at the k-jet of U at ζ 0 represented by the parametrized curve defined by the identity map of C. In other words, the value of eval id C (ϕ * ω) at ζ 0 ∈ U is the value of ω at the k-jet on Y represented by the parametized complex curve germ ϕ : U → Y at ϕ(ζ 0 ).
When for some local coordinates y 1 , · · · , y n of Y the k-jet differential ω is written as where ϕ is represented by (ϕ 1 , · · · , ϕ n ) with respect to the local coordinates y 1 , · · · , y n of Y , so that if y j is locally equal to log F j for some global meromorphic function F j on Y (for 1 ≤ j ≤ n), the logarithmic derivative lemma of Nevanlinna (p.51 of [Ne25]) can be applied to d ℓ dζ ℓ ϕ j (ζ) = d ℓ dζ ℓ log F j (ϕ(ζ)).

Approach of Vector Fields and Lie Derivatives
1.1. Moduli Space of Hypersurfaces.
The moduli space of all hypersurfaces of degree δ in P n is the same as the complex projective space P N of complex dimension N = δ+n n − 1. The defining equation for the universal hypersurface X in P n × P N is The number of indices ν ∈ (N ∪ {0}) n+1 with |ν| = δ is δ+n n = N + 1. For α ∈ P N we use X (α) to denote X ∩ (P n × {α}).
Lemma 1.2. X is a nonsingular hypersurface of P n × P N of bidegree (δ, 1).
Remark 1.3. Though X is nonsingular at every point (y, α) ∈ P n × P N which belongs to X , the hypersurface X (α) in P n which corresponds to α and is equal to X ∩ (P n × {α}) may have singularities.
Lemma 1.4. Let ℓ be a positive integer and let L be a homogeneous polynomial of degree ℓ in the variables Proof. The expression -valued global holomorphic vector field on P n × P N , because the tangent bundle of P N is generated by global holomorphic vector fields of the form The hypersurface X is defined by f = |ν|=δ α ν z ν . From ∂f ∂α ν = z ν and ν + e q = µ + e p it follows that ∂α µ on P n × P N is tangential to the hypersurface X of P n × P N Remark 1.5. The use of L is to make sure that we have a line-bundlevalued global holomorphic tangent vector field on P n × P N . We will only use the case ℓ = 1.
Lemma 1.6. For any global holomorphic vector field ξ on P n there exists a global holomorphic vector fieldξ on P n × P N such that (i)ξ is tangential to X , and (ii)ξ is projected to ξ under the natural projection from P n × P N onto the second factor P n .
Since H 1 (P n , O Pn ) vanishes, it follows from the exact cohomology sequence of the Euler sequence that ξ is of the form n j,k=0 a j,k z j ∂ ∂z k for some complex numbers a j,k .
It now suffices to set Lemma 1.7. T X ⊗ O Pn (1) is globally generated.
It is equivalent to look at the tangent bundle of the pullbackX of is equal to L(s) ∂ ∂αν . Thus we conclude that the global holomorphic sections of T X × O Pn (1) generate ∂ ∂αν for ν = (δ, 0, · · · , 0). By Lemma 1.6 the global holomorphic sections of T X × O Pn (1) also generate ∂ ∂z j for 0 ≤ j ≤ n. We thus conclude that global holomorphic sections of T X ×O Pn (1) generate a codimension 1 vector subspace of the tangent space of P n × P N at (y, α). Since X is nonsingular at (α, y), it follows that global holomorphic sections of T X ⊗ O Pn (1) generate T X ⊗ O Pn (1).
Lemma 1.8. Let q be a nonnegative integer. Global holomorphic sections of T X ⊗ O Pn (q + 1) ⊗ O P N (q) generate all of its q-jets of X .
Proof. Global holomorphic section of O Pn (q) ⊗ O P N (q) generate all of its q-jets. Thus we can use the product of a global holomorphic section of O Pn (q) ⊗ O P N (q) and a global holomorphic section of T X ⊗ O Pn (1) to generate any prescribed q-jet of X .

Lie Derivatives
Let X be a complex manifold and ξ be a holomorphic vector field on X. Let ϕ ξ,t be a 1-parameter local biholomorphism defined by the vector field ξ so that ∂ ∂t ϕ * ξ,t g t=0 = ξ(g) for any local holomorphic function g on X. For any k-jet differential ω on X, we define the Lie derivative Lie ξ (ω) of ω with respect to ξ by it follows that d (Lie ξ (ω)) = Lie ξ (dω) . Let η be a holomorphic ℓ-jet differential on X. The Leibniz product formula holds for the Lie derivatives of the product of ω and η so that Lie ξ (ωη) = Lie ξ (ω)η + ωLie ξ (η).
Let (w 1 , · · · , w n ) be a local coordinate system of X. Fix some 1 ≤ i ≤ n. If ω = d k w i and ξ = n j=1 g j (w) ∂ ∂w j , then Lemma 1.10. Let k be a positive integer. Let X be a complex manifold and D be a complex hypersurface in X. Let ω be a holomorphic k-jet differential on X which vanishes at points of D to order p. Let ξ be a meromorphic vector field on X whose only possible poles are those of order at most q at D. If p ≥ q + k, then Lie ξ (ω) is a holomorphic k-jet differential on X which vanishes at points of D to order p − (q + k).
We will use the convention that the binary tree, as a collection of indices, will be denoted by a lower case Gothic letter and its indices are denoted by the corresponding lower case Latin letter. When k = 0, we use the convention that p (0) is just the empty set.
Let f = ν α ν z ν be a polynomial of homogeneous degree δ. Let Φ (0) ν ≡ 1 and inductively for k ≥ 0, where for the differentiation α ν is regarded as a constant and Φ (k) ν is of homogeneous weight k when ξ (j) ℓ is given the weight j and is independent of (z 0 , · · · , z n ). Moreover, the coefficients of Φ (k) ν in ξ (1) , · · · , ξ (k) are polynomials in ν = (ν 0 , · · · , ν n ) of degree at most k with universal coefficients. As a polynomial in ν = (ν 0 , · · · , ν n ), the degree of Proof. To prove the Lemma, we define Φ (k) ν by (2.2.2) and verify (2.2.1) 1 and (2.2.1) k+1 for k ≥ 1. The verification is as follows. Clearly, and is homogeneous of weight 1. To verify (2.2.2) when k is replaced by k + 1, we apply d to both sides of (2.2.2). The effect of applying d to z ν is to replace the factor z ν by Φ (1) by the chain rule. We now consider the question of weights and homogeneous degrees. For k ≥ 1 the first term on the right-hand side of (2.2.1) k+1 is the product of the factors Φ ν which are respectively homogeneous of weights 1 and k by induction hypothesis. The second term on the right-hand side of (2.2.1) k+1 is the sum of a product of two factors which are respectively homogeneous of weights i + 1 and k − i by induction hypothesis. This finishes the verification of (2.2.1) k+1 and the homogeneity of Φ on the right-hand side as a polynomial in ν is of degree no higher than that of Φ (k) ν which is no higher than k. Thus for the induction process of going from Step k to Step k + 1, if the degree of is at most k in ν = (ν 0 , · · · , ν n ).

Construction by Induction.
Let p (k) be a binary tree of indices of order k. We denote by λ (k) a multi-index of n + 1 components with total degree δ − k. For 1 ≤ j ≤ k and for the choice of each γ 1 , · · · , γ j being 0 or 1, we denote by λ (k;γ 1 ,··· ,γ j ) the multi-index Recall that e pγ 1 ,··· ,γ i is the index of n + 1 components whose only nonzero component is the p γ 1 ,··· ,γ i -th component which is 1.
For 0 ≤ k ≤ n − 1, for any multi-index λ (k) of n + 1 components with total degree δ − k, and for any binary tree p (k) of order k which has level-wise homogeneous branches, we are going to explicitly construct by induction on k, is a meromorphic vector field on the parameter space with coordinates α ν (for multi-indices ν of n + 1 components of total degree δ) which is a linear combination of ∂ ∂αν (for |ν| = δ) whose coefficients are rational functions of and which satisfies Here we regard Θ (k) λ (k) ,p (k) as a vector field on the space with variables α ν for |ν| = δ while the variables z 0 , · · · , z n and ξ are regarded as constants. It is the same as regarding Θ (k) λ (k) ,p (k) as a vector field on the space with variables α ν for |ν| = δ and the variables z 0 , · · · , z n and ξ The construction is as follows.
For k = 0 with the convention that p (0) is the empty set, we simply set and is independent of (z 0 , · · · , z n ) and is a polynomial in the n + 1 components of λ (0) of degree ≤ j. Moreover, it follows from f = ν α ν z ν and the definition of Φ for 0 ≤ j ≤ n − 1. Suppose the construction has been done for the step k and we are going to construct for the step k + 1. Define Lemma 2.4. For any integer k with 0 ≤ k ≤ δ the following two identities hold.
Proof. We prove by induction on k ≥ 0. Since the positive integer δ is fixed once for all, the induction on k ≥ 0 is the same as descending induction on the total degree δ − k of λ (k) . In the case k = 0 the statement is simply (2.3.2). To go from Step k to Step k + 1, we have .

We have Θ
q is assigned the weight ℓ. Moreover, for k ≤ j ≤ n, as a function of the n + 1 components of the multi-index λ (k) , the function Ψ Proof. We prove the Lemma by induction on 0 ≤ k ≤ n − 1.

By induction assumption Ψ
is a polynomial of the n + 1 components of the multi-index λ (k+1;p 1 ) of degree at most j − k. Now which means that the multi-index λ (k+1;p 0 ) is a translate of the multiindex λ (k+1;p 1 ) by the multi-index e p 0 − e p 1 and is independent of the n + 1 components of the multi-index λ (k+1) . Thus the difference is a polynomial of the n + 1 components of the multi-index λ (k+1) of degree at most j − k − 1. The last statement follows from Lemma 2.4.
Lemma 2.6. If λ (k) is a multi-index of n + 1 components with total degree δ − k and p (k) is any binary tree of order k which has level-wise homogeneous branching, then where the nodes of p (k) are denoted by p γ 1 ,··· ,γ j with 1 ≤ j ≤ k and each γ ℓ taking on the value 0 or 1 for 1 ≤ ℓ ≤ j. As a consequence, Proof. First we make the following simple observation. Let G (ν 0 , · · · , ν n ) be a polynomial in ν 0 , · · · , ν n of degree no more than M. For any p = q define Then (∆ p,q G) (ν 0 , · · · , ν n ) is a polynomial in ν 0 , · · · , ν n of degree no more than M − 1, because we can write and clearly each of the two terms is a a polynomial in ν 0 , · · · , ν n of degree no more than M − 1. As a consequence, is of degree zero in ν 0 , · · · , ν n and ∆ r 1 ,s 1 · · · ∆ r N+1 ,s N+1 G is identically zero for any polynomial G (ν 0 , · · · , ν n ) in ν 0 , · · · , ν n of degree no more than N.
This finishes the verification of (2.6.3) by induction.
for any λ (k) of n + 1 components and total degree δ − k. We rewrite it as is regarded as a polynomial in the n + 1 components of ν. Note that by (2.6.1) the left-hand side ν as a polynomial in ν is no more than k.
Since by Lemma A as a polynomial in ν 1 , · · · , ν n the degree of We have a polynomial of degree no more than k − 2 in the n + 1 components of ν and It follows from (2.6.4) that (2.6.5) Ψ (1) Finally (2.6.5) yields by induction on k for any multi-index λ (k) of n + 1 components with total degree δ − k and for any binary tree p (k) of order k which has level-wise homogeneous branching.
Remark 2.7. The reason for explicitly computing the function Ψ is to determine the pole set of the vector field Θ and is independent of (z 0 , · · · , z n ) and is a polynomial in the n+1 components of λ (k) of degree ≤ j.

Generation of Vector Fields in the Parameter Direction.
We now look at the special case of Lemma 2.8(c) with k = n, theñ We can choose homogeneous coordinates in P n so that (z 0 , z 1 , · · · , z n ) (y) = (1, 0, · · · , 0). We choose also s 1 = · · · = s n = 0 and r j = 0 for 1 ≤ j ≤ n. Then at y we end up withΘ For the choice of λ (n) we can choose any multi-index of total degree δ − n. When we worry about the generation of the vector fields by global sections, for differentiations in the direction of the parameters α = {α ν } |ν|=δ at the origin, we can capture in inhomogeneous coordinates the differentiation with respect to all coefficients for monomials of degree at least n, because we must include n ℓ=1 e r ℓ with r j = 0 for 1 ≤ j ≤ n in ν which is equal to λ (n) + n ℓ=1 e r ℓ .

Example of Vector Fields on Jet Spaces of Low
Order.
Proposition 2.11. Let 0 ≤ p = q ≤ n and 0 ≤ r = s ≤ n. Let µ be a multi-index of total weight δ − 2. Then for any λ with |λ| = δ − 1. Choose µ with |µ| = δ − 2. Apply the above equation to λ = µ + e r and get Since the right-hand side is independent of r, we can replace r by s and take the difference to get Remark 2.12. We can rewrite the vector field 1 z r To illustrate the situation of vector fields on jet spaces of low order, we do the case of the next order.
Thus for any multi-index µ of total degree δ − 3, the vector field We can now formulate the case of higher-order jets.
In the above Proposition the vector field Θ µ;r 1 ,··· ,r k ;s 1 ,··· ,s k is a linear combination of the partial differentiation operators ∂ ∂α µ+er i 1 +···+er ip +es j 1 +···+es j k−p for 0 ≤ j ≤ k. The process of generating such vector fields is not independent of coordinate transformations from the general linear group GL (n + 1, C). Suppose we have the coordinate transformation from the element a = (a jℓ ) 0≤j,ℓ≤n of the general linear group GL (n + 1, C). Then When the generation of the vector field Θ in the coordinate system (z 0 , · · · , z n ) gives the procedure applied to the coordinate system (w 0 , · · · , w n ) gives When we transform back to the coordinate system (z 0 , · · · , z n ), we get Θ a;µ;r 1 ,··· ,rn;s 1 ,··· ,sn = µ,ν Now We would like to show that when k = n, the dimension of the quotient space a∈GL(n+1,C);|µ|=δ−n; r 1 ,··· ,rn;s 1 ,··· ,sn CΘ a;µ;r 1 ,··· ,rn;s 1 ,··· ,sn    is no more than n over C. For this we need only show that modulo the linear space generated by all such Θ a;µ;r 1 ,··· ,rn;s 1 ,··· ,sn , every generator ∂ ∂αν can be expressed as a linear combination of n fixed ∂ ∂α ν [ℓ] ( where ν [ℓ] is a multi-index of total weight δ. We use the linear transformations defined by a ∈ GL (n + 1, C) simply to make sure that, for any given point, we are free to do the checking in an appropriate coordinate system which depends on the pointj.
For the convenience of bookkeeping we let M be an integer > δ and introduce a new weight ν M for any multi-index ν of total degree δ which is defined as follows.
Remark 2.14. The reason why in the above argument we fail to get generation of all vectors in parameter space is that we can only expect to get generation all vectors in parameter space up to codimension n for (n − 1)-jets. The vector fields have to be tangential to the space J (vert) n−1 (X ) of vertical (n − 1)-jets of X which is of codimension n in the product J n−1 (P n ) × P N of the space J n−1 (P n ) of (n − 1)-jets of P n and the parameter space P N .

Generation of Vectors in Vertical Directions
Now we construct holomorphic vector fields which generate the vertical directions modulo the horizontal directions.
To unify the notations, we use T to denote any one of This means that ν is homogeneous of weight k − ℓ and is independent of z 0 , · · · , z n .
Given any multi-index ν of n + 1 components and total degree δ, we choose a multi-index λ (n−1) ν of n+1 components and total degree δ−n+1 such that λ (n−1) ν ≤ ν in the sense that the j-th component of λ (n−1) ν ≤ ν is no more than ν j for 0 ≤ j ≤ n. Though the binary tree p (n−1) of order n − 1 has level-wise homogeneous branches, yet λ (n−1;γ 1 ,··· ,γ j ) ν does depend on the choices of the values 0 or 1 for γ 1 , · · · , γ j and 1 ≤ j ≤ n. The dependence is as follows. If we denote r j by r j,0 and s j = r j,1 , then for k ≤ j ≤ n for any multi-index λ (k) of n + 1 components and total degree δ − k. So we can set By Lemma 2.8, The function and is a polynomial in the variables z 0 , · · · , z n of degree ≤ n − 1 − k and in the n + 1 components of ν of degree ≤ j. The vector field s ℓ is a polynomial in z 0 , · · · , z n of degree ≤ n − 1 and is independent of ξ (ℓ) on ν is only through the partial differentiation with respect to α µ with µ depending on ν.
To make sure that a modification of T annihilates f , we modify T to T + ν Ξ ν . To make sure that our constructed vector field annihilates df , we use ν + ν Ξ (2) ν Θ (2) ν .
In general, to make sure that we have the annihilation of all d j f for 0 ≤ j ≤ n − 1, we need to write The main point is to control the pole order of the vector fields and make the pole order bounded and independent of δ. That is the reason why we want to remove z ν by using the vector fields Θ (k) λ (k) ,p (k) . We now count the degree in z 0 , · · · , z n and the weight in ξ (ℓ) j after we clear the denominators. We need to multiply j . The worst that can occur is , it follows that the worst situation is that after multiplication by the above factor to clear the denominator we end up with a weight of j 0 + j 1 + · · · + j ℓ plus the weight of the factor which is no greater than the weight j 0 + j 1 + · · · + j ℓ + n(n − 1) 2 ≤ n(n − 1).
2.16. Vector Fields in Terms of Differentiation with Respect to Inhomogeneous Coordinates.
The introduction of homogeneous coordinates is simply for the notational convenience of our discussion. We now return to inhomogeneous coordinates by specializing to z 0 ≡ 1. First of all we would like to go back to the coordinates which is the same as z 0 , z 1 , · · · , z n , . We are going to use the chain rule for the transformation of vector fields.
j (z 0 , · · · , z n , dz 0 , · · · , dz n , · · · , d n z 0 , · · · , d n z n ) is a rational function which is homogeneous of weight 0 when d ℓ z p is assigned weight 1 and is homogeneous of weight k when d ℓ z p is assigned weight ℓ. Thus is of weight −1 when d ℓ z p is assigned weight 1 and is homogeneous of weight k − ℓ when d ℓ z p is assigned weight ℓ. It follows from weight considerations that ∂ξ where δ p,j is the Kronecker delta. We conclude that, so far as the independence of the constructed vector fields are concerned, it makes no difference whether we are using the coordinate system Now we pass from the homogeneous coordinates to the inhomogeneous coordinates. It is equivalent to restricting all the objects to the linear subspace are concerned, the linear subspace is part of the line defined by setting some coordinates equal to constant and the argument is not affected. Since the pole order of d ℓ z j is no more than ℓ + 1, we have the following proposition. A point of the space J vert n−1 (X ) of vertical (n − 1)-jets is represented by a nonsingular complex curve germ in P n precisely when the value of z 1 dz 0 − z 0 dz 1 is nonzero at it for some homogeneous coordinate system z 0 , · · · , z n of P n .
Proposition 2.17. (Global Generation on Jet Space by Slanted Vector Fields at Points Representable by Regular Curve Germs) Let P 0 be a point of the space J vert n−1 (X ) of vertical (n − 1)-jets such that P 0 can be represented by a nonsingular complex curve germ in P n . Then the meromorphic vector fields on P n × P N tangential to of pole order ≤ n 2 + 2n + n(n − 1) = n(2n + 1) (along the infinity hyperplane of P n ) generate at P 0 the tangent space of the total space of fiber-direction (n − 1)-jets, where d k f is taken with α ν regarded as constants. In terms of inhomogeneous coordinates, the statement is equivalent to that of the formulation in terms of homogeneous coordinates on the restriction to Proof. Since P 0 can be represented by a nonsingular complex curve germ in P n , there exists some homogeneous coordinate system z 0 , · · · , z n such that z 0 = 0 and z 1 dz 0 −z 0 dz 1 = 0 at it. Since one of z 0 and z 1 must be nonzero at P 0 , we assume without loss of generality that z 0 is nonzero We then apply a translation to the the affine coordinates (x 1 , · · · , x n ) to make sure that x ℓ = 0 at P 0 for 1 ≤ ℓ ≤ n. Then we apply a linear transformation to the affine coordinates (x 1 , · · · , x n ) that x ℓ = 0 at P 0 for 1 ≤ ℓ ≤ n and dx 1 = dx 2 at P 0 . We are going to set r j = 1 and s j = 2 for 1 ≤ j ≤ n. Also we will restrict the vector fields to z 0 = 1 and z ℓ = x ℓ for 1 ≤ ℓ ≤ n so that 2 = 0 at P 0 and z ℓ = 0 at P 0 for 0 ≤ ℓ ≤ n. The above construction now gives the generation of the tangent bundle of J vert n−1 (X ) at P 0 . Remark 2.18. In the global generation of the tangent bundle of X in Lemma 1.7 there is no reference to the tangent vector be representable by nonsingular complex curve germ, because a tangent vector is not representable by a nonsingular complex curve germ must be zero and the identically zero global vector field already generates the zero tangent vector. However, a higher order jet which cannot be represented by a nonsingular complex curve germ need not be zero. The condition of representability by a nonsingular complex curve germ can be technically suppressed by formulating global generation over some suitably defined projectivization of the jet space which includes only those jets which have well-defined images in the projectivization of the tangent bundle.
For the hyperbolicity of generic hypersurface X of sufficiently high degree, the generation by slanted vector fields of low vertical pole order only at jets representable by nonsingular complex curve germs offers no difficulty, because any nonconstant holomorphic map ϕ from the affine complex line C to X must be nonzero tangent vector at some point ζ 0 of C and we need use slanted vector fields of low vertical pole order at the jet represented by ϕ at ϕ (ζ 0 ).
The slanted vector fields on J vert n−1 (X ) of low vertical pole order constructed in (2.15) for the the proof of Proposition 2.17 start out with the vector field T which is any one of The constructed slanted vector fields on J vert n−1 (X ) of low vertical pole order are actually restrictions J vert n−1 (X ) of vector fields on J n−1 (P n )×P N of low vertical pole order which are tangential to J vert n−1 (X ). We formulate below a proposition about the slanted vector fields on J n−1 (P n ) × P N of low vertical pole order which are tangential to J vert n−1 (X ) before their restrictions to J vert n−1 (X ). We want generation of the tangent space of J n−1 (P n ) × P N only at points P 0 represented by a nonsingular complex curve germ in P n × {α} which is disjoint from X . In some steps we need to modify the proof of Proposition 2.17 by replacing the defining function f of X by its product with a polynomial g of degree n in x 1 , · · · , x n in order to make sure that is satisfied at P 0 , where d k (f g) is taken with α ν regarded as constants.
Proposition 2.19. Let α ∈ P N and P 0 be a point of the space J n−1 (P n )× P N such that P 0 can be represented by a nonsingular complex curve germ in P n ×{α} which is disjoint from X . Then the meromorphic vector fields on J n−1 (P n )×P N of pole order ≤ n 2 +2n+n(n−1) = n(2n+1) (along the infinity hyperplane of P n ) which are tangential to J vert n−1 (X ) generate at P 0 the tangent space of J n−1 (P n ) × P N .

Remark on Homogeneous Weights of Vector Fields.
We would like to consider the effect on the weight of a jet differential after the application of the vector fields which we have constructed. The weight of d j x ℓ is j. The coordinates x ℓ has weight zero and does not contribute at all to the computation of weights. When we consider the vector field which starts with to clear the denominator we have to multiply the result by the factor which means that the action of the vector field after clearing out the denominator preserves the weight of the jet differential. Moreover, by the explicit construction of the slanted vector fields of low vertical pole order, we cannot apply them to (n − 1)-jet differentials to get k-jet differentials for k < n − 1.
By applying slanted vector fields of low vertical pole order (from Proposition 2.19), at points of X where the (n − 1)-jet differentials (assumed to be holomorphically extendible to holomorphic jet differentials on neighboring hypersurfaces X (α) ) are not zero pointwise, we can lower the multiplicity of the common zero-set of all holomorphic (n − 1)-jet differentials vanishing on an ample divisor as functions of d ℓ x j (1 ≤ j ≤ n, 1 ≤ ℓ ≤ n − 1) with x 1 , · · · , x n−1 being regarded as local coordinates of X to yield already at this point the conclusion that, for the degree δ of X sufficiently large, there is a proper subvariety of X which contains the image of all nonconstant holomorphic maps from C to X, as explained in (2.21) below, under the assumption that, for some 0 < θ < θ ′ ≤ 1, there is a holomorphic (n − 1)-jet differential on X whose weight m is of order at most comparable to δ θ and whose order of vanishing at the intersection of X and the infinity hyperplane of P n is of order at least comparable to δ θ ′ .
2.21. Non Zariski Density of Entire Curves from Existence of Appropriate Jet Differential .
We have vector fields on J k (P n ) × P N with low pole-order which are tangential to the space of vertical (n − 1)-jets of X . We can use either homogeneous coordinates z 0 , · · · , z n of P n or inhomogeneous coordinates x 1 , · · · , x n . The difference is very low pole order and such a very low pole order does not affect our argument. Then we use the affine coordinates ξ (ℓ) j = d ℓ log z j or d ℓ x j . A k-jet differential of weight m is simply a polynomial in ξ (ℓ) j homogeneous of weight m, which is pulled back to z 0 = 1, z j = x j when the affine coordinates are used. Assume that the k-jet differential is holomorphically extendible to holomorphic jet differentials on neighboring hypersurfaces X (α) . We can either use Lie derivatives with respect to the vector fields or we could just use differentiation of functions on the variables z 0 , · · · , z n (or x 1 , · · · , x n ) and ξ (ℓ) j (or dx j ). Even after we clear the denominator of the vector field used, with differentiation with respect to the vector field we end up with another polynomial in ξ (ℓ) j homogeneous of another weight m ′ . Suppose 0 < θ < θ ′ ≤ 1 and we have a holomorphic (n − 1)-jet differential ω (α) on X (α) (assumed to be holomorphically extendible to holomorphic jet differentials on neighboring hypersurfaces X (α) ) whose weight m is of order at most comparable to δ θ and whose order of vanishing at the intersection of X (α) and the infinity hyperplane of P n is of order at least comparable to δ θ ′ . Let Z be the set of X (α) where all coefficients of the (n − 1)-jet differential ω vanishes.
At a point y of X − Z the (n − 1)-jet differential ω of weight m, as a function on the space J n−1 (X) of (n − 1)-jets of X (α) , has vanishing order at most m at every point of J n−1 (X) above y, because at least one of the coefficients of ω is nonzero. Because of θ < θ ′ , at any point P 0 , above y, of J n−1 (X) which is represented by a nonsingular complex curve germ in X (α) , we can apply slanted vector fields of low vertical pole order (from Proposition 2.19) up to as many times as the weight m of ω to get a holomorphic (n−1)-jet differential ω (P 0 ) which vanishes on an ample divisor of X (α) and has nonzero value at P 0 .
By the Schwarz lemma of the vanishing of pullbacks of holomorphic jet differentials vanishing on ample divisor to C by an entire curve, we conclude, from ϕ * ω (P 0 ) ≡ 0 for any P 0 above y representable by nonsingular curve germ in X (α) , that y cannot be equal to ϕ(ζ 0 ) for some ζ 0 ∈ C where the differential dϕ of ϕ is nonzero. Hence the image of any nonconstant holomorphic map ϕ from C to X (α) must be contained in the proper subvariety Z of X (α) .

Hyperbolicity from Existence of Appropriate Jet Differential.
In (2.21), if the holomorphic (n − 1)-jet differential ω on X (α) is assumed to satisfy in addition the condition that, for some 0 < θ ′′ < θ ′ , at every point of X (α) some coefficient of ω has vanishing order no more than e θ ′′ , then at any point P 0 of J n−1 (X) representable by nonsingular complex curve germ, we can apply slanted vector fields of low vertical pole order (from Proposition 2.19) up to as many times as the weight m of ω plus e θ ′′ to get a a holomorphic (n − 1)-jet differential ω (P 0 ) which vanishes on an ample divisor of X (α) and has nonzero value at P 0 . As in (2.21), we can now conclude that there is no nonconstant holomorphic map from C to X (α) .

Extendibility of Jet Differentials Generic Fiber to Neighborhing Fibers
In both (2.21) and (2.22) there is an assumption about the holomorphic extendibility of the jet differential on X (α) in question to neighboring fibers X (α) . Such a condition is automatically satisfied for a genericα in P N because of the following general statement.
Letπ : Y → S be a flat holomorphic family of compact complex spaces and L → Y holomorphic vector bundle. Then there exists a proper subvariety Z of S such that for s ∈ S − Z the restriction map is surjective for some open neighborhood U s of s in S.

Construction of Holomorphic Jet Differentials
We are going to construct holomorphic jet differentials. One crucial ingredient is the use of the Koszul complex to show that a homogeneous polynomial of low degree in n + 1 homogeneous coordinates and their differentials up to order n − 1 cannot locally belong the ideal generated by a second homogeneous polynomial and its differentials up to order n − 1 when the second homogeneous polynomial is a homogeneous polynomial of high degree in the n + 1 homogeneous coordinates.
Lemma 3.1. Let Y be a compact complex manifold and Z be a subvariety of pure codimension at least 2 in Y . Let F be a locally free sheaf on Y . Then the restriction map Proof. This is a standard removability result for cohomology groups. Let {U j } j be a finite cover of Y by Stein open subsets U j . Since (U j ℓ − Z) , F = 0 for 1 ≤ r ≤ codim Y Z − 2 and for any j 0 , · · · , j p , by Leray's theorem the following natural isomorphism gives the computation of the sheaf cohomology by Cech cohomology. Since the restriction map is bijective for any j 0 , · · · , j p , it follows that the map defined by restriction is an isomorphism. The lemma follows from the following natural isomorphism gives the computation of the sheaf cohomology by Cech cohomology.
Lemma 3.2. Let ℓ and a ≤ N be positive integers. Let Z be a linear subspace of P N and let such that the zero-set of F 1 , · · · , F a in P N − Z is a submanifold of codimension a in P N − Z which is a complete intersection. Assume that H q (P N − Z, O P N (r)) = 0 for 1 ≤ q < a for any integer r. Then The homomorphisms in the Koszul complex is defined as follows. Take symbols e 1 , · · · , e a . We use e i 1 ∧ · · · ∧ e i k (1 ≤ i 1 < · · · < i k ≤ a) as a local basis for O ⊕( a k ) P N (−kℓ + p) to represent an element and define in such a representation. Since the zero-set of F 1 , · · · , F a in P N − Z is a submanifold of codimension a in P N − Z which is a complete intersection, it follows that the Koszul complex is exact on P N − Z.
We are going to prove by descending induction on b for 1 ≤ b ≤ a − 1 that H q (P N − Z, Ker φ b ) = 0 for 1 ≤ q ≤ b. The case b = a − 1 follows from the assumption of the lemma and Ker φ a−1 = O ⊕( a a ) P N (−aℓ + p). For 1 ≤ b < a − 1 the exact sequence and for 1 ≤ q ≤ b − 1 we conclude, from in the assumption of the lemma and Lemma 3.3. Let w 0 , w 1 be two transcendental variables (representing two local holomorphic functions). Then which is homogeneous of degree j + 1 in all the variables and of total weight j in the differentials d ℓ w k for 0 ≤ ℓ ≤ j and k = 0, 1 when the weight of d ℓ w k is assigned to be ℓ.
Proof. The case j = 0 of the claim is clear. The induction process of the claim going from Step j to Step j + 1 simply follows from w 0 and the observations that (i) the differential of a homogeneous polynomial in the variables is a homogeneous polynomial in the variables d ℓ w k (0 ≤ ℓ ≤ j, k = 0, 1) of the same degree, and (ii) the differential of a polynomial in the variables which is of homogeneous weight a is a polynomial in the variables which is of homogeneous weight a + 1 when the weight of d ℓ w k is assigned to be ℓ.
Lemma 3.4. (Injectivity of Pullback Map for Jet Differentials) Let 1 ≤ k ≤ n − 1 and let f be a polynomial of degree δ in inhomogeneous coordinates x 1 , · · · , x n of P n so that the zero-set of f defines a complex manifold X in P n . Let Q be a non identically zero polynomial in the variables d j x ℓ (0 ≤ j ≤ k, 1 ≤ ℓ ≤ n). Assume that Q is of degree m 0 in x 1 , · · · , x n is m 0 and is of homogeneous weight m in the variables d j x ℓ (1 ≤ j ≤ k, 1 ≤ ℓ ≤ n) when the weight of d j x ℓ is assigned to be j. If m 0 + 2m < δ, then Q is not identically zero on the space of k-jets of X.
Proof. Suppose Q is identically zero on the space of k-jets of X. We are going to derive a contradiction.
Since Q is of homogeneous weight m in the variables d j x ℓ (1 ≤ j ≤ k, 1 ≤ ℓ ≤ n) when the weight of d j x ℓ is assigned to be j, it follows that the degree of Q in the variables d j x ℓ (1 ≤ j ≤ k, 1 ≤ ℓ ≤ n) is at most m. We introduce the homogeneous coordinates z 0 , z 1 , · · · , z n of P n so that Let N = (k + 1)(n + 1) − 1 and relabel the variables as the N + 1 homogeneous coordinates w 0 , · · · , w N of P N . Let P = z m 0 +2m 0 Q. Since the degree of Q in the variables d j x ℓ (1 ≤ j ≤ k, 1 ≤ ℓ ≤ n) is at most m, by Lemma 3.3 we conclude that P is a polynomial in the variables w 0 , · · · , w N and is homogeneous of degree m 0 + 2m.
We are going to apply Lemma 3.2. In our application We set m = n. The homogeneous polynomials F 1 , · · · , F m of degree δ in the N + 1 homogeneous coordinates w 0 , · · · , w N of P N are The linear subspace Z in P N is defined by z 0 = z 1 = · · · = z n = 0 which is of complex codimension n+1 in P N and is therefore of complex dimension N − (n + 1) = k(n + 1) − 1.
We know that, ifẐ is a subvariety of P N , then for any Stein open subset U of P N the cohomology group H q U −Ẑ, O P N vanishes for 0 ≤ q ≤ codim P NẐ − 2 = n − 1, where codim P NẐ means the complex codimension of Z in P N . Thus, Since Q is identically zero on the space of k-jets of X, it follows that P locally belongs to the ideal generated by for some homogeneous polynomials g 0 , · · · , g k of the variables where the total degree of g j is m 0 +2m−δ. We arrive at a contradiction, because m 0 +2m−δ is negative and a polynomial cannot have a negative degree.
Now we count the number of unknowns and the number of equations.
Lemma 3.5. Let X be a hypersurface of degree δ in in P n . Let S be a hypersurface in X defined by a homogeneous polynomial g of degree s in the homogeneous coordinates of P n . Then for q ≥ δ + s + n, In particular, dim Γ (S, O S (q)) ≤ s δ (n + q − 2) n−2 (n − 2)! for q ≥ δ + s + n.
Proof. First of all, for any nonnegative integer ℓ we have dim Γ (P n , O Pn (ℓ)) = ℓ + n ℓ = ℓ + n n , because it is equal to the number of possibilities of choosing ℓ elements out of n + 1 elements with repetition allowed which is the same as choosing ℓ elements out of n + 1 + ℓ − 1 = ℓ + n elements without repetition. From the exact sequence where φ f is defined by multiplication by f , it follows that Hence dim Γ (X, O X (ℓ + δ)) = n + ℓ + δ n − n + ℓ n From (3.5.1) we have the exact sequence H p (P n , O Pn (ℓ + δ)) → H p (X, O X (ℓ + δ)) → H p+1 (P n , O Pn (ℓ)) .
From the vanishing of H p (P n , O Pn (ℓ + δ)) for 1 ≤ p < n it follows that H p (X, O X (ℓ + δ)) = 0 for 1 ≤ p < n − 1. From the exact sequence where φ g is defined by multiplication by g and from H 1 (X, O X (ℓ)) = 0 for n ≥ 3 it follows that for ℓ ≥ δ. We are going to use the following identity for binomial coefficients Lemma 3.6. Let y 1 , · · · , y r be independent transcendental variables. Let 1 = n 1 ≤ n 2 ≤ · · · ≤ n r be integers. Let a m be the number of all monomials y k 1 1 · · · y kr r such that r j=1 n j k j = m. Let a m be the number of elements in A m . Then where ⌊u⌋ denotes the largest integer not exceeding u.
Proof. Let A m be the set of all monomials y k 2 2 · · · y kr r such that r j=2 n j k j = m. Since n 1 = 1, A m is the same as the set of all monomials y k 1 1 · · · y kr r such that r j=1 n j k j = m and a m is the number of elements of A m . Let B m be the set of all monomials y k 2 2 · · · y kr r such that r j=2 k j ≤ m n 1 . Let C m be the set of all monomials y k 2 2 · · · y kr r such that r j=1 k j ≤ m.
the conclusion of the Lemma follows.
Lemma 3.7. Let f be a polynomial of degree δ in the variables x 1 , · · · , x n . Let κ ℓ be the smallest nonnegative integer such that (f x 1 ) κ ℓ d ℓ x 1 can be expressed as a polynomial P ℓ of on the space of ℓ-jets of the zero-set of f . Then κ 1 = 1 and Moreover, as a polynomial of the degree of P ℓ in the variables x 1 , · · · , x n is at most κ ℓ (δ − 1). The integers κ ℓ in can be estimated by κ ℓ ≤ ℓ! Proof. We use induction on ℓ for ℓ ≥ 1. The case ℓ = 1 is clear, because f xr dx r and we can set f xr dx r whose degree in x 1 , · · · , x r is obviously at most δ − 1. From d ℓ f = 0 it follows that on the space of ℓ-jets of the zero-set of f the jet differential f x 1 d ℓ x 1 can be written as a polynomial Q ℓ in d ℓ x 2 , · · · , d ℓ x n when d j x r is given the weight j. As a polynomial in dx 1 , · · · , d ℓ−1 x 1 the total weight of Q ℓ is no more than ℓ. As a polynomial in x 1 , · · · , x n the degree of Q ℓ is at most δ − 1. Thus inductively on ℓ we conclude that we need only to multiply f x 1 d ℓ x 1 by a power of f x 1 not exceeding max ℓ−1 j=1 κ j s j s 1 + 2s 2 + · · · + (ℓ − 1)s ℓ−1 ≤ ℓ to yield a polynomial P ℓ of on the space of ℓ-jets of the zero-set of f . Moreover, the degree of P ℓ in x 1 , · · · , x n is at most κ ℓ (δ − 1).
The integers κ ℓ can be estimated by κ ℓ ≤ ℓ!, because, when r 1 + 2r 2 + · · · + (ℓ − 1)r ℓ−1 ≤ ℓ, we have r j ≤ ℓ j , and from κ j ≤ j! for 1 ≤ j ≤ ℓ − 1 it follows that Proposition 3.8. (Jet Differential from Polynomial in Differentials of Inhomogeneous Coordinates) Let X be a nonsingular hypersurface of degree δ in P n defined by a polynomial f (x 1 , · · · , x n ) of degree δ in the affine coordinates x 1 , · · · , x n of P n . Suppose ǫ, ǫ ′ , θ 0 , θ, and θ ′ are numbers in the open interval (0, 1) such that nθ 0 + θ ≥ n + ǫ and θ ′ < 1 − ǫ ′ . Then there exists an explicit positive number A = A(n, ǫ, ǫ ′ ) depending only on n, ǫ, and ǫ ′ such that for δ ≥ A and any nonsingular hypersurface X in P n of degree δ there exists a non identically zero O Pn (−q)-valued holomorphic (n − 1)-jet differential ω on X of total weight m with q ≥ δ θ ′ and m ≤ δ θ . Here, with respect to a local holomorphic coordinate system w 1 , · · · , w n−1 of X, the weight of ω is in the variables d j w ℓ (1 ≤ j ≤ n − 1, 1 ≤ ℓ ≤ n − 1) with the weight j assigned to d j w ℓ . Moreover, for any affine coordinates x 1 , · · · , x n of P n , when f x 1 = 1 defines in a nonsingular hypersurface in X, the (n − 1)-jet differential ω can be chosen to be of the form Q fx 1 −1 , where Q is a polynomial in which is of degree m 0 = δ θ 0 in x 1 , · · · , x n and is of homogeneous weight m = δ θ in when the weight of d j x ℓ is assigned to be j.
Proof. Let x 1 , · · · , x n and z 0 , · · · , z n be respectively the homogeneous and inhomogeneous coordinates of P n so that x j = z j z 0 for 1 ≤ j ≤ n. Let f be a polynomial of degree δ in x 1 , · · · , x n so that the zero-set of f in P n is X.

Consider a non identically zero polynomial Q in
which is of degree m 0 in x 1 , · · · , x n and is of homogeneous weight m in d j x 1 , · · · , d j x n (1 ≤ j ≤ n − 1) when the weight of d j x ℓ is assigned to be j. We impose the condition m 0 + 2m < δ so that according to Lemma 3.4 the pullback of Q to the space of (n − 1)-jets of {f = 0} is not identically zero. According to Lemma 3.6 the degree of freedom in the choice of the polynomial Q is at least where the first factor m 0 + n n is the number of mononials of degree ≤ m 0 in n variables x 1 , · · · , x n and the second factor m n−1 + n(n − 1) − 1 n(n − 1) − 1 , is the number of mononials of homogeneous degree m n−1 in the n(n−1) variables The key point of this proof is that though we have all the variables d j x 1 , · · · , d j x n for 0 ≤ j ≤ n − 1, we do not have to worry about the dependence resulting from the relations d j f = 0 (0 ≤ j ≤ n − 1).
Let H Pn be the hyperplane of P n defined by x n = 0. We now want the meromorphic (n−1)-jet differential defined by Q to be holomorphic on X and and, moreover, to vanish at X ∩ H Pn of order q. First of all, on the space of (n − 1)-jets of X, we can use the relation To do this, according to Lemma 3.7 we can multiply Q by (f x 1 )Ñ with N = 2m n−1 j=1 κ j , because, in a monomial of weight m, the degree of The degree of (f x 1 ) N Q in x 1 , · · · , x n is now m 0 + N(δ − 1) and the weight of (f x 1 ) N Q in d j x ℓ (1 ≤ j ≤ n − 1, 2 ≤ ℓ ≤ n) is homogeneous and equal to m when the weight of d j x ℓ is assigned to be j.
We let S be the divisor in X defined by f x 1 −1 = 0. The hypersurface in P n defined by f x 1 − 1 = 0 is of degree δ − 1. We observe that for a generic polynomial f of degree δ, the divisor S in X is nonsingular, because it is the case when f equals the Fermat hypersurface Then F x 1 = δx δ−1 1 and the 2 × n matrix whose rows are nonzero multiplies of the gradients of F and F x 1 has rank 2 unless either x 1 = 0 or x 2 = · · · = x n = 0, which is impossible, because on S one has |x 1 | = δ −1 δ−1 from F x 1 = 1 and the condition x 2 = · · · = x n = 0 implies |x 1 | = 1 = δ −1 δ−1 when F = 0. To prove this Lemma we need only prove it for a generic f and then remove the genericity assumption for f by using the semi-continuity of the dimension of the space of global holomorphic sections over a fiber in a holomorphic family of compact complex manifolds and a holomorphic vector bundle.
The polynomial Q defines a meromorphic (n − 1)-jet differential on X which we again denote by Q. We now count the pole order of the jet differential Q on X at X ∩ H Pn . For the counting of this pole order, we introduce another set of inhomogeneous coordinates ζ 1 , · · · , ζ n of P n defined by x 1 x n , · · · , ζ n−1 = x n−1 x n , ζ n = 1 x n so that Since by Lemma 3.3 and ζ n j+1 d j x n = ζ n j+1 d j 1 ζ n are polynomials in d k ζ r (0 ≤ k ≤ j, 1 ≤ r ≤ n). Thus ζ n 2m Q is a polynomial in d k ζ r (0 ≤ k ≤ j, 1 ≤ r ≤ n). The pole order of the jet differential Q at X ∩ H Pn is at most m 0 + 2m.
We are going to show that we can choose the coefficients of the polynomial Q so that the (n − 1)-jet differential Q is zero at points of S. This would imply that the (n − 1)-jet differential is holomorphic on X and vanishes to order δ − m 0 − 2m at X ∩ H Pn . The reason is the following. For some proper subvariety Z of X ∩ H Pn the function ζ δ−1 0 (f x 1 − 1) is holomorphic and nowhere at points of X ∩ H Pn − Z. Thus is holomorphic on X − Z and vanishes to order at least δ − 1 − 2m along X ∩ H Pn − Z. What we want follows from Hartogs' extension theorem because Z is of complex codimension at least 2 in X. Now on J n−1 (X) S (which is the part of the space of (n − 1)-jets of X lying over S) the jet differential Q equals to the jet differential because f x 1 = 1 holds on S. Since the degree of (f x 1 ) N Q in x 1 , · · · , x n is m 0 +N(δ −1) and the weight of (f x 1 ) N Q in d k z r (1 ≤ k ≤ j, 2 ≤ r ≤ n) is homogeneous and equal to 2m, it follows from Lemma 3.5 that the number of linear equations, with the coefficients of Q as unknowns, needed for (f x 1 ) N Q to vanish at all points S is no more than the product For the existence of a nontrivial Q with the required vanishing at all points of S, it suffices to have In particular, it suffices that (m 0 + 1) n m n−1 n(n−1)−1 n! (n(n − 1) − 1)! is greater than We choose m 0 = δ θ 0 , m = δ θ , and q = δ θ ′ . Since the three positive numbers θ 0 , θ, and θ ′ are all strictly less than 1, measured in terms of powers of δ as δ becomes dominantly large, the order of (m 0 + 1) n m n−1 n(n−1)−1 n! (n(n − 1) − 1)! is at least δ nθ 0 +(n(n−1)−1)θ and the order of (δ − 1) δ (m 0 + (n − 1)! 2m (δ − 1)) n−2 (m + (n − 1) 2 − 1) (n−1) 2 −1 (n − 2)! ((n − 1) 2 − 1)! is at most δ 2+(n−2)(1+θ)+((n−1) 2 −1)θ . Since by assumption nθ 0 + θ ≥ n + ǫ, it follows that So there exists a positive number A depending only on n and ǫ such that (m 0 + 1) n m n−1 n(n−1)−1 n! (n(n − 1) − 1)! is greater than We can also assume that A is chosen so that δ−δ θ 0 −2δ θ ≥ δ θ ′ for δ ≥ A to make sure that the (n − 1)-jet differential 1 is holomorphic on X and vanishes to order at least q at X ∩ H Pn .
Remark 3.9. Proposition 3.8 is the same as Proposition 4.6 on p.446 of [Si02] and also the same as Proposition 2 on p.558 of [Si04].

Hyperbolicity from Slanted Vector Fields and Vanishing Order of Low-Degree Polynomials
Since the holomorphic jet differential ω (α) vanishing on an ample divisor of X (α) from the construction in Proposition 3.8 is of the form x 1 −1 , the discussion in (2.21) and (2.22) shows that the hyperbolicity of generic hypersurface of high degree is now reduced to the control of the vanishing order of the pullback to X (α) of a polynomial of low degree in d ℓ x j for 0 ≤ ℓ ≤ n − 1 and 1 ≤ j ≤ n. In this section we deal with this question of the control of the vanishing order of such pullbacks and finish the proofs of Theorem 0.1 and Theorem 0.2.
Lemma 4.1. (Vanishing Order of Restriction of Polynomials) Let W be a nonempty connected open subset of P N such that X (α) is regular for every α ∈ W . Let m be a positive integer and for α ∈ W let g (α) be a homogeneous polynomial of degree m in the homogeneous coordinates z 0 , · · · , z n of P n whose coefficients g (α) µ 0 ,··· ,µn (for µ 0 + · · · + µ n = m) are holomorphic functions of α on W , which is not identically zero on P n for α ∈ W . Let σ(α) be a holomorphic local section of X over W . Then there is a proper subvariety Z g,σ of W such that for α ∈ W − Z g,σ the vanishing order of the restriction g (α) X (α) of g (α) to X (α) at the point σ(α) of X (α) is no more than m. As a consequence, for any relatively compact nonempty subsetW of W there exists a proper subvarietyZ g ofW such that for α ∈W −Z g the vanishing order of the restriction g (α) X (α) of g (α) to X (α) at any point of X (α) is no more than m.
Proof. We prove first the first conclusion involving the local holomorphic section σ of X over W and then prove the second statement without the local holomorphic section σ of X over W later. Since the set of all α ∈ W such that the vanishing order of the restriction g (α) is > m is a subvariety of W which may be all of W , it suffices to show that there exists some point α of W such that the vanishing order of the restriction g (α) X (α) of g (α) to X (α) at the point σ(α) of X (α) is no more than m. We pick some point α * of W and for notational simplicity, when we need to replace W by a smaller open neighborhood W ′ of α * in W , we use the same symbol W for the new smaller neighborhood W ′ . When we need to replace α * by some other point, we use the same symbol α * for the new point.
We now make the reduction to the special case that σ(α) is always the origin in some inhomogeneous coordinates system of P n for α ∈ W . Choose an inhomogeneous coordinate system x 1 , · · · , x n of P n centered at σ(α * ) and let (σ 1 (α), · · · , σ n (α)) be the coordinates of σ(α) with respect to x 1 , · · · , x n . We can assume without loss of generality that (σ 1 (α), · · · , σ n (α)) belongs to the affine part C n of P n for α ∈ W (after replacing W by a smaller open neighborhood of α * in W if necessary). For α ∈ W let ν 0 ,··· ,νn x ν 1 1 · · · x νn n .
Now we can assume without loss of generality that σ(α) equal to the origin x 1 = · · · = x n = 0 for all α ∈ W . Let p be the minimum vanishing order of g (α) on P n at x 1 = · · · = x n = 0 for α ∈ W , which is achieved at some point α * of W . We have 0 ≤ p ≤ m, because the vanishing order p is for g (α * ) on P n and not for the restriction of g (α * ) on X (α * ) . If p = 0, then g (α * ) is nonzero at the origin x 1 = · · · = x n = 0 and as a consequence the restriction of g (α * ) to X (α * ) is nonzero at the origin x 1 = · · · = x n = 0. So the vanishing order of the restriction of g (α * ) on X (α * ) at the origin x 1 = · · · = x n = 0 is just 0. So we can assume without loss of generality that 1 ≤ p ≤ m.
By using a C-linear transformation of the inhomogeneous coordinates x 1 , · · · , x n and, after dividing g (α) by a local holomorphic function on P N and replacing W by another nonempty open subset of P N , we can assume without loss of generality that g (α) is of the form (4.1.1) where each A j (α, x 1 , · · · , x n−1 ) is a polynomial in x 1 , · · · , x n−1 of degree ≤ m − j whose coefficients are holomorphic functions on W such that A j (α, x 1 , · · · , x n−1 ) vanishes at x 1 = · · · = x n−1 = 0 for 0 ≤ j ≤ p.
The expression (4.1.1) is simply writing g (α) as a polynomial in x n , where a C-linear transformation of x 1 , · · · , x n is used to make sure that the coefficient of x p n in the initial polynomial of degree p, at the origin in the inhomogeneous coordinates, is a non identically zero function of α, which is reduced to 1 after dividing g (α) by it. The condition that A j (α, x 1 , · · · , x n−1 ) vanishes at x 1 = · · · = x n−1 = 0 for 0 ≤ j < p simply means that the vanishing order at the origin of the inhomogeneous coordinates is indeed p and not less than p. The condition that A j (α, x 1 , · · · , x n−1 ) vanishes at x 1 = · · · = x n−1 = 0 for j = p simply means that its constant term has been separated out to be the first term x p n after division by its coefficient which is a function of α.
Let Y = ℓ q ℓ Y ℓ be the divisor of the restriction of g (α) to X , where Y ℓ is an irreducible hypersurface in X ∩ (P n × W ). Since the p times differentiation ξ p g (α) of g (α) with respect to the vector field ξ is nonzero at every point of X ∩ ({0} × W ), it follows that the multiplicity of Y at every point of , otherwise we let Y ′ ℓ be the empty set. Then for α ∈ W which is not in the image of the projection of ℓ Y ′ ℓ to W , the vanishing order of the restriction g (α) X (α) at the origin Note that in this proof of the first statement we already fully use the special form of X defined by f (α) = ν 0 +···+νn=δ α ν 0 ,··· ,νn x ν 1 1 · · · x νn n in our use of the vector field ξ = ∂ ∂x n − ν 0 +···+νn= δ, νn≥1 ν n α ν 0 ,··· ,νn ∂ ∂α ν 0 ,··· ,ν n−1 ,νn−1 , which comes from the proof of Lemma 1.6 by setting z 0 = 1, z j = x j for 1 ≤ j ≤ n and setting a j,k = 0 for (j, k) = (0, n) and a 0,n = 1. One can also easily get ξ directly from ∂ ∂x n f (α) = ν 0 +···+νn= δ, νn≥1 n n α ν 0 ,··· ,νn x ν 1 1 · · · x ν n−1 n−1 x νn−1 n and then from replacing the monomial x ν 1 1 · · · x ν n−1 n−1 x νn−1 n by ∂ ∂α ν 0 ,··· ,ν n−1 ,νn−1 f (α) = x ν 1 1 · · · x ν n−1 n−1 x νn−1 n . Now we come to the second statement without the local holomorphic section σ of X over W . Let V be the set of all points (P, α) of P n × W such that the vanishing order at P of the restriction g (α) X (α) of g (α) to X (α) is > m. Then V is a subvariety of X ∩ (P n × W ). If the second statement does not hold, then the projection of V to W induced by P n × W → W is surjective. We can find a local holomorphic section σ of V over some nonempty open subset W ′ of W . This contradicts the first statement involving the local holomorphic section σ of X over the nonempty open subset W ′ of P N .
Remark 4.2. (1) For use of the Lemma 4.1 in our proof of hyperbolicity of generic hypersurface of high degree, we need only a very weak version of it, for example, with the conclusion that the vanishing order of the restriction of g (α) to X (α) is ≤ c n m 1+εn + c ′ n at any point of its zero-set in X (α) for some positive integers c n and c ′ n which depend only on n, where ε n is some small positive number depending only on n.
(2) The generic condition on α in Lemma 4.1 cannot be removed. A simple example to illustrate this point is the Fermat hypersurface n j=1 x δ j = 1 of degree δ in P n , for which the complex hyperplane x n = 1 touches the Fermat hypersurface to order δ, because the equation x n = 1 when pulled back to the Fermat hypersurface yields n−1 j=1 x δ j which vanishes to order δ at the point (x 1 , · · · , x n−1 , x n ) with x j = 0 for 1 ≤ j ≤ n − 1 and x n = 1. x 1 −1 be the holomorphic jet differential on X(a) constructed in (3.8). There exists some proper subvariety Z of P N such that for a ∈ P N − Z and any P 0 ∈ J n−1 X (α) representable by a nonsingular complex curve germ in X (α) , there exist slanted vector fields of low vertical pole order ξ 1 , · · · , ξ ℓ (with ℓ explicitly of order δ ε ) such that Lie ξ 1 · · · Lie ξ 1 ω (α) defines a holomorphic jet differential on X (α) which is nonzero at the (n − 1)-jet P 0 .
Proof. From our construction in (3.8) the k-jet differential Q (α) is a polynomial of low degree in both the inhomogeneous coordinates x 1 , · · · , x n and their differentials d ν x 1 , · · · , d ν x n (for 1 ≤ ν ≤ n − 1) so that for a generic a the pullback to X (α) of the k-jet differential ω (α) := Q (α) x 1 − 1 is a polynomial in x 1 , · · · , x n of high degree (i.e., comparable to the degree δ of f ) and dividing by it contributes to high vanishing order at the infinity hyperplane of P n . We introduce the new notation F (α) here to emphasize that in the following argument the coordinate x 1 plays no role and what matters is the contribution of division by F (α) to high vanishing order at the infinity hyperplane of P n . We are interested in the vanishing order of the pullback to X (α) of the k-jet differential Q (α) F (α) on X (α) and how, if the hypersurface X (α) is generic, at any given point of X (α) , an effectively small number of Lie differentiations can reduce it to no vanishing.
First of all, for notational simplicity we introduce the following notation for a monomial of differentials of inhomogeneous coordinates. Take monomial 1≤j≤n, 1≤ℓ≤k d ℓ x j ν ℓ,j of differentials of the inhomogeneous coordinates x 1 , · · · , x n of P n up to order k, where each ν ℓ,j is a nonnegative integer. We use ν to to denote the indexed collection (ν ℓ,j ) 1≤j≤n, 1≤ℓ≤k of nonnegative integers and use d ν x to denote the monomial 1≤j≤n, 1≤ℓ≤k itself. To the monomial d ν x of differentials of inhomogeneous coordinates up to order k define its multi-degree as the k-tuple of nonnegative integers (m k , m k−1 , · · · , m 1 ), where m ℓ = n j=1 ν ℓ,j is its degree as a monomial in the variables d ℓ x 1 , · · · , d ℓ x n when the other differentials are regarded as coefficients. We use the notation mdg (d ν x) to denote the multi-degree (m k , m k−1 , · · · , m 1 ). Since the multi-degree of d ν x depends only on the indexed collection ν of nonnegative integers, we denote the multi-degree (m k , m k−1 , · · · , m 1 ) also simply by mdg(ν).
We say that a k-tuple (m k , m k−1 , · · · , m 1 ) is represented as a multidegree in Q (α) if there exists some multi-index ν such that (m k , m k−1 , · · · , m 1 ) is equal to mdg(ν) and the polynomial Q (α) ν of inhomogeneous coordinates x 1 , · · · , x n of P n is not identically zero.
Take an arbitrary point y of X (α) . For a multi-index ν with Q (α) ν not identically zero as a polynomial of inhomogeneous coordinates x 1 , · · · , x n of P n , let q y,ν be the vanishing order at y of the restriction of Q (α) ν to X (α) . This vanishing order q y,ν is effectively bounded from above by Lemma 4.1 for a generic α. For a k-tuple (m k , m k−1 , · · · , m 1 ) which is represented as multi-degree in Q (α) , let q y,m k ,m k−1 ,··· ,m 1 be the minimum of q y,ν over all multi-indices ν with mdg(ν) = (m k , m k−1 , · · · , m 1 ). Note that the number q y,m k ,m k−1 ,··· ,m 1 remains unchanged under any C-linear change of the inhomogeneous coordinates x 1 , · · · , x n , even though Q (α) ν as well as q y,ν may be changed under a C-linear change of the inhomogeneous coordinates x 1 , · · · , x n .
We now pull back Q (α) to X (α) to get ι * Q (α) and express ι * Q (α) in terms of the coordinates x 1 , · · · , x n−1 and their differentials. We will show that the result of applying an effective number of well chosen slanted vector fields ξ 1 , · · · , ξ p to ι * Q (α) will be a holomorphic jet differential on X (α) which is pointwise non identically zero at y (for a generic α ∈ P n with the generic condition not depending on y).
We are going to use an inhomogeneous coordinate system x 1 , · · · , x n of P n centered at y as follows. First of all, we choose x n such that the hyperplane {x n = 0} is the tangent hyperplane of X (α) at the point y. Secondly, we generically choose C-linear functions x 1 , · · · , x n−1 so that q y,m * k ,m * k−1 ,··· ,m * 1 = q y,ν * , where ν * is the indexed collection of nonnegative integers such that In order to express ι * Q (α) in terms of the coordinates x 1 , · · · , x n−1 and their differentials, we use the implicit function theorem to express x n in terms of x 1 , · · · , x n−1 on X (α) to get a local holomorphic function x n = g α,u (x 1 , · · · , x n−1 ) in a neighborhood of y in P n . Since {x n = 0} is the tangent hyperplane of X (α) at the point y, it follows that ∂gα,y ∂x 1 vanishes at y.
For any λ * -tuple of nonnegative integers t = (t λ ) 1≤λ≤λ * with 0 ≤ t λ ≤ m * λ for 1 ≤ λ ≤ λ * let ν * t be the indexed collection of nonnegative integers such that The vanishing order at y of the restriction Since the vanishing order at y of ι * Q (α) ν * is equal to q y,m * k ,m * k−1 ,··· ,m * 1 , by Proposition (2.17) on the global generation on jet space by slanted vector fields at points representable by regular curve germs, we can choose slanted vector fields ξ j for 1 ≤ j ≤ q of low vertical pole order with q = q y,m * k ,m * k−1 ,··· ,m * 1 such that ξ 1 · · · ξ q ι * Q (α) ν * is nonzero at y and Lie ξ λ Lie ξ λ+1 · · · Lie ξq d ℓ x j on X equals d ℓ x j at y for 1 ≤ λ ≤ q, 1 ≤ ℓ ≤ k and 1 ≤ j ≤ n − 1, because slanted vector fields generate the tangent bundle of the space of vertical jet differentials of X . To choose such slanted vector fields of low vertical pole order, we need only make sure that their components along the local coordinates x 1 , · · · , x n−1 of X (which are now regarded as local coordinates of the k-jet space of X by pullback) give a nonzero value of ξ 1 · · · ξ q ι * Q (α) ν * at y and their components along the coordinates d ℓ x j for 1 ≤ ℓ ≤ k and 1 ≤ j ≤ n−1 in the k-jet space of X are all zero at y. Since the vanishing order at y of ι * Q (α) ν * is equal to q, we have the vanishing of ξ λ ξ λ+1 · · · ξ q ι * Q (α) ν * for 2 ≤ λ ≤ q.
Since the vanishing order at y of ι * Q (α) ν * t is no more than q and since ∂gα,y ∂x 1 vanishes at y, we have the vanishing of We now apply the Lie differentiation by ξ 1 , · · · , ξ q to ω (α) = Q (α) F (α) and conclude that at y the coefficient of d ν * x in Lie ξ 1 Lie ξ 2 · · · Lie ξq ω (α) is equal to the value of ξ 1 · · · ξ q ι * Q (α) ν * at y which is nonzero.

Proof of Theorem 0.1
For a generic hypersurface X (α) of sufficiently high degree δ, by Proposition 4.3 for every P 0 in the (n − 1)-jet space J n−1 X (α) representable by nonsingular complex curve germ there exists a holomorphic (n − 1)-jet differential ω on X (α) which vanishes on an ample divisor of X (α) and which has nonzero value at P 0 .
We need to verify that there exists no nonconstant holomorphic map ϕ : C → X (α) . Suppose the contrary and we are going to derive a contradiction. Since ϕ is nonconstant, there exists some ζ 0 ∈ C with the differential dϕ of ϕ is nonzero. Let P 0 be the (n − 1)-jet of X (α) at ϕ (ζ 0 ) represented by the parametrized complex curve germ ϕ : C → X (α) . Since ω is a holomorphic (n − 1)-jet differential on X (α) vanishing on an ample divisor of X (α) , it follows that the pullback ϕ * ω is identically zero on C, contradicting the value of ω at P 0 being nonzero (c.f., Proposition 5.4 and Remark 5.5 below).

Proof of Theorem (0.2)
The proof is analogous to the proof of Theorem 0.1. The difference is that (i) the holomorphic jet differential ω used in the proof of Theorem 0.1 is replaced by the log-pole jet differential on P n constructed in Theorem 5.13, (ii) the use of Lemma 4.1 in the proof of Proposition 4.3 is modified by Remark 5.14 concerning the vanishing order of log-pole jet differential constructed as the direct image of jet differential, (iii) the use of Proposition 2.17 is replaced by the use of Proposition 2.19 (iv) the use of the Schwarz lemma of the vanishing of the pullback to C of a holomorphic jet differential vanishing on an ample divisor is replaced by Proposition 5.10 concerning the general Schwarz lemma for log-pole jet differentials.

Essential Singularities, Varying Coefficients, and Second Main Theorem
Besides the Little Picard Theorem of nonexistence of nonconstant holomorphic maps from C to P 1 −{0, 1, ∞} there is a stronger statement which is the Big Picard Theorem of no essential singularity at ∞ for any holomorphic function from C − ∆ r 0 to C − {0, 1} for r 0 > 0. The Little Picard Theorem corresponds to our theorem on the hyperbolicity of a generic hypersurface X (α) of sufficiently high degree in P n (for α in P N outside a proper subvariety Z of P N ). Corresponding to the Big Picard Theorem, there is a statement concerning the extendibility across a holomorphic map C − ∆ r 0 → X (α) to a holomorphic map In this section we are going to prove such a theorem on removing the essential singularity at ∞ of a holomorphic map from C − ∆ r 0 to a generic hypersurface of sufficiently high degree.
The more quantitative version of the Big Picard Theorem was introduced by Nevanlinna [Ne25] in his Second Main Theorem in his theory of value distribution theory. In this section we discuss the Second Main Theorem from log-pole jet differentials which is more in the context of Cartan's generalization of Nevanlinna's Second Main Theorem to holomorphic maps from the affine complex line to P n and a collection of hyperplanes in general position [Ca33].
The hyperbolicity of a generic hypersurface of high degree δ in P n can be reformulated as the nonexistence of n + 1 entire holomorphic functions on P with some ratio nonconstant which satisfy a generic homogeneous polynomial of degree δ with constant coefficients. Our solution of the hyerbolicity problem for a generic hypersurface of high degree makes use of the universal hypersurface X in P n × P N and the variation of the hypersurface X (α) in P n with α ∈ P N . The variation of X (α) corresponds to the varying of the constant coefficients of the homogeneous polynomial equation for the n + 1 entire functions. In this section we will discuss the problem of nonexistence of entire functions satisfying polynomial equations with slowly varying coefficients and also the more general result for removing essential singularities for holomorphic functions on C minus a disk for this setting.

I. Removal of Essential Singularities
To extend our methods from maps C → X (α) to maps C − ∆ r 0 → X (α) for some r 0 > 0, we need a corresponding extension of Nevanlinna's logarithmic derivative lemma (p.51 of [Ne25]). For such an extension of Nevanlinna's logarithmic derivative lemma, we need the following trivial multiplicative version of the Heftungslemma [AN64].
Lemma 5.1. (Trivial Multiplicative Version of Heftungslemma) Let r 0 > 0 and F be meromorphic on C − ∆ r 0 . Let r 0 < r 1 . Then there exists some function G holomorphic and nowhere zero on C∪{∞}−∆ r 1 such that F G is meromorphic on C. Moreover, when F is holomorphic, G can be chosen so that F G is also holomorphic on C − {0}.
For Y = P n and η being the Fubini-Study form, we drop η in the notation T (r, r 1 , ϕ, η) and simply use T (r, r 1 , ϕ) when there is no confusion. When a holomorphic map ϕ from C − ∆ r 0 to P n is given by holomorphic functions [F 0 , · · · , F n ] on C−∆ r 0 without common zeroes, its characteristic function is We would like to compare it with the characteristic function T r, r 1 , The verification of for 1 ≤ j ≤ n is as follows.
From twice integration of Laplacian with g = log N k=0 |F k | 2 in (0.12), we have from which we conclude that 1 2π Finally from it follows that T r, r 1 , The verification of for 1 ≤ j ≤ n is as follows. From r ρ=r 1 |ζ|<ρ The verification of the inequality comparing characteristic functions of maps and meromorphic functions is a straightforward modification of the proof of Lemma (2.1.2) on p.426 of [Si95], where the map is from C instead of from C − ∆ r 0 .
The logarithmic derivative lemma holds for meromorphic functions on C − ∆ r 0 for r 0 > 0 in the following form.
for r > r 1 .
Proof. By 5.1 there exists some function G holomorphic and nowhere zero on C ∪ {∞} − ∆ r 1 such that F G is meromorphic on C. Let Thus, because G holomorphic and nowhere zero on C ∪ {∞} − ∆ r 1 . The required statement follows from and log + |H| ≤ log + |F | + O(1). By 5.1 on the trivial multiplicative version of Heftungslemma, for some holomorphic nowhere-zero function G 0 on C∪{∞}−∆ r 2 with r 0 < r 2 < r 1 such that H(ζ) := ζ ℓ F 0 (ζ)G 0 (ζ) is holomorphic on C (with coordinate ζ) and is nonzero at because F 0 , F 1 , · · · , F n are assumed to have no common zeroes on C − ∆ r 0 . Thus, and ω be a holomorphic jet differential on X vanishing on some ample divisor D of X. Let r 1 > r 0 > 0 and ϕ : C−∆ r 0 → X be a holomorphic map. Let eval id C (ϕ * ω) denote the function on C − ∆ r 0 whose value at ζ is the evaluation of the jet differential ϕ * ω at the jet defined by the identity map of C − ∆ r 0 at ζ. Then either eval id C (ϕ * ω) is identically zero on C − ∆ r 0 or T (r, r 1 , ϕ, η) = O (log r) .
Proof. Let k be the order of jet differential ω and m be its weight. Let L D be the line bundle associated to the ample divisor D. Let e −χ D be a smooth metric of L D whose curvature form η D is strictly positive definite on X. Let s D be a holomorphic section of L D whose divisor is D. Let Φ = eval id C (ϕ * ω). We assume that Φ is not identically zero. We apply twice integration of Laplacian in (0.12) to Since ω is holomorphic on X and vanishing on D, it follows that Here we have the inequality instead of an identity, because of possible contribution from the zero-set of ω s D . At this point enters Bloch's technique of applying the logarithmic derivative lemma by using the logarithm of global meromorphic functions as local coordinates. As functions on the k-jet space J k (X) of X (with the right-hand side being global functions and the left-hand side being only local functions due to the transition functions of the line bundles L D ), for some C > 0 and a finite collection F (λ) j,ℓ of global meromorphic functions on X, where the product is taken over the indices ν j,ℓ , j, ℓ with the ranges 1 ≤ j ≤ n, 1 ≤ ℓ ≤ k and 1≤j≤n, 1≤ℓ≤k ℓ ν j,ℓ = m. By Nevanlinna's logarithmic derivative lemma (extended to C outside a disk centered at the origin) given in (5.3), 2π θ=0 log + d ℓ log F (λ) j,ℓ re iθ dθ = O (log T (r, r 1 , ϕ, η D ) + log r) .
Remark 5.5. The argument in Proposition (5.4) is simply a modification of the case of the usual Schwarz lemma on pullbacks of jet differentials (see e.g., Theorem 2 on p.1140 of [SY97]) when the holomorphic map ϕ is from the entire affine complex line C to X. For this case, the pullback ϕ * ω is always identically zero on C for the following reason. We can replace ϕ by the composite ψ of ϕ with the exponential map from C to C to rule out the case of T (r, ψ, η) = O (log r) so that eval id C (ψ * ω) vanishes identically on C. Since any k-jet of C at any point ζ 0 of C can realized by some holomorphic map σ from C to itself, from the vanishing of eval id C (σ * ω) vanishes identically on C it follows that ψ * ω is identically zero on C, which implies that ϕ * ω is identically zero on C.
Lemma 5.6. (Extension of Holomorphic Maps with Log Order Growth Characteristic Function Across Infinity Point) Let ϕ be a holomorphic map from C − ∆ r 0 to P n given by holomorphic functions [F 0 , · · · , F n ] on C − ∆ r 0 without common zeroes. If T (r, r 1 , ϕ) = O(log r) , then ϕ can be extended to a holomorphic map from C ∪ {∞} − ∆ r 0 to P n .
Proof. By the comparison of characteristic functions of maps defined outside a disk given in (5.2),we have for 1 ≤ j ≤ n. By the trivial multiplicative version of the Heftungslemma given in (5.1), there exists some holomorphic nowhere zero for 1 ≤ j ≤ n we conclude that G j F j F 0 is a rational function on C. Hence F j F 0 can be extended to a meromorphic function on C∪{∞}−∆ r 0 . Thus ϕ can be can be extended to a holomorphic map from C ∪ {∞} − ∆ r 0 to P n .
Proposition 5.7. Let X be a compact complex manifold. Let ω 1 , · · · , ω N be holomorphic k-jet differentials of total weight m on X with each vanishing on some ample divisor of X. Assume that at any point P 0 of J k (X) which is representable by a nonsingular complex curve germ, at least one of ω 1 , · · · , ω N is nonzero at P 0 for some 1 ≤ j ≤ N. Then any holomorphic map ϕ from C − ∆ r 0 to X can be extended to a holomorphic map from C ∪ {∞} − ∆ r 0 to X.
Proof. We can assume without loss of generality that ϕ is not a constant map so that at some point ζ 0 of C − ∆ r 0 the differential of ϕ is nonzero at ζ 0 . Let P 0 be the element of J k (X) at the point ϕ(ζ 0 ) of X defined by the nonsingular complex curve germ represented by ϕ at ζ 0 . By assumption, there exists some 1 ≤ j 0 ≤ N such that ω j 0 is nonzero at P 0 . It follows that the function eval id C (ϕ * ω j 0 ) associated to ϕ * ω j 0 as described in (0.13) is nonzero at ζ 0 . Let η be a Kähler form of X and let r 1 > r 0 . By (5.4) we have T (r, r 1 , ϕ, η) = O(log r) , which implies that the holomorphic map ϕ from C − ∆ r 0 to X can be extended to a holomorphic map from C ∪ {∞} − ∆ r 0 to X.
Theorem 5.8. For any integer n ≥ 3 there exists a positive integer δ n with the following property. For any positive integer δ ≥ δ there exists a proper subvariety Z in the moduli space P N of all hypersurfaces of degree δ in P n (where N = n+δ n ) such that for α ∈ P n − Z and any holomorphic map ϕ : C − ∆ r 0 → X (α) (where r 0 > 0) can be extended to a holomorphic map C ∪ {∞} − ∆ r 0 → X (α) , where X (α) is the hypersurface of degree δ in P n corresponding to the point α in the moduli space P N . Before the introduction of the language of geometry of manifolds, hyperbolicity problems were formulated in terms of entire functions satisfying functional equations. For example, a theorem of Borel states that if entire function ϕ 1 , · · · , ϕ n satisfy e ϕ 1 + · · · + e ϕn = 0, then ϕ j − ϕ k is constant with 1 ≤ j < k ≤ n. In the formulation in terms of functional equations satisfied by entire functions, the hyperbolicity of a generic hypersurface of high degree δ states that no n + 1 entire holomorphic functions ϕ 0 (ζ), · · · , ϕ n (ζ), with the ratios ϕ j ϕ ℓ not all constant, can satisfy a homogeneous polynomial equation ν 0 +···+νn=δ α ν 0 ,··· ,νn ϕ ν 1 1 · · · ϕ ν 1 1 ≡ 0 of degree δ whose constant coefficients α ν 0 ,··· ,νn are generic.
For this question of polynomial equations with slowly varying coefficients under additional assumption of osculation, we present here two results. The first one, corresponding to hyperbolicity, is that when the map ζ → α(ζ) is slowly moving compared to the map ζ → ϕ(ζ) ∈ P n , no such pair of curves ζ → α(ζ) ∈ P N and ζ → ϕ(ζ) ∈ P n exist.
The second result, corresponding to the Big Picard Theorem, concerns extension across ∞ when the pair of maps ζ → ϕ(ζ) ∈ P n and ζ → α(ζ) ∈ P N are only defined for ζ ∈ C − ∆ r 0 instead of on C.
Since the Schwarz lemma is the crucial tool for the hyperbolicity problem, for the more general case of slowly varying coefficients we need a variation of the Schwarz lemma for it. We are going to present it in the form which is more than we need by allowing log-pole jet differentials rather than just holomorphic jet differentials so that it can be used later in this article in the proof of Second Main Theorems for log-pole jet differentials (see Thoerem 5.15 and Theorem 5.16 below).
A log-pole jet differential means that locally it is of the form where x 1 , · · · , x n are local holomorphic coordinates, G λ and F 1 , · · · , F µ λ are local holomorphic functions. Each d ℓ log F ν contributes νℓ times the divisor of F to the log-pole divisor (with multiplicities counted) of the log-pole jet differential.
Proposition 5.10. (General Schwarz Lemma for Log-Pole Differential on Subvariety of Jets and Map with Slow Growth for Pole Set) Let X be a compact complex algebraic manifold of complex dimension n and Y be a complex subvariety of the space J k (X) of k-jets. Let π k : J k (X) → X be the natural projection map. Let D and E be nonnegative divisors of X whose associated line bundles L D and L E respectively have smooth metrics e −χ D and e −χ D with smooth (1, 1)-forms η D and η E as curvature forms such that D + E is an ample divisor of X and its curvature form η D + η E for the metric e −χ D −χ E is strictly positive on X. Let s D (respectively s E ) be the holomorphic section of L D (respectively L E ) whose divisor is D (respectively E). Let F be a nonnegative divisor of X and Supp F be its support. Let ω be a function on Y such that s E (s D ) −1 ω is holomorphic on Y −π −1 k (Supp F ). Assume that for some finite open cover {U j } J j=1 of X, there exists a log-pole k-form ω j on U j (for 1 ≤ j ≤ J), whose log pole is contained in F with multiplicities counted, such that on Y ∩ π −1 k (U j ) the function ω agrees with the function on π −1 k (U j ) defined by s D (s E ) −1 ω j for 1 ≤ j ≤ J. Let r 1 > r 0 > 0. Let ϕ : C − ∆ r 0 → X be a holomorphic map such that the image of the map J k (ϕ) : J k C − ∆ r 0 → J k (X) induced by ϕ is contained in Y . Let G j (ζ) be the function eval id C (ϕ * j ω) associated to ϕ ω j as explained in (0.13). If G j (ζ) is not identically zero on C − ∆ r 0 , then In particular, if for some ε > 0 one assumes that N (r, r 1 , ϕ, F ) + T (r, r 1 , ϕ, |η E |) ≤ (1 − ε) (T (r, r 1 , ϕ, η D )) , then either G j (ζ) is identically zero for all 1 ≤ j ≤ J) or Proof. We assume that G j 0 (ζ) is not identically zero for some 1 ≤ j 0 ≤ J. We apply twice integration of Laplacian with in (0.12) to Since ω j is holomorphic on Y − π −1 k (Supp F ), it follows that T (r, r 1 , ϕ, η D ) − T (r, r 1 , ϕ, η E ) − N (r, r 1 , ϕ, F ) Here we have the inequality instead of an identity, because of possible contribution from the zero-set of of global meromorphic functions on X, where the product is taken over the indices ν j,ℓ , j, ℓ with the ranges 1 ≤ j ≤ n, 1 ≤ ℓ ≤ k and 1≤j≤n, 1≤ℓ≤k ℓ ν j,ℓ = m. By Nevanlinna's logarithmic derivative lemma (extended to C outside a disk centered at the origin) given in (5.3), 2π θ=0 log + d ℓ log F (λ) j,ℓ re iθ dθ = O (log T (r, r 1 , ϕ, η D + η E ) + log r) .
Proof. We apply Proposition 5.10 with k = n − 1 to the the space J n−1 (X ) of (n − 1)-jets of the universal hypersurface X with subvariety Y equal to the space J vert n−1 (X ) of vertical (n − 1)-jets of X . The assumption ν 0 +···+νn=δ α ν 0 ,··· ,νn (ζ) d j dζ j (ϕ 0 (ζ) ν 0 · · · ϕ n (ζ) νn ) ≡ 0 for 0 ≤ j ≤ n − 1 implies that for every ζ ∈ C the element of J n−1 (X ) represented by the parametrized complex curve germ ϕ at ζ belongs to Y = J vert n−1 (X ). By Proposition 4.3 we have a proper subvariety Z of P N and for α ∈ P N − Z and 1 ≤ j ≤ J a holomorphic family of (n − 1)-jet differentials ω (α) j on X (α) vanishing on the infinity hyperplane of P n (extendible to a meromorphic family over all of P N ) such that, at any point P 0 of J n−1 X (α) with α ∈ P N − Z which is representable by a nonsingular complex curve germ, at least one ω (α) j is nonzero at P 0 for some 1 ≤ j ≤ J.
Since for each 1 ≤ j ≤ J the holomorphic family ω (α) j for α ∈ P N −Z can be extended to a meromorphic family for α varying in all of P N , we can find a divisor E j in P N such that for all 1 ≤ j ≤ J the pole-set of ω (α) j as a meromorphic vertical (n−1)-jet differential on X is contained in the intersection of X and P n × E j with multiplicities counted. For 1 ≤ j ≤ J, because ω (α) j vanishes on an ample divisor of X (α) for α ∈ P N − Z, we can find a divisor D j in X such that D j + E j is an ample divisor of X and the zero-set of ω (α) j as a meromorphic vertical (n − 1)-jet differential on X contains D j with multiplicities counted.
Since ψ : C → P N is nonconstant and T (r, ψ) = o (T (r, ϕ) + log r) , it follows that the differential dϕ is nonzero for some ζ 0 ∈ C. Denote by P 0 the point in J n−1 X (α) represented by the nonsingular complex curve germ ϕ at ζ 0 . Some ω (α) j 0 has nonzero value at P 0 . From Proposition 5.10 applied to ω The analogue of the Big Picard Theorem about removable essential singularities is the following result.
Proof. We use the same notations as in the proof of Theorem 5.11 except the domains for the maps ϕ : C − ∆ r 0 → X and psi : C − ∆ r 0 → P N are now different. Without loss of generality we can assume that the map ϕ is nonconstant, otherwise the extendibility of ϕ and ψ is clear. Denote by P 0 the point in J n−1 X (α) represented by the nonsingular complex curve germ ϕ at ζ 0 . Some ω (α) j 0 has nonzero value at P 0 . From Proposition 5.10 applied to ω (α) j 0 , it follows that T (r, ϕ) = O(log r) . Now the extendibility of ϕ and ψ to respectively holomorphic maps C − ∆ r 0 → X and C − ∆ r 0 → P N follows from Lemma 5.6.

III. Second Main Theorem from Log-Pole Jet Differential
Nevanlinna's Second Main Theorem is a quantitative version of the Little Picard Theorem. The hyperbolicity of generic hypersurface of high degree corresponds to the Little Picard Theorem. We now discuss the analogue of Nevanlinna's Second Main Theorem for any regular hypersurface of high degree from our approach of jet differentials.
In contrast to the use of holomorphic jet differentials vanishing on an ample divisor in the hyperbolicity problem, the jet differentials used for the Second Main Theorem are log-pole jet differentials vanishing on ample divisor. Our method fits in with Cartan's proof of the Second Main Theorem for entire holomorphic curves in P n and a collection of hyperplanes in P n in general position given in [Ca33].
We will first show how to construct log-pole jet differentials on P n which vanishes on an appropriate ample divisor of P n and whose log pole-set is contained in the hypersurface. We then present two Second Main Theorems for log-pole jet differentials, with the second one dealing with the situation of slowly moving targets. Then we show how Cartan's proof can be recast in our setting of Second Main Theorem for log-pole jet differentials vanishing on an appropriate ample divisor.
Second main theorems are useful only when the estimates are reasonably sharp. In the case at hand, because of our construction of jet differentials is so far away from the conjectured optimal situation, the discussion about Second Main Theorems can only serve as pointing out a connection between Second Main Theorems and jet differentials and their construction.
Theorem 5.13. (Existence of Log Pole Jet Differential) Let 0 < ε 0 , ε ′ 0 < 1. There exists a positive integerδ n such that for any regular hypersurface X of degree δ ≥δ n in P n there exists a non identically zero log-pole n-jet differential ω on P n of weight ≤ δ ε 0 which vanishes with multiplicity at least δ 1−ε ′ 0 on the infinity hyperplane of P n and which is holomorphic on P n − X. In particular, the log-pole divisor of ω is no more than λ times X with λ ≤ nδ ε 0 .
Let f (x 1 , · · · , x n ) be a polynomial in terms of the inhomogeneous coordinates x 1 , · · · , x n of P n which defines X. LetX be the regular hypersurface in P n+1 defined by the polynomial F = f (x 1 , · · · , x n ) − x δ n+1 in the inhomogeneous coordinates x 1 , · · · , x n+1 of P n+1 . We apply Proposition 3.8 to F (instead of to f ) to get an n-jet differentialω of the form Q Fx 1 −1 which vanishes to order ≥ δ θ ′ at the infinity hyperplane of P n+1 , where Q is a polynomial in which is of degree m 0 = δ θ 0 in x 1 , · · · , x n+1 and is of homogeneous weight m = δ θ in d j x 1 , · · · , d j x n+1 (1 ≤ j ≤ n) when the weight of d j x ℓ is assigned to be j.
We choose a nonzero integer ℓ such that one nonzero term of Q x ℓ n+1 is of the form where Q 0 is a polynomial in the variables d j x 1 , · · · , d j x n (0 ≤ j ≤ n) with constant coefficients and ν 1 , · · · , ν n are nonnegative integers.
The complex manifoldX is a branched cover over P n with cyclic branching of order δ at X under the projection mapπ :X → P n induced by (x 1 , · · · , x n , x n+1 ) → (x 1 , · · · , x n ). Let ω be the direct image ofω x ℓ n+1 = Q x ℓ n+1 (f x 1 − 1) underπ. The n-jet differential ω on P n can be computed as follows. First we express d ℓ x n+1 (by induction on ℓ) as a polynomial of the variables x n+1 , d log x n+1 , d 2 log x n+1 , · · · , d ℓ log x n+1 By applying Proposition 5.10, we have the following two Second Main Theorems for log-pole jet differentials, with second one dealing with the case of slowly moving targets.
Theorem 5.15. (Second Main Theorem from Log-Pole Jet Differentials) Let X be an n-dimensional compact complex manifold with an ample line bundle L. Let D 1 , · · · , D p , E 1 , · · · , E q be divisors of L. Let ω be log-pole jet differential on X with vanishing on D = D 1 + · · · + D p such that the log-pole set of ω is contained in E = E 1 + · · · + E q with multiplicities counted. Then for any holomorphic map ϕ from the affine complex line C to X such that the image of ϕ is not contained in E and the pullback ϕ * ω not identically zero, pT (r, ϕ, L) ≤ N(r, ϕ, E) + O (log T (r, ϕ, L)) .
Theorem 5.16. (Second Main Theorem for Jet Differential with Slowly Moving Targets) Let S ⊂ P N be a complex algebraic manifold and X ⊂ Pn × S be a complex algebraic manifold. Let π : X | → S be the projection induced by the natural projection Pn × P N → P N to the second factor. Let L S be an ample line bundle of S. Let L be a line bundle of X such that L+π −1 (L S ) is ample on X. Let D 1 , · · · , D p , E 1 , · · · , E q be divisors of L. Let D = D 1 + · · · + D p and E = E 1 + · · · + E q , For a ∈ S let X (α) = π −1 (a) and D (α) = D| X (α) and E (α) = E| X (α) . Let Z be a proper subvariety of S. For a ∈ S − Z let ω (α) be a log-pole jet differential on X (α) such that ω (α) vanishes on the divisor D (α) and the log-pole set of ω (α) is contained in divisor E (α) with multiplicity counted. Assume that ω (α) is holomorphic in a for a ∈ S − Z and is meromorphic in a for a ∈ S. Let ϕ be a holomorphic map from the affine complex line C to X such that the image of π • ϕ is not contained in Z and T (r, π • ϕ, L S ) = o (T (r, ϕ, L + π −1 (L S ))), then qT r, ϕ, L + π −1 (L S ) ≤ N(r, ϕ, D) + o T r, ϕ, L + π −1 (L S ) .
In the following remark we discuss how Cartan's proof of his Second Main Theorem for hyperplanes in general position can be interpreted in the setting of the Second Main Theorem for log-pole jet differentials.
Remark 5.17. Cartan's Second Main Theorem for hyperplanes in P n for hyperplanes in general position given in [Ca33] is simply the special case of Theorem 5.15 with ω = Wron(dx 1 , · · · , dx n ) F 1 · · · F q in inhomogeneous coordinates x 1 , · · · , x n of P n , where F 1 , · · · , F q are the degree-one polynomial in x 1 , · · · , x n which define the q hyperplanes in P n in general position.
The denominator F 1 · · · F q in ω gives the vanishing order q at the infinity hyperplane of P n . The key argument here is that from the general position assumption of the zero-sets of F 1 , · · · , F q we can locally write ω as a constant times Wron (dF ν 1 , · · · , dF νn ) F 1 · · · F q = Wron (d log F ν 1 , · · · , d log F νn ) F ν n+1 · · · F νq in a neighborhood U of a point when F j is nowhere zero on U for j not equal to any of the indices ν 1 , · · · , ν n .