This Is a Reprint of the Following Article: Yum-Tong Siu, Hyperbolicity Problems in Function Theory

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This is a reprint of the following article: Yum-Tong Siu, Hyperbolicity problems in function theory. Five decades as a mathematician and educator, 409–513, World Sci. Publ., River Edge, NJ, 1995.

HYPERBOLICITY PROBLEMS IN FUNCTION THEORY

YUM-TONG SIU

Introduction Chapter 1. Negative curvature in Dimension One 1. Nonexistence of Metric on C of Curvature with Uniform Negative Upper Bound 2. The Conformal Factor From the PoincaréMetric of the Punctured Disk 3. The First Main Theorem and the Defect Relation of Nevanlinna 4. Interpretation by Forms with Logarithmic Poles Chapter 2. logarithmic Derivative Lemma and Wronskians 1. Alternative Description of the Characteristic Function 2. The Logarithmic Derivative Lemma and the Wronskian 3. Cartan’s Second Main Theorem With Truncated Counting Func- tion 4. The Use of the Second Main Theorem with Truncated Counting Function 5. Relation with the Argument of the Borel Lemma 6. Use of Wronskian in the Defect Relation for Slowly Moving Targets Chapter 3. Negative Curvature in Higher Dimensions 1. Motivation for the Form of the Metric 2. The Case of Hyperplanes in the Complex Projective Space 3. The Use of the General Position Assumption 4. Meromorphic Connections of Low Pole Order 5. The Method of Dual Curves Chapter 4. Brody’s Reparametrization and Maps of Finite Order 1. Brody’s Reparametrization 2. Complements of Three Qradric Curves Chapter 5. Jet Differentials with Pole Logarithmic Singularities 1. Second Main Theorem for Jet Differentials with Logarithmic Pole Singularities 2. Complements of Three Plane Curves

1 Chapter 6. Special 2-Jet Differentials 1. Statement of Result and Outline of the Method 2. Construction of Special 2-Jet Differentials 3. Schwarz Lemma for Special Jet Differentials 4. The Proof of Algebraic Degeneracy 5. The Deformation Approach Chapter 7. The Theorem of Bloch 1. The Proof of Bloch, Ochiai, Kawamata, Wong, and Green-Griffiths 2. McQuillan’s Proof

Introduction The main purpose of this article is to present some of the important techniques which have been introduced over the years for solving hyperbolicity problems in function theory. To make the techniques more transparent, we will stress the main ideas and the key points instead of formulating them in the most general setting. The uniformization theorem for Riemann surfaces says that a simply con- nected Riemann surface is biholomorphic to the Riemann sphere, the Gaussian plane C, or the open unit disk in C. A Riemann surface whose universal cover is biholomorphic to the open unit disk in C is known as hyperbolic. An alternative characterization is that a Riemann surface M is hyperbolic if and only if there exists no nonconstant holomorphic map from C to M. In the compact case, the hyperbolic compact Riemann surfaces are precisely those whose genus is at least two. A simple case of a hyperbolic noncompact Riemann surface is the Riemann sphere minus three points. For the higher dimensional case, in general a complex manifold M is called hyperbolic if there exists no nonconstant holomorphic map from C to M. There are several notions of hyperbolicity and the one that is used here is more precisely known as Brody-hyperbolicity. In recent years interest in hyperbolicity problems in function theory has increased because of its formal and not-yet-understood relation with number theory which was pointed out by Osgood [Os85] and Vojta [Vo87]. According to this formal relationship the existence of nonconstant holomorphic maps from C corresponds to the existence of an infinite number of rational points. It led Lang and Vojta to conjecture that a compact nonsingular algebraic variety defined over Q which is hyperbolic has only a finite number of rational points. For the noncompact case the conjecture is that a nonsingular affine algebraic variety defined over Q has a finite number of integral points. However, in this article we will mention this formal relation only tangentially. We will mainly focus on the following hyperbolicity conjecture due to Koba- yashi and Zaidenberg and on Bloch’s theorem and related problems.

2 Conjecture (0.1). If H is a generic complex hypersurface of degree δ ≥ 2n + 1 in Pn, then Pn − H is hyperbolic. Here the genericity of H means that, as an element of the moduli space D of all complex hypersurfaces of degree δ in Pn, H must be in a suitable Zariski open subset of D for the statement in the conjecture to hold. There is the following compact version of the conjecture. Conjecture (0.2). If H is a generic complex nonsingular hypersurface of degree at least 2n in Pn, then H is hyperbolic. Relation of the Two Conjectures. These two conjectures are related in the following way. Let H be a complex hypersurface in Pn defined by the homogeneous polynomial f(z0, ··· , zn) = 0 of degree δ. Let X be the complex δ hypersurface in Pn+1 defined by zn+1 = f(z0, ··· , zn). Consider the projection map π : X → Pn defined by [z0, ··· , zn, zn+1] → [z0, ··· , zn]. The map π makes X a branched cover of δ sheets over Pn with cyclic branching of order δ along H. If g : C → Pn − H is a nonconstant holomorphic map, then g can be lifted to a nonconstant holomorphic mapg ˜ from C to X such that π ◦ g˜ = g. So if one can prove that X is hyperbolic, then Pn − H is hyperbolic. This does not necessarily mean that Conjecture (0.2) implies Conjecture (0.1), because, when H is generic as a complex hypersurface of Pn, X may not be generic as a hypersurface of Pn+1. Since Conjecture (0.2) has not been proved yet, it is not known what kind of genericity is needed for the statement of Conjecture (0.2). If it turns out that the genericity requirement for Conjecture (0.2) is satisfied δ when the defining equation is zn+1 = f(z0, ··· , zn) with f(z0, ··· , zn) generic, then Conjecture (0.2) implies Conjecture (0.1). When the number of components of H is at least three, there are results on Conjecture (0.1) for n = 2 very close to the conjecture itself with explicit description of the condition of genericity (Detloff-Schumacher-Wong [DSW93] and [DSW94]). The proof of Conjecture (0.1) becomes more difficult when the number of components of H becomes smaller. The most difficult case is when H is irreducible. The only result so far for an irreducible H is the result of [SY94] for n = 2 which requires the degree of H to be much higher than the conjectured 5. For general n and an irreducible hypersurface H in Pn, there are examples constructed in [AdS90], [AzS80], [BGr77], [MN94], [Na89], and [Z89] with H hyperbolic. The hypersurfaces H in those examples, of course, are not generic. They are defined by a homogeneous polynmomial of high degree but with a very few number of nonzero terms. These constructions, because of the relation between Conjectures (0.1) and (0.2), also give examples with Pn − H hyperbolic. Bloch’s theorem is the analog of Conjecture (0.2) with the complex projective space replaced by an abelian variety. An abelian variety is a compact quotient of Cn by a lattice in Cn which can be embedded as a complex submanifold

3 of a complex projective space. There are different proofs of Bloch’s theorem. The analog of Conjecture (0.1) for an abelian variety is still open and a precise statement for it is given in Conjecture (0.4). Bloch’s Theorem (0.3). Let A be an abelian variety and X be a subvariety of A which is not a translate of an abelian subvariety of A. Then there exists no nonconstant holomorphic map from C to X whose image in X is Zariski dense in X. Bloch’s original statement is that if X is a compact complex manifold of complex dimension n on which there are n+1 linearly independent holomorphic 1-forms, then the image of any holomorphic map from C to X must be contained in a proper subvariety of X. There are now a number of different proofs of Bloch’s theorem. We will present the two main methods of proof in Chapter 7 of this article. One is based on the original proof of Bloch [Bl26] with some additional arguments taking care of a number of unsatisfactory details in the original proof. Conjecture (0.4). Let A be an abelian variety and X be an ample divisor in A. Then A − X is hyperbolic. Here an ample divisor means that the holomorphic line bundle associated to the divisor has a Hermitian metric whose curvature form is positive definite. From the very beginning of the theory of hyperbolicity and the value distribution theory of Nevanlinna some of the powerful basic tools are the negative curvature argument, the logarithmic derivative lemma, and the Wronskian. In the first three Chapters of this article we will present such basic techniques and their numerous variations. Then in later Chapters we will discuss more recent results on the problem of hyperbolicity of the complements of hypersurfaces in Pn (Chapters 4, 5, and 6) and Bloch’s theorem (Chapter 7). The convention of summing over two repeated indices with each in subscript and superscript position without using the summation sign is used in some places in the article and its use will be clear from the context. I would like to thank Sai-Kee Yeung who read a draft of this article and made useful comments on it. I would also like to thank Pit-Mann Wong for a number of conversations about his work on hyperbolicity.

Chapter 1. Negative Curvature in Dimension One 1. Nonexistence of Metric on C of Curvature with Uniform Negative Upper Bound The method that is usually used to prove the nonexistence of a nonconstant holomorphic map from C to a complex manifold depends on the nonexistence of Hermitian metric on C of curvature with uniform negative upper bound.

4 Lemma (1.1.1) (On the Nonexistence of Hermitian Metric on C of Curvature with Uniform Negative Upper Bound). Let g be a real-valued nonnegative function on C which locally is of the form h|f|2, where h is a local smooth positive function and f is a local holomorphic function. Then there cannot exist any positive number c such that ∂z∂z log g ≥ cg on C in the sense of distribution. There are a number of ways to prove Lemma (1.1.1). One way is to apply Green’s Theorem, perform one integration, and then use a simple Calculus R r dρ R Lemma. For a function or a (1,1)-form η let Ir(η) = η. For a R ρ=0 ρ |z|<ρ 1 2π iθ function g let Ar(g) = 2π θ=0 g(re )dθ. The fundamental theorem of calculus gives 1 d d (1.1.1.2) (r I (g)) = 2πA (g). r dr dr r r Lemma (1.1.2) (Green’s Theorem). If g is a function on {|z| ≤ r} which is locally the product of a smooth function and |f|2 for some local holomorphic 1 1 1 function f, then Ir( π ∂z∂zg) = 2 Ar(g) − 2 g(0), where ∂z∂zg is interpreted as a distribution. Proof. The divergence theorem gives Z Z 2π ∂ ∆g = g(reiθ)rdθ. |z| 1. Applying it with f = S and then with f = rS0, we get (rS0)0 ≤ rµ−1(rS0)ν ≤ rµ−1(r(rµ−1Sν ))ν . Q.E.D.

5 When there is no risk of confusion, we drop ², δ or just δ from the notation k²,δ and simply write k or k² instead. The following corollary is a consequence of (1.1.1.2), and Lemma(1.1.3) applied with S(r) = Ir(g).

Corollary (1.1.4). log Ar(g) ≤ ² log r + (1 + δ) log Ir(g) k²,δ for any smooth nonnegative function g on C whose zero-set is discrete in C. Corollary (1.1.5). Let g be a continuous subharmonic function on C which is smooth and strictly subharmonic outside a discrete set of C. Then 2 log A ( ∂ ∂ g) ≤ ² log r + (1 + δ) log(A (g) − g(0)) k . r π z z r ²,δ

2 2 Proof. By Corollary (1.1.4) log Ar( π ∂z∂zg) ≤ ² log r+(1+δ) log Ir( π ∂z∂zg) k²,δ which by Green’s Theorem (Lemma(1.1.2)) is equal to ² log r+(1+δ) log(Ar(g)− g(0)). Q.E.D. Proof of Lemma (1.1.1). By using another coordinate system of C we can assume without loss of generality that near the origin g is a positive-valued smooth function. By Green’s Theorem (Lemma (1.1.2)), the concavity of the logarithmic function, and the Calculus Lemma (Lemma (1.1.3)) c 1 1 I (g) ≤ A (log g) − log g(0) π r 2 r 2 1 1 ≤ log A (g) − log g(0) 2 r 2 1 1 d d 1 = log (r I (g)) − log g(0) 2 2πr dr dr r 2 1 1 1 ≤ (² log r + (1 + δ) log I (g)) − log 2π − log g(0) k . 2 r 2 2 ²,δ Hence

(1.1.4.1) Ir(g) ≤ ² log r + O(1) k² for any positive number ², where O(1) means the Landau symbol denoting a bounded term.R From ∂z∂z log g ≥ cg we know that log g is subharmonic on C. 1 2π iθ Let G(z) = 2π θ=0 g(e z)dθ. By the sub mean value property of subharmonic functions, G(z) ≥ g(0). From the deﬁnition of Ir(g) it follows that π I (g) = I (G) ≥ g(0)r2 r r 2 which contradicts (1.1.4.1). Q.E.D. 2. The Conformal Factor From the Poincar´eMetric of the Punctured Disk

6 The theory of hyperbolicity and more generally the value distribution theory of Nevanlinna is based on a conformal factor used in the Hermitian metric of P1 or a general compact complex manifold. In this section we would like to explain the motivation for this conformal factor.

Though P1 with the usual Fubini-Study metric is positively curved, one can use a conformal change of the metric (i.e. a change of the metric by a scalar factor) to get a metric with pole singularities so that the curvature of the metric outside the pole singularities is bounded from above by a negative number. If there is a holomorphic map f from C to P1 minus the pole singularities, then the pullback to C of the metric would give a contradiction. If f is not assumed to miss the pole singularities, then we can modify the argument to show that f must actually assume the pole singularities frequently enough. The use of diﬀerent pole singularites tells us that the values of f are distributed in the target space P1 in a rather uniform manner.

Certain branched coverings X of P1 are hyperbolic compact Riemann surfaces and therefore have smooth Hermitian metrics with constant negative curvature. We can push forward to P1 locally the sum of the metric of X on all the sheets and get a metric on P1 of curvature with uniform negative upper bound outside a set of pole singularities which is the set of branching points in P1. A metric on X of curvature with uniform negative upper bound can be constructed by taking the sum of the absolute value square of a basis of holomorphic 1-forms on X. The reason why there are nontrivial holomorphic 1-forms on X while there is none on P1 is because of the the presence of enough branching points. Locally at one branching point of order n where the local coordinate z is n zero the branching is given by z → w = z . Locally the form dw on P1 is pulled back to nzn−1dz which gives a zero at z = 0 and we can use the zeroes so created from the pullback to X of a meromorphic 1-form on P1 to get a holomorphic 1-form on X if we have enough branching points. Or equivalently, instead of n−1 dw pulling back dw to get nz dz, we can consider the pullback of w(n−1)/n to get ndz. The larger the branching order n is, the easier it is to construct holomorphic 1-forms and get a metric on the branched cover of curvature with uniform negative upper bound. As n → ∞ the limiting case of the branched covering z → w = zn is the exponential map which is a map with infinitely many sheets. We can interpret the limiting situation in two ways. The first way is to change the branching point n z n of z → w = z from the origin to −n to consider the branching z → w = (1+ n ) whose limit as n → ∞ is w = ez. Another way is to consider the pullback of dw n dw w(n−1)/n in the case of the branching z → w = z . The limiting case of w(n−1)/n dw z is w which corresponds to the map z → w = e . Thus to get the maximum benefit of branching, one should consider the case of branching of order n with n → ∞. In other words, the conformal change in the metric of P1 should be obtained by locally pushing forward a metric of constant negative curvature

7 by the map z → w = ez. The Poincarémetric of constant negative curvature √ dzdz¯ on the left half plane with coordinate z = x + −1y is x2 which under the z dwdw¯ map z → w = e is pushed forward to |w|2(log |w|)2 . This motivates the use of 1 the conformal factor |w|2(log |w|)2 for the pole singularity at w = 0. Note that, since the map z → w = ez makes the left half-plane the universal cover of the dwdw¯ punctured unit disk, the metric |w|2(log |w|)2 is the metric on the puncture unit disk induced by the Poincarémetric of its universal covering. 3. The First Main Theorem and the Defect Relation of Nevanlinna Let M be a compact complex manifold and ω be a smooth positive (1,1)-form on M. Let f : C → M be a holomorphic map. We define the characteristic ∗ function T (r, f, ω) as Ir(f ω). When there is no possibility of confusion, we simply write T (r, f, ω) as T (r, f) or as Tf (r), especially in the case when M is Pn. Let S be a complex hypersurface (or more generally a divisor) in M. We ∗ ∗ define the counting function N(r, f, S) as Ir(f S) with f S interpreted as a (1,1)-current on C. Let L be a holomorphic line bundle over M with a Hermitian metric whose curvature form θL is positive definite. Let s be a holomorphic section of L over M. Let |s| denote the pointwise norm of s and Zs denote the zero-set (or more precisely the divisor) of s. Define the proximity function m(r, f, Zs) as Ar(log |s|). Since |s| is bounded on the compact manifold M, the proximity function Ar(log |s|) is always bounded from above as r → ∞. Assume that f(0) does not belong to the zero-set of s. By applying Green’s √ −1 ¯ 2 Theorem to 2π ∂∂ log |s| = −θL + Zs, we get

Ir(−θL + Zs) = Ar(log |s|) − log |s|(0).

The equivalent form

T (r, f, θL) = m(r, f, Zs) + N(r, f, Zs) + O(1) is known as the First Main Theorem of value distribution theory. We always have N(r, f, Zs) ≤ T (r, f, θL) + O(1). The defect δ(f, Zs) is deﬁned as

m(r, f, Zs) N(r, f, Zs) lim infr→∞ = lim infr→∞(1 − ). T (r, f, θL) T (r, f, θL)

When the image of f is disjoint from Zs, the defect δ(f, Zs) is 1.

Geometrically the characteristic function T (r, f, θL) measures the “growth” behavior of f(z) as z → ∞. The counting function N(r, f, Zs) measures, in a weighted manner, the number of times f hits the hypersurface Zs

8 in the disk of radius r centered at 0. The proximity function N(r, f, Zs) measures, in an appropriate sense, the number of times when the hypersurface Zs is a “near miss” for the map f on the boundary of the disk. The First Main Theorem of Nevanlinna tells us that, in an appropriate sense, the number of times a hypersurface is hit by a map is equal to the order of growth of the map if we compensate it with the counting of all the “near misses”. We now consider the special case when the dimension of the compact complex manifold M is 1. By rescaling the Hermitian metric of L by a small positive factor, we assume that |s| < 1 on all of M. We are going to use the idea 1 of the conformal factor |w|2(log |w|)2 discussed above which in our case will be 1 |s|2(log |s|)2 . We have 1 2 ∂¯log = − ∂¯log |s|2 (log |s|2)2 log |s|2 and √ √ −1 1 2 −1 (1.3.1.1) ∂∂¯log = − ∂∂¯log |s|2 2π (log |s|2)2 log |s|2 2π

2∂ log |s|2 ∧ ∂ log |s|2 + (log |s|2)2 2 2Ds ∧ Ds = (θ − Zs) + log |s|2 L |s|2(log |s|2)2 where D is the covariant diﬀerential operator so that locally Ds = ∂s + As for some smooth (1,0)-form A. Hence √ −1 1 2 2Ds ∧ Ds (1.3.1.2) ∂∂¯log = (1 + )(θ − Zs) + . 2π |s|2(log |s|2)2 log |s|2 L |s|2(log |s|2)2

2 2 1 2 2 Locally we have |Ds| = |∂s + As| ≥ 2 |∂s| − |As| . Assume that the zero-set of s is nonsingular and without multiplicity. Let ω be a volume form of M. Then 2 2 |Ds| + c1|s| ω ≥ c2ω for some positive constants c1 and c2. So we have the lower bound

Ds ∧ Ds c ω c ω (1.3.1.3) ≥ 2 − 1 . |s|2(log |s|2)2 |s|2(log |s|2)2 (log |s|2)2

When the metric of L is rescaled by a constant so that |s| becomes λ|s|, the covariant diﬀerentiation is not aﬀected by the rescaling and c1 is unchanged, but c2 2 2 2 2 c2 is changed to λ2 . With this rescaling for λ < 1 we have (log |s| ) ≥ (log λ )

9 c1ω c1ω ² on all of M and (log |s|2)2 ≤ (log λ2)2 which can be made less than 2 θL for any prescribed ² > 0 when 0 < λ < 1 is chosen small enough. We can assume that 2 ² 0 < λ < 1 is chosen small enough so that 0 > log |s|2 > − 2 . Putting everything together, we have √ −1 1 c ω ∂∂¯log ≥ (1 − ²)θ − Zs + 2 . 2π |s|2(log |s|2)2 L |s|2(log |s|2)2 √ −1 ¯ Let θKM = 2π ∂∂ log ω. Then θKM is the curvature form of the metric of the canonical line bundle KM of M deﬁned by the volume form ω. Thus √ −1 ω c ω ∂∂¯log ≥ (1 − ²)θ + θ − Zs + 2 . 2π |s|2(log |s|2)2 L KM |s|2(log |s|2)2 Pulling back via f and applying Green’s Theorem, we obtain 1 ω (1.3.1.4) A (log f ∗( )) ≥ I (f ∗((1 − ²)θ 2 r |s|2(log |s|2)2 r L

c ω +θ − Zs + 2 )). KM |s|2(log |s|2)2 By the concavity of the logarithmic function we have ω ω (1.3.1.5) A (log f ∗( )) ≤ log A (f ∗( )). r |s|2(log |s|2)2 r |s|2(log |s|2)2

By the Corollary to the Calculus Lemma (Corollary(1.1.4)) ω (1.3.1.6) log A (f ∗( )) ≤ ² log r r |s|2(log |s|2)2

ω +(1 + δ) log I (f ∗( )) k . r |s|2(log |s|2)2 ²,δ

From (1.3.1.4), (1.3.1.5), and (1.3.1.6) it follows that ω (1.3.1.7) ² log r + (1 + δ) log I (f ∗( )) r |s|2(log |s|2)2

c ω ≥ I (f ∗((1 − ²)θ + θ − Zs + 2 )) k , r L KM |s|2(log |s|2)2 ²,δ from which we conclude that

−T (r, f, θKM ) + ²T (r, f, θL) + ² log r δ(f, Zs) ≤ lim infr→∞ . T (r, f, θL)

10 If αθL + θKM ≥ 0 on M, since ² is an arbitrary positive number, it follows that δ(f, Zs) ≤ α. The inequality (1.3.1.7) is a weak form of the Second Main Theorem of value distribution theory. The inequality δ(f, Zs) ≤ α is known as the defect relation.

We now consider the case of M = P1. Let a1, ··· , aq be a number of points ⊗q in P1. Let H be the hyperplane section line bundleP of P1 and L be H and s be the holomorphic section of L whose divisor is q a . Then 2 θ + θ ≥ 0 j=1 j q L KP1 Pq on P1. From m(r, f, Zs) = j=1 m(r, f, aj) and T (r, f, θL) = qT (r, f, θH ) it follows that q X 2 δ(r, f, a ) = qδ(r, f, Zs) ≤ q = 2. j q j=1 This is the classical defect relation of Nevanlinna. The arguments for the defect relation work also for the case of a holomorphic map from Cn to a compact complex manifold M of complex dimension n. The modiﬁcation needed is that one uses the inequality for the arithmetic and geometric means in the step corresponding to (1.3.1.3). The defect for the zero-set of a holomorphic section of L again is bounded by any real number α satisfying

αθL + θKM ≥ 0 when the zero-set consists of nonsingular hypersurfaces in normal crossing. This value distribution theory for the higher equidimensional case (where both the domain and the target manifolds have the same dimension) was developed by Stoll [St53], Griffiths and others ([CG72], [GK73]). However, when the dimension of the target manifold is greater than that of the domain manifold, the value distribution theory becomes very difficult, except in the case of holomorphic maps from C to Pn when the defect is considered only for hyperplanes in Pn. Insurmountable difficulties occur already in the case of a holomorphic map from C to P2 when the defect is for a geneal nonsingular complex curve (see §1 of Chapter 3). 4. Interpretation by Forms with Logarithmic Poles The key point in the derivation of (1.3.1.7) is the use of the conformal factor 1 |s|2(log |s|2)2 . Instead of interpreting it as motivated by the Poincarémetric on the punctured disk, we can also interpret it in terms of forms with logarithmic poles. In the case of holomorphic maps from C to P1 this alternative interpretation is no more than another way of looking at the same formula. However, when one tries to transplant the arguments to the case of holomorphic maps from C to higher dimensional manifolds, different interpretations give rise to completely different arguments and formulas.

ω dwdw Locally the expression |s|2(log |s|2)2 in (1.3.1.7) looks like |w−a|2(log |w−a|2)2 dw 2 1 near a point a in {a1, ··· , aq}. We can rewrite it as | w−a | (log |w−a|2)2 . The form dw w−a has a pole at a and, since its indeﬁnite integral log(w − a) has logarithmic singularity at a, such a pole is known also as a logarithmic pole. In the dimension

11 one case there is no difference between a logarithmic pole and a simple pole. However, in the higher dimensional case there is a difference. For example, in dw1 the case of complex dimension two with variables w1 and w2, the form has w2 a simple pole along w2 = 0 but it does not have a logarithmic pole like the form dw1 which is equal to the differential of log w1. w1 dw Consider the form ϕ = q on P1 with logarithmic poles at the finite Πj=1(w−aj ) points a1, ··· , aq. It has a zero of order q − 2 at the infinity point of P1. Let ⊗(q−2) t be the holomorphic section of H over√P1 with a zero of order q − 2 −1 ¯ ω at the infinity point. Then instead of using 2π ∂∂ log |s|2(log |s|2)2 to derive √ −1 ¯ |ϕ|2 (1.3.1.7), we can get the same conclusion by using 2π ∂∂ log |t|2(log |s|2)2 . In both interpretions the number 2 in the resulting defect relation is the Chern class of the tangent bundle of P1 which is interpreted either in terms of the curvature computed from the volume form ω or as the degree of the divisor of a meromorphic 1-form on P1.

Chapter 2. Method of Logarithmic Derivative Lemma and Wron- skians 1. Alternative Description of the Characteristic Function To derive the Second Main Theorem, besides the negative curvature argument, there is another argument involving the logarithmic derivative lemma. For the argument using the logarithmic derivative lemma and also for using arguments involving the Wronskian, another description of the characteristic function is more convenient and more natural in that setting. We now give alternative descriptions of the characteristic function.

For a holomorphic map f from C to Pn we can use the homogeneous coordinates of Pn and represent f by in the form [f0, ··· , fn] by n + 1 holomorphic functions f (0 ≤ j ≤ n) without common zeroes on C. Let ω be the Fubini- j √ P ∗ −1 ¯ n 2 Study form on Pn whose pullback f ω by f is equal to 2π ∂∂ log( j=0 |fj| ). From Green’s Theorem (Lemma (1.1.2)) we have

1 Xn I (f ∗ω) = A (log( |f |2)) + O(1). r 2 r j j=0

Since Xn 1 2 1 2 max log |fj| ≤ log( |fj| ) ≤ log((n + 1) max log |fj| ) 0≤j≤n 2 2 0≤j≤n j=0 1 ≤ max log |fj| + log(n + 1), 0≤j≤n 2

12 it follows that

∗ (2.1.0.1) Ir(f ω) = Ar( max log |fj|) + O(1). 0≤j≤n

∗ So up to a bounded term the characteristic function T (r, f) = Ir(f ω) can be described by Ar(max0≤j≤n log |fj|). Note that the important point about the alternative description of the characteristic function in (2.1.0.1) is the taking of the maximum. The main task of value distribution theory is to bound the characteristic function in terms of the counting function, i.e., bounding Ar(max0≤j≤n log |fj|) in terms of the counting function. If no maximum is taken in Ar(max0≤j≤n log |fj|), i.e., in the case of only one single meromorphic function g instead of a set of holomorphic functions f0, ··· , fn without common zeroes, we can easily bound A (log |g|) by the counting function for the zeroes r √ −1 ¯ 2 of g by simply using Green’s Theorem (Lemma (1.1.2)) for 2π ∂∂ log |g| and get

(2.1.0.2) Ir(Zg) − Ir(P g) = Ar(log |g|) − log |g|(0), where Zg and P g are respectively the zero divisor and pole divisor of g. This is the Poisson-Jensen formula at the origin. So we have the desired bound

Ar(log |g|) ≤ N(r, g, 0) + O(1).

As we will see in §3 of Chapter 2, one way of handling the difficulty from the taking of maximum in Ar(max0≤j≤n log |fj|) is the use of the logarithmic derivative lemma for meromorphic functions (introduced in §2 of Chapter 2). In order to later use the logarithmic derivative lemma for meromorphic functions, we need the relation between the characteristic function of a holomorphic map from C to Pn and the characteristic functions of the meromorphic functions which are the quotients of the components of the map. For a meromorphic function g on C we define the characterisitic function + T (r, g) of g to be Ar(log |g|) + N(r, g, ∞), where N(r, g, ∞) means Ir(P g) with P g being the pole divisor of g. We define T (r, g) only when the origin is not a pole of g. According to Corollary (2.1.3) below the function T (r, g) is independent of the choice of the origin up to a bounded term as long as the origin is not a pole of g. 1 Lemma (2.1.1). For a meromorphic function g on C, T (r, g ) = T (r, g)+O(1). + + 1 Proof. From log |g| = log |g| − log | g | it follows that 1 A (log+ |g|) − A (log+ | |) = A (log |g|) r r g r

13 √ −1 = I ( ∂∂¯log |g|) − log |g|(0) r 2π = N(r, g, 0) − N(r, g, ∞) − log |g|(0)

1 and T (r, g) = T (r, g ) − log |g|(0). Q.E.D.

Lemma (2.1.2). Let Fj (0 ≤ j ≤ n) be entire functions on C without common zeroes. Let f : C → P be a holomorphic map deﬁned by [F ,F , ···,F ]. Then n P 0 1 n T (r, Fj ) ≤ T (r, f) + O(1) for 1 ≤ j ≤ n and T (r, f) ≤ n T (r, Fj ) + O(1). F0 j=1 F0 Proof. By (2.1.0.1) we know that

T (r, f) = Ar( max log |Fj|) + O(1) 0≤j≤n µ ¶ F1 Fn = Ar(log |F0|) + Ar max(1, log | |, ··· , log | |) + O(1) F0 F0 √ −1 ¯ 2 + Fj = Ir( ∂∂ log |F0| ) − log |F0|(0) + Ar( max log | |) + O(1) 2π 1≤j≤n F0

+ Fj = N(r, F0, 0) − log |F0|(0) + Ar( max log | |) + O(1). 1≤j≤n F0

Since the entire functions F0,F1, ···,Fn have no common zeroes, it follows that

Xn F N(r, F , 0) ≤ N(r, j , ∞) 0 F j=1 0 and Xn F F T (r, f) ≤ (N(r, j , 0) + A (log+ | j |)) + O(1) F r F j=1 0 0 Xn F = T (r, j ) + O(1). F j=1 0

Fix 1 ≤ j ≤ n. Then

+ Fj = Ar(log | |) + Ar(log |F0|) − log |F0|(0) F0

= Ar (max(log |Fj|, log |F0|)) − log |F0|(0)

14 µ ¶

≤ Ar max (log |Fj|) − log |F0|(0) 0≤j≤n = T (r, f) + O(1). Q.E.D.

Corollary (2.1.3). Let F0 and F1 be entire functions on C without common zeroes. Let f : C → Pn be a holomorphic map deﬁned by [F0,F1]. Then T (r, f) = T (r, F1 ) + O(1). In particular, for a meromorphic function g on C F0 up to a bounded term the function T (r, g) is independent of the choice of the origin when the origin is not a pole of g. Later for one proof of Bloch’s theorem (§2 of Chapter 7) we will need the following estimate of the characteristic function due to Valiron [Va31].

Lemma (2.1.4). Let ϕ and aj (0 ≤ j ≤ m) be meromorphic functions on C Pm ν Pm such that ν=0 aν ϕ ≡ 0 on C. Then T (r, ϕ) ≤ ν=0 T (r, aν ) + O(1).

Proof. Without loss of generality we can assume that am and ϕ are not identically zero. We have Ã¯ ¯! ¯mX−1 ¯ ¯ aν ¯ m log+ |ϕ| = log+(|ϕ|m) = log+ ¯ ϕν ¯ ¯ a ¯ ν=0 m µ ¯ ¯¶ ¯ a ¯ mX−1 ≤ log+ max ¯ ν ¯ + log+( |ϕ|ν ) 0≤ν≤m−1 ¯a ¯ m ν=0 µ ¯ ¯¶ ¯ ¯ + ¯ aν ¯ + m−1 ≤ log max ¯ ¯ + log (m max(1, |ϕ| )) 0≤ν≤m−1 am ¯ ¯ ¯ 1 ¯ mX−1 ≤ log+ ¯ ¯ + log+ |a | + (m − 1) log+ |ϕ| + log m. ¯a ¯ ν m ν=0 It follows that ¯ ¯ ¯ 1 ¯ mX−1 log+ |ϕ| ≤ log+ ¯ ¯ + log+ |a | + O(1). ¯a ¯ ν m ν=0 We have to handle the counting function for the pole divisor of ϕ. Let b be an entire function on C whose zero divisor is the maximum of the pole divisors of a0, ··· , am−1 so that ba0, ··· , bam−1 are entire functions. Clearly we have Pm−1 Pm−1 −m+ν N(r, b, 0) ≤ ν=0 N(r, aν , ∞). It follows from bam = − ν=0 baν ϕ that N(r, ϕ, ∞) ≤ N(r, bam, 0). Thus

mX−1 N(r, ϕ, ∞) ≤ N(r, am, 0) + N(r, b, 0) ≤ N(r, am, 0) + N(r, aν , ∞) ν=0

15 and 1 mX−1 T (r, ϕ) ≤ T (r, ) + T (r, a ) + O(1) a ν m ν=0 Xm = T (r, aν ) + O(1). ν=0 Q.E.D. Later in Chapter 4 we will need to represent maps with ﬁnite-order characteristic functions by entire functions of ﬁnite order. We now present the result on representability here. For any nonnegative integer q let E(z, q) be the Weierstrass factor (1 − Pq zν P∞ z) exp( ν=1 ν ). Let Z = ν=1 aν be a nonnegative divisor in C whose sup- port does not contain the origin, i.e., no aν is 0. Denote by WZ,q(z) the canonical product Π∞ E( z , q). ν=1 aν q+² Lemma (2.1.5). Let q be a nonnegative integer. If Ir(Z) = O(r ) for any q+² ² ≥ 0, then T (r, WZ,q) = O(r ) for any ² > 0.

Pq ξν ξq+1 −(q+1) Proof. Since the quotient of log(1+ξ)+ ν=1 ν by 1+ξ is equal to ξ (1+ Pq ξν 1 ξ)(log(1 + ξ) + ν=1 ν ) has limit q as ξ → ∞, it follows that there exists a q+1 constant C depending only on q such that log E(z, q) ≤ C |z| . Let n(t) be R q q 1+|z| Z when Z is regarded as a (1,1)-current, i.e., n(t) is the number of aν |z| 0, it follows that n(t) = O(tq−²) for any ² > 0. For some constants C and C0 we have X∞ 1 sup log |W (z)| ≤ C rq+1 Z,q q |a |q(r + |a |) |z|=r ν=1 ν ν Z ∞ Z ∞ q+1 dn(t) q+1 n(t)dt = Cqr q ≤ Cq(q + 1)r q+1 t=0 t (t + r) t=0 t (t + r) Z r Z ∞ q n(t) q+1 n(t) ≤ Cq(q + 1)(r q+1 dt + r q+2 dt) t=0 t t=r t Z r Z ∞ q+² q It(Z) q+1 t ≤ Cq(q + 1)(Ir(Z) + (q + 1)r q+2 dt + Cr q+2 dt) t=0 t t=r t Z ∞ q 0 1 q+² q+² ≤ Cq(q + 1)(Ir(Z) + (q + 1)r C 2−² dt + C(1 − ²)r ) = O(r ), t=0 t n(t) where in one of the steps integration by parts is done by using dIt(Z) = t dt. + Since WZ,q(z) is an entire function, it follows that T (r, WZ,q) = O(sup|z|=r log |WZ,q(z)|) = O(rq+²) for any ² > 0. Q.E.D.

16 For a nonnegative number p an entire function f on C is said to be of order at most p if for any positive number ² there exists a constant C² such that p+² |f(z)| ≤ exp(|z| ) + C² on C.

Lemma (2.1.6). Suppose f : C → Pn is a holomorphic map and q is a nonnegative integer such that T (r, f) = O(rq+²) for any ² > 0, then f can be represented by entire functions F0, ···,Fn of order at most q.

Proof. Let G0, ···,Gn be any entire functions on C without common zeroes such that [G0, ···,Gn] defines f. For any 1 ≤ j ≤ n it follows from Corollary (2.1.3) Gj Gj q+² that N(r, , ∞) ≤ T (r, ) ≤ T (r, f) = O(r ) for any ² > 0. Let Pj be the G0 G0 Gj pole divisor of . Let Z be the zero-set of G0. Since G0, ···,Gn have no common G0 Pn q+² zeroes, it follows that Z ≤ j=1 Pj as divisors. Hence Ir(Z) = O(r ) for any q+² ² ≥ 0 and by Lemma(2.1.5) we have T (r, WZ,q) = O(r ) for any ² > 0. The Gj requirement of the Lemma is satisfied with F0 = WZ,q and Fj = WZ,q for G0 1 ≤ j ≤ n. Q.E.D. 2. The Logarithmic Derivative Lemma and the Wronskian The argument of the logarithmic derivative lemma is parallel to the negative curvature argument used for the weak form of the Second Main Theorem and the defect relation, but is formulated in terms of a single meromorphic function on C. Lemma (2.2.1) (The Logarithmic Derivative Lemma). Let f be a mero- + f 0 + morphic function on C. Then Ar(log | f |) ≤ O(log T (r, f) + log r) k. Proof. Recall that for the negative curvature argument we use the conformal 1 1 factor log |s|2(log |s|2)2 . The main contribution comes from log (log |s|2)2 . We would like to use a similar computation. However, in the case at hand, we 1 cannot consider log (log |f|2)2 , because now f is a meromorphic function and there is no analog of the rescaling of the Hermitian metric of the line bundle to 1 make |s| < 1. So the expression log (log |f|2)2 is meaningless. One possibility is to 1 1 d d use the modified expression u = log 1+(log |f|2)2 . However, for r dr (r dr Ar(u)) ≤ ² 1+δ r Ar(u) in the chain of inequalities given above, among others things we need the nonnegativity of u. In order to have a computation similiar to that used for the negative curvature argument, we consider u = (1 + (log |f|2)2)1/2. We have

2 2 1/2 2 2 −3/2 2 ∂z∂z(1 + (log |f| ) ) = (1 + (log |f| ) ) |∂z log f| 2 2 −1/2 2 2 +(1 + (log |f| ) ) (log |f| )∂z∂z log |f| . (The argument works also when we use u = (1 + (log |f|2)2)α for some other α 1 positive not equal to 2 , but the computations are more involved.) We are going to apply Corollary (1.1.5) to u. However, u is not subharmonic, because it is unbounded at a zero or at a pole of f. So some small technical modiﬁcation is

17 needed for the application of Corollary (1.1.5). Since the technical modification does not in essence affect the rest of the proof, we do not carry out the technical modification here and, strictly speaking, a complete rigorous proof is slightly more complicated than what follows. By Corollary (1.1.5) we have

2 f 0 log A ( (1 + (log |f|2)2)−3/2| |2) ≤ ² log r r π f 2 2 1/2 +(1 + δ) log Ar((1 + (log |f| ) ) ) k²,δ.

By the concavity of the logarithmic function the left-hand side dominates

+ f 0 + which gives Ar(log | f |) ≤ O(log T (r, f) + log r) k. Q.E.D. Remark (2.2.2). A consequence of the logarithmic derivative lemma is that f 0 + if f is nowhere zero holomorphic, T (r, f ) = O(log T (r, f) + log r), because f 0 f 0 f 0 in that case f is holomorphic and T (r, f ) = Ar( f ) + O(1). In general, if f 0 + the meromorphic function f has zeroes or poles, T (r, f ) = O(log T (r, f) + f 0 f 0 f 0 log r) does not hold, because N(r, f , ∞) ≤ T (r, f ) + O(1) and N(r, f , ∞) = N¯(r, f, 0) + N¯(r, f, ∞) and each of N¯(r, f, 0) and N¯(r, f, ∞) can be of the order T (r, f). Here N¯(r, f, 0) and N¯(r, f, ∞) mean respectively the counting function of f for the points 0 and ∞ with multiple zeroes or poles counted only with multiplicity one.

The Wronskian W (f1, ··· , fk) of meromorphic functions f1, ··· , fk on a domain Ω in C is defined as the determinant of the k × k matrix (ν−1) (ν) th (fµ )1≤µ,ν≤k, where fµ denotes the ν derivative of fµ. The main property of the Wronskian is that W (f1, ··· , fk) ≡ 0 on Ω if and only if there exist Pk cj ∈ C (1 ≤ j ≤ k) not all zero such that j=1 cjfj ≡ 0 on Ω. The “if” part is clear. For the “only if” part, let 1 ≤ ` ≤ k be the smallest integer such that W (f1, ··· , f`) ≡ 0. Without loss of generality we can assume that ` = k. We can Pk (µ) find mermorphic functions cj on Ω such that j=1 cjfj ≡ 0 (0 ≤ µ ≤ k − 1) Pk 0 (µ) on Ω with ck ≡ 1. By differentiating the equation once, we get j=1 cjfj ≡ 0 0 0 (0 ≤ µ ≤ k − 2) on Ω, where cj is the derivative of cj. Since ck ≡ 0 and the

18 (µ−1) 0 determinant of (fj )1≤j,µ≤k−1 is not identically zero, it follows that cj ≡ 0 for 1 ≤ j ≤ k − 1. Using the Wronskian, we can formulate the logarithmic derivative lemma for maps from C to Pn. Lemma (2.2.3) (The Logarithmic Derivative Lemma for Maps into Pn). Let f0, ··· , fn be entire functions on C without common zeroes. Let f : C → Pn be deﬁned by f0, ··· , fn. Then ¯ ¯ ¡ + −1 ¢ + Ar log ¯(f0 ··· fn) W (f0, ··· , fn)¯ = O(log T (r, f) + log r).

−1 Proof.(f0 ··· fn) W (f0, ··· , fn) is equal to the determinant of   1 1 ··· 1 f 0 f 0 f 0  0 1 ··· n   f0 f1 fn   ······     ······     ······  f (n−1) f (n−1) f (n−1) 0 1 ··· n f0 f1 fn which is equal to the determinant of

 0 0 0 0  f1 f0 fn f0 f − f ··· f − f  1 0 n 0   ·····     ·····  .  ·····  f (n−1) f (n−1) f (n−1) f 0 1 − 0 ··· n − 0 f1 f0 fn f0

Let Fj = fj/f0 for 1 ≤ j ≤ n. We claim that in the computation of the determinant we can start from the ﬁrst row and replace successively the νth (ν) (ν) row by the row vector F1 /F1, ··· ,Fn /Fn. Suppose this has been veriﬁed up to and including the (ν − 1)st row. Then we use µ ¶ (ν) (ν) (ν) (ν) νX−1 (λ) (ν−λ) fj f0 (Fjf0) f0 (ν) ν Fj f0 − = − = Fj /Fj + fj f0 Fjf0 f0 λ Fj f0 λ=1 and get the νth by subtracting from it a linear combination of the preceding rows ¡ ¢ f (ν−λ) with the λth row multiplied by ν 0 . Now the desired conclusion follows µ ¶ λ f0 (ν) (ν−1) 0 (ν−1) F F F F 0 from j = j + j j , Lemma (2.1.2), and repeated application of Fj Fj Fj Fj the Logarithmic Derivative Lemma (Lemma (2.2.1)). Q.E.D. 3. Cartan’s Second Main Theorem With Truncated Counting Func- tion

19 Let g = (g1, ··· , gp): C → Pp−1 be a holomorphic map whose image is not contained in a hyperplane so that the Wronskian

µ k ¶ d g` W (g1, ··· , gp) = det k dz 0≤k,`−1

|Fα1 (x)| ≥ |Fα2 (x)| ≥ · · · ≥ |Fαq (x)|.

We pick the bottom p − 1 values Fαq−p+2 (x), Fαq−p+3 (x), ···, Fαq (x) and then any Fαj (x) from the top q − p + 1 values with 1 ≤ j ≤ q − p + 1 and we solve g1(x), ··· , gp(x) in terms of Fαj (x),Fαq−p+2 (x), Fαq−p+3 (x), ···, Fαq (x). After we solve the equations by Cramer’s rule we replace |Fαq−p+2 (x)|, |Fαq−p+3 (x)|,

···,|Fαq (x)| by |Fαj (x)| and get an inequality

(2.3.2.1) |gi(x)| ≤ K|Fαj (x)| for 1 ≤ i ≤ p and 1 ≤ j ≤ q − p + 1 for some constant K. Since the permutation (α1, ··· , αq) of (1, ··· , q) changes with the point x, we would like to take the maximum so that we can get a statement independent of the permutation. Let v(x) be the maximum of log |Fβ1 (x) ··· Fβq−p (x)| as {β1, ··· , βq−p} ranges over the collection of all subsets of q − p elements in {1, ··· , q}. Then from (2.3.2.1) it follows that

(q − p) log |gj(x)| ≤ v(x) + (q − p) log K.

20 We now use the following deﬁnition of the characteristic function

T (r) = Ar (log max(|g1|, ··· , |gp|)) as described in (2.1.0.1). Then

(2.3.2.2) (q − p)T (r) ≤ Ar(v) + (q − p) log K.

We would like to link v to H. Let {β1, ··· , βq−p} be a subset of {1, ··· , q} and {α1, ··· , αp} be its complement. The Wronkian W (g1, ··· , gp) is equal to some constant cα1···αp times the Wronskian W (Fα1 , ··· ,Fαp ). The purpose of this step is that a choice of {α1, ··· , αp} enables us to take the maximum to get the relation of H with the characteristic function. Thus 1 F ··· F H = 1 q cα1···αp W (Fα1 , ··· ,Fαp )

1 −1 = Fβ1 ··· Fβq−p (Qα1···αp ) , cα1···αp

−1 where Qα1···αp = (Fα1 ··· Fαp ) W (Fα1 , ···, Fαp ) is the p×p determinant whose th (ν−1) (µ, ν) entry is Fαµ /Fαµ . Hence

Fβ1 ··· Fβq−p = cα1···αp HQα1···αp and

v ≤ log |H| + max log |Qα1···αp | + C, where the maximum is taken over the collection of all subsets {α1, ··· , αp} of p elements in {1, ··· , q}. By the logarithmic derivative lemma (Lemma (2.2.3)), we have X + Ar(max log |Qα1···αp |) ≤ Ar( log |Qα1···αp |) = O(log T (r)) k.

Hence

(2.3.2.3) Ar(v) ≤ Ar(log |H|) + O(log T (r)) k.

From (2.1.0.2) we have

Ar(log |H|) ≤ N(r, H, 0) + log |H(0)|.

Thus (2.3.2.2) and (2.3.2.3) yield

(2.3.2.4) (q − p)T (r) ≤ N(r, H, 0) + log |H(0)| + O(log T (r)) k.

We now estimate N(r, H, 0) by looking at the pole set of H−1. At any given point x we have at least q − p + 1 functions among F1, ··· ,Fq nonzero at

21 x, because the q hyperplanes are in general position and no more than p − 1 of them can contain any given point. We choose a subset {β1, ··· , βq−p} of

{1, ··· , q} so that each Fβj is nonzero at x for 1 ≤ j ≤ q − p. Since

−1 W (Fα1 , ··· ,Fαp ) 1 H = cα1,···,αp , Fα1 ··· Fαp Fβ1 ··· Fβq−p

−1 it follows that the poles of H can only come from Fα1 ··· Fαp . When the vanishing order of Fαj is k ≥ p, the contribution of Fαj to the vanishing order of W (Fα1 , ···, Fαp ) is at least k − p + 1. Hence we have

Xq N(r, H, 0) ≤ Np−1(r, Fj, 0), j=1 which together with (2.3.2.4) yields the theorem. Q.E.D. 4. The Use of the Second Main Theorem with Truncated Counting Function The Second Main Theorem with truncated counting function can be used to show that certain hypersurfaces in Pn deﬁned by a homogeneous polynomial of high degree but only very few nonzero terms are hyperbolic (see Ma- suda and Noguchi [MN94]). Let X be the hypersurface in Pn deﬁned by Pq p the homogeneous polynomial g(x0, ··· , xn) = µ=0(sµ) of degree pδ, where νµ,0 νµ,n Pn sµ = aµx0 ··· xn , 0 6= aµ ∈ C, and j=0 νµ,j = δ. Theorem (2.4.1). For p > (q + 1)(q − 1) and with good appropriate choices of the indices (νµ,0, ··· , νµ,n) for 0 ≤ µ ≤ q the hypersurface X is hyperbolic. We will not specify the exact requirements for the relative sizes of q and δ and the choices of (νµ,0, ··· , νµ,n). One can formulate such requirements from the discussion of the proof given below.

Discussion of the Proof of Theorem (2.4.1). Let f : C → Pn be a nonconstant holomorphic map whose image is contained in X. Consider the map p p Φ: Pn → Pq−1 defined with the homogeneous coordinates [(s1) , ··· , (sq) ]. We assume that the indices (νµ,0, ··· , νµ,n) for 1 ≤ µ ≤ q are so chosen that Φ is a well-defined holomorphic map. Let Hµ (1 ≤ µ ≤ q) be the coordinate 0 hyperplanes of Pq−1. Let H be the hyperplance in Pq−1 defined by the vanishing of the sum of the homogeneous coordinates of Pq−1. Since the image of f lies in X, the pullback by Φ ◦ f of the defining function of H0 is the same as ∗ p 0 the pullback by f of −(f s0) . A point P of H is assumed by Φ ◦ f at some ∗ p point z0 of C if and only if −(f s0) vanishes at z0 and, in that case, it must automatically vanish to order at least p and so that the point P of H0 is assumed by Φ ◦ f with multiplicity at least p. As a consequence the truncated counting 0 q−1 0 function Nq−1(r, Φ ◦ f, H ) ≤ p N(r, Φ ◦ f, H ). The same argument holds for

22 0 Hj (1 ≤ j ≤ N) instead of H . Unless the image of Φ ◦ f is contained in a hyperplane of Pq−1, we know from the Second Main Theorem with truncated counting function that

Xq 0 T (r, Φ ◦ f) ≤ Nq−1(r, Φ ◦ f, H ) + Nq−1(r, Φ ◦ f, Hj) + O(log T (r, Φ ◦ f)) j=1 q q − 1 X ≤ (N (r, Φ ◦ f, H0) + N (r, Φ ◦ f, H )) + O(log T (r, Φ ◦ f)) p q−1 q−1 j j=1 q − 1 ≤ (q + 1)T (r, Φ ◦ f) + O(log T (r, Φ ◦ f)), p which is a contradiction if p > (q + 1)(q − 1). So we conclude that the image of Pq p f is contained in the zero-set of j=1 λj(sj) = 0 for some λj ∈ C (1 ≤ j ≤ q) Pq p not all zero. We can now replace g byg ˆ := j=1 λj(sj) which has one fewer nonzero terms than g. Continuing this kind of argument, we conclude that the image of f must be in a number of diﬀerent hypersurfaces. With a good choice of the indices (νµ,0, ··· , νµ,n) for 0 ≤ µ ≤ q we can conclude the intersection of all such hypersurfaces is zero dimenaional and there cannot exist any nonconstant holomorphic map from C to X. 5. Relation with the Argument of the Borel Lemma The argument in §4 is very much related to the argument of the Borel lemma which is used to show that there is no nonconstant holomorphic map from C into the complement of 2n + 1 hyperplanes in Pn in general position [Gr77]. As a matter of fact the Borel lemma argument can be regarded in a way as the limiting case of the above argument with p going to inﬁnity.

Theorem (2.5.1) (The Borel Lemma). Let f0, ··· , fn be nowhere zero entire functions on C satisfying f0 + ··· + fn ≡ 0. Then after relabelling the set f0, ··· , fn, one can divide up the set {0, ··· , n} into q disjoint subsets {`0, ··· , `1 − 1}, {`1, ··· , `2 − 1}, ··· , {`q−1, ··· , `q − 1} with 0 = `0 < `1 < ··· < ` = n + 1 and one can ﬁnd constants c (0 ≤ µ < q and ` < j < ` ) such q µ,j P µ µ+1 that f = c f for ` < j < ` and 1 + `µ+1−1 c = 0. j µ,j `µ µ µ+1 j=`µ+1 µ,j The usual way to prove the Borel lemma is to use the Wronskian. Divide the equation f0 + ··· + fn ≡ 0 by fn and get

(2.5.1.1) h0 + ··· + hn−1 ≡ 1

fj Pn−1 with hj = − . We would like to show that λjhj ≡ 0 for some complex fn j=0 numbers λj (0 ≤ j < n) not all zero so that the proof is ﬁnished by induction on n. We diﬀerentiate the equation (2.5.1.1) (n−1) times and use Cramer’s rule to solve this system of n equations for the n unknowns hj (0 ≤ j < n) and we get

23 (µ−1) hν hj as the quotient of the two determinants D1,j = ( )1≤µ 2 and p > (n − 1)(n + 1).

I would like to remark that we need 2n + 1 hyperplanes in Pn in general position instead of n + 2 for the complement to be hyperbolic. Though the missing of n + 2 hyperplanes Hj (1 ≤ j ≤ n + 2) forces the image to be inside 0 0 a hyperplane H , it may happen that the intersections Hj ∩ H (1 ≤ j ≤ n + 2) in H0 may not provide us with n + 1 hyperplanes of H0 in general position and we can find a nonconstant holomorphic map from C to H0 which misses 0 0 the hyperplanes H0 ∩ H , ··· ,Hn ∩ H . As an example, when n = 2 we can 0 take H be the line passing through the two points H1 ∩ H2 and H3 ∩ H4 and 0 the set Hj ∩ H (1 ≤ j ≤ 4) just consists of two points and we can find a 0 4 nonconstant holomorphic map from C to H minus the two points ∪j=1(Hj ∩ H0). Recently Min Ru [R94] worked out the linear algebra by using Nochka weights [Noc82] and obtained the following necessary and sufficient condition for a collection of hyperplanes in Pn to have the property that their complement is hyperbolic. The condition is that for any positive dimensional linear subspace not contained in any one of the hyperplanes the intersection of the hyperplanes with the subspace contains at least three distinct hyperplanes which are linearly independent. 6. Use of Wronskian in the Defect Relation for Slowly Moving Targets

24 Let f(z) be a meromorphic function on C. Instead of a constant a we consider a meromorphic function a(z) on C. Let Z(f −a) be the zero-set of f −a. We can also deﬁne the counting function N(r, f, a) = Ir(Z(f−a)), the proximity 1 + m(r, f−a ) function m(r, f) = Ar(log |f|), and the defect δ(f, a) = lim infr→∞ T (r,f) . The counting function N(r, f, ∞) simply means the counting function for the pole set of f. In the situation of a nonconstant function a(z), the target a(z) for f(z) to hit is moving. By a slowly moving target a we mean a meromorphic function a(z) such that T (r, a) = o(T (r, f)). Let the set of all slowly moving targets be K. Let a1, ··· , aq be q elements of K. For a meromorphic function + F on C we denote Ar(log |F |) by m(r, F ). Theorem (2.6.1) (Defect Relation for Slowly Moving Targets Pq [Ste86]). ν=1 δ(f, aν ) ≤ 2. Proof. In this proof, for notational simplicity we will suppress the notation k, because it will be clear from the context when it is needed.

n1 nq For s let L(s) be the vector space over C spanned by a1 , ··· , aq with nj ≥ 0 Pq ¡q+s−1¢ ¡q+s−1¢ and j=1 nj = s. The dimension of L(s) is no more than s = q−1 q−1 which is of the order s . Choose a basis β1, ··· , βn of L(s) over C and a basis b1, ··· , bk of L(s + 1) over C. The key point is the introduction of the function P [f] = W (b1, ··· , bk, β1f, ··· , βnf), where W (·) is the Wronskian.

There are two important properties of P [f]. The first one is P [f − aj] = P [f] which means that aj can be treated as a constant so far as the nonlinear n f 0 differential operator P [·] is concerned. The second one is P [f] = f Q( f ), where Q is a differential polynomial over the set K of slowly moving targets. The second property enables us to use the logarithmic derivative lemma. The first property is obtained by expanding the determinant

P [f − aj] = W (b1, ··· , bk, β1(f − aj), ··· , βn(f − aj)) and using the fact that −β`aj ∈ L(s + 1) and is therefore a linear combination f 0 of b1, ··· , bk over C. The second property uses βν f = fQ0( f ) with Q0 = βν (µ) f 0 0 0 f 0 0 f 0 f 0 0 and inductively (βν f) = (fQµ−1( f )) = f Qµ−1( f ) + fQµ−1( f )( f ) = f 0 0 0 fQµ( f ) with Qµ(x) = xQµ−1(x) + x Qµ−1(x). We now calculate the counting function of P [f] at ∞ and its proximity function at ∞ so that we get its characteristic function as the sum of the two. We have b b P [f] = f n+kW ( 1 ··· k , β , ··· , β ). f f 1 n The contribution to the pole-counting of P [f] comes from the pole of f through n+k bν f , from the zero of f through f , and from the poles of bν and βν . Since P [f] is also equal to W (b1, ··· , bk, β1f, ··· , βnf), the contribution from the

25 zeroes of f to the pole-counting of P [f] is already oﬀset by the factor f n+k. The contribution to the pole-counting of P [f] from the poles of bν and βν is of the order o(Tf (r)), because aj is a slowly moving target. Hence we have N(r, P [f], ∞) ≤ (n + k)N(r, f, ∞) + o(Tf (r)). We now estimate the proximity function of m(r, P [f]). f 0 f 0 m(r, P [f]) = m(r, f nQ( )) ≤ n m(r, f) + m(r, Q( )). f f For any meromorphic function g the logarithmic derivative lemma gives g0 g0 g0 T (r, ) = N(r, , ∞) + m(r, ) + O(1) ≤ 2T (r, g) + o(T (r, g)), g g g

g0 1 because N(r, g , ∞) ≤ N(r, g, ∞) + N(r, g , ∞). Moreover, T (r, g0) = N(r, g0, ∞) + m(r, g0) + O(1) g0 ≤ 2N(r, g, ∞) + m(r, g) + m(r, ) + O(1) g ≤ 2T (r, g) + o(T (r, g)).

g0 (ν) Thus by induction on ν we have m(r, ( g ) ) = o(T (r, f)), because g0 g0 g0 g0 m(r, ( )(ν)) ≤ m(r, ( )(ν−1)) + m(r, ( )(ν−1)/( )(ν)) g g g g g0 ≤ o(T (r, g)) + o(T (r, ( )(ν−1))) g g0 ≤ o(T (r, g)) + o(T (r, )) ≤ o(T (r, g)). g

f 0 Hence m(r, Q( f )) = o(T (r, f)) and m(r, P [f]) ≤ n m(r, f) + o(T (r, f)). Finally T (r, P [f]) = m(r, P [f]) + N(r, P [f], ∞) ≤ n m(r, f) + (n + k)N(r, f, ∞) + o(T (r, f)) ≤ (n + k)T (r, f) + o(T (r, f)). P We now get the defect relation by estimating q log+ 1 . The j=1 |f(z)−aj (z)| main point is that not all f(z) − aj(z) (1 ≤ j ≤ q) are small at the same time and we make more precise the statement by considering the minimum distances 1 between aj(z) (1 ≤ j ≤ q). Let d(z) = mini6=j 2 |ai(z) − aj(z)|. Let Ej be the set of all z such that |f(z) − aj(z)| < d(z). The estimation of the upper bound 1 for is done by using the ﬁrst property P [f − aj] = P [f] of P [f]. We |f(z)−aj (z)| have n P [f] = P [f − aj] = (f − aj) Qj,

26 0 f 1 |Qj | 1/n where Qj = Q( ). On Ej we use ≤ ( ) and outside of Ej f−aj |f(z)−aj (z)| |P [f]| we use 1 ≤ 1 . It follows that |f(z)−aj (z)| d(z)

q X 1 1 1 1 1 log+ ≤ log+ + log+ |Q (z)| + q log+ , |f(z) − a (z)| n |P [f]| n j d(z) j=1 j because Ei is disjoint from Ej for i 6= j. From the deﬁnition of d(z) we have 1 X 1 log+ ≤ log+ + O(1). d(z) |ai(z) − aj(z)| i6=j

Now ai(z) − aj(z) is again a slowly moving target, because

T (r, ai − aj) = m(r, ai − aj) + N(r, ai − aj, ∞) + O(1)

≤ m(r, ai) + m(r, aj) + N(r, ai, ∞) + N(r, aj, ∞) + O(1)

≤ T (r, ai) + T (r, aj) + O(1).

Hence m(r, 1 ) = o(T (r, f)) and m(r, 1 ) ≤ o(T (r, f)) and we have ai−aj d

q Z X 1 1 1 1 2π 1 m(r, ) ≤ m(r, ) + m(r, Q ) + q log+ dθ f − a n P [f] n j d(z) j=1 j θ=0 1 n + k ≤ T (r, P [f]) + o(T (r, f)) ≤ T (r, f) + o(T (r, f)). n n

We are free to choose s. We claim that there exists a sequence of s going k to infinity such that n approaches 1. Otherwise there exists some κ > 1 such s that the quotient of dimC L(s + 1) by dimC L(s) is at least κ for s sufficiently q−1 large, contradicting the estimate that dimC L(s) is of the order s as s goes to infinity. P It follows from q m(r, 1 ) ≤ n+k T (r, f) + o(T (r, f)) and lim k = 1 j=1 f−aj n n Pq that ν=1 δ(f, aν ) ≤ 2 + ² for any positive ². Q.E.D. Min Ru and Stoll [RS91] combined the techniques of Steinmetz [Ste86] and Cartan [C33] to generalize Steinmetz’s result to the defect relations with slowly ∗ moving targets for the higher dimensional case. We denote by Pn the dual of ∗ complex projective space Pn in the sense that a point of Pn is a hyperplane in Pn.

Theorem (2.6.2). Let q ≥ n + 2. Let f : C → Pn and gν = [gν0, ··· , gνn]: ∗ C → Pn be holomorphic maps so that T (r, gν ) = o(T (r, f)) (1 ≤ ν ≤ q). Assume that for some point a ∈ C the q hyperplanes gν (a) of Pn are in general position. Suppose that f is not linearly degenerate in the sense that there does ∗ not exists h = [h0, ··· , hn]: C → Pn such that the pairing < h, f > of h and f

27 is identically zero on C and hν belong to the ﬁeld generated over C by gλµ for Pq 1 ≤ λ ≤ q and 1 ≤ µ ≤ n. Then the defect relation ν=1 δ(f, gν ) ≤ n+1 holds.

Chapter 3. Negative Curvature Arguments in Higher Dimensions In the ﬁrst three sections of this Chapter we will use negative curvature arguments to give another proof of Corollary (2.3.2) due to Ahlfors [A41] with later reinterpretation and enhancement by Wu [Wu70], Cowen-Griﬃths [CG76], and others. Then in the last two sections we will discuss respectively the method of meromorphic connections and the method of dual curves, both closely related to negative curvature arguments. 1. Motivation for the Form of the Metric First let us review the proof of the negative curvature argument in dimension 1 (§3 of Chapter 1) to see what the trouble is when one tries to transplant the argument to the higher dimensional case of a holomorphic map from C to a compact complex manifold of complex dimension greater than 1. For the higher dimensional case, we are unable to get the lower bound in (1.3.1.3) for the last term on the right hand-side of (1.3.1.2). As a result, the 2 2 ∗ |Ds| factor |Ds| in the numerator of Ir(f ( |s|2(log |s|2)2 )) prevents us from domi- 2 ∗ 1 ∗ |Ds| nating Ar(log f ( |s|2(log |s|2)2 )) by Ir(f ( |s|2(log |s|2)2 )). To remedy the situa- ∗ 1 tion we should put the same numerator in Ar(log f ( |s|2(log |s|2)2 )) and consider 2 √ 2 ∗ |Ds| −1 ¯ ∗ |Ds| Ar(log f ( |s|2(log |s|2)2 )). For that formula we have to use 2π ∂∂ log f ( |s|2(log |s|2)2 ). Then we end up with the boundary term ∗ 2 Ar(log |f Ds| ) which we have to handle. The need to handle the boundary ∗ 2 term Ar(log |f Ds| ) leads us to consider the formula from √ 2 −1 ¯ ∗ |DDs| 2π ∂∂ log f ( |Ds|2(log |Ds|2)2 ) obtained by replacing s by Ds. We then have ∗ 2 to handle the boundary term Ar(log |f DDs| ). We continue this process and √ ν+1 2 −1 ¯ ∗ |D s| get a sequence of formulas from 2π ∂∂ log f ( |Dν s|2(log |Dν s|2)2 ). This sequence of formulas is to be added up and we end up with the consideration of the single formula √ −1 1 ∂∂¯log f ∗( ). 2π |s|2(log |s|2)2(log |Ds|2)2(log |D2s|2)2(log |D3s|2)2 ···

2. The Case of Hyperplanes in the Complex Projective Space In general we have the problem that the product in the denominator does not terminate in a finite number of steps. For the case where M is the complex projective space and the divisors are hyperplanes, a suitable modification of this process gives us a finite product. We let f : C → Cn+1 − 0 be the lifting of the holomorphic map from C to Pn. Consider the holomorphic map Fk defined

28 by f ∧ f 0 ∧ · · · ∧ f (k) from C to the Grassmanian of all (k + 1)-subspaces in n+1 0 (k) C . Note that to get Fk from f ∧ f ∧ · · · ∧ f we have to remove the 0 (k) zeroes of f ∧ f ∧ · · · ∧ f . Let Zk be the zero-set of the differential of Fk with multiplicities counted. As the analog of f ∗Dks we consider the following function. For a hyperplane n+1 D in C with unit normal vector a, let ϕk(D, z) be the square of the length of the orthogonal projection of a onto the (k + 1)-dimensional subspace of Cn+1 √ 0 (k+1) −1 ¯ 2 spanned by the vectors f(z), f (z), ···, f (z). Let Ω˜ k = ∂∂ log |Fk| . √ 2π ˜ −1 Write Ωk = Ωk 2 dz ∧dz¯. For notational simplicity we suppress the arguments D and s in ϕk(D, z) and just use ϕk for the time being. The analog of the inequality (1.3.1.1) is √ −1 ¯ 1 2ϕk+1 ˜ ˜ (3.2.1.1) ∂∂ log 2 µ ≥ 2 µ Ωk − ²Ωk, 2π log ( ) ϕk log ( ) ϕk ϕk where µ is a large positive number so that the small positive number ² goes to 1 1 0 as µ goes to infinity. The factor log2( µ ) is the analog of the factor log |f ∗s|2 ϕk when a large positive numbere µ is used to replace the rescaling of the norm of 1 the line bundle L to make the absolute value of log |f ∗s|2 small. The difficulty is, of course, the factor Ω˜ k in (3.2.1.1). If there were no such factor, we would be able to get a proof of the defect relation by considering the formula for √ −1 ¯ 1 ∂∂ log n−1 2 µ . 2π ϕ0Π log ( ) k=0 ϕk

˜ Now, because of the factor Ω√k we have to put each Ωk into the numerator −1 ¯ of the fraction and worry about 2π ∂∂ log Ωk. We have √ −1 (3.2.1.2) ∂∂¯log Ω = Ω˜ − 2Ω˜ + Ω˜ + Z , 2π k k−1 k k+1 k which one derives first for z not in Zk−1 ∪ Zk ∪ Zk√+1 and then for z near Zk by −1 ¯ arguing that the only singularity of the current 2π ∂∂ log Ωk comes from the zero-set of the differential of Fk. We use the telescoping effect in the equation (3.2.1.2) and get √ nX−1 −1 nX−1 (n − k) ∂∂¯log Ω = −(n + 1)Ω˜ + (n − k)Z 2π k 0 k k=1 k=1 √ −1 nX−1 = −(n + 1) ∂∂¯log |f(z)|2 + (n − k)Z . 2π k k=1

29 This telescoping eﬀect forces us to use the formula for

√ n−1 n−k −1 ¯ Πk=0 Ωk ∂∂ log n−1 2 µ . 2π ϕ0Π log ( ) k=0 ϕk

Now we use a ﬁnite number of hyperplanes D1, ··· ,Dq in general position and let n−1 n−k Πk=0 Ωk Ψ = q n−1 2 µ . Π (ϕ0(Dj)Π log ( )) j=1 k=0 ϕk(Dj ) Using √ −1 ¯ 1 ∗ ∂∂ log = Ω˜ 0 − Zf Dj, 2π ϕ0(D) we have

√ q −1 X nX−1 nX−1 (3.2.1.3) ∂∂¯log Ψ ≥ (q − n − 1)Ω˜ + Φ Ω˜ − ² Ω˜ , 2π 0 jk k k j=1 k=0 k=0 where 2ϕk+1(Dj) Φjk = 2 µ . ϕk(Dj) log ( ) ϕk(Dj ) Since n−1 n 1 Πk=0 Φjk = 2 q n−1 2 µ Π (ϕ0(Dj)Π log ( )) j=1 k=0 ϕk(Dj ) Pq Pn−1 is very close to Ψ, in order to use the term j=1 k=0 ΦjkΩk on the right- hand side of (3.2.1.3) to take care of the boundary term Ar(Ψ) (which would come from the left-hand side of (3.2.1.3) upon an application of Green’s The- orem), we should apply the inequality on arithmetic and geometric means to Pq Pn−1 j=1 k=0 ΦjkΩk. We ﬁrst use the inequality on arithmetic and geometric means for the sum- Pq Pq mation j=1 Φjk with k ﬁxed. The easiest way is just to get j=1 Φjk ≥ q 1/q q 1/q Πj=1(Φjk) . However, after we multiply by Ωk, we end up with Πj=1(Φjk) Ωk q n−k which is not a fractional power of (Πj=1Φjk)Ωk . So we have to get something Pq q 1/(n−k) like j=1 Φjk ≥ constant · Πj=1(Φjk) to achieve that. At this point we have to use the assumption that the set of hyperplanes Dj (1 ≤ j ≤ q) are in general position. 3. The Use of the General Position Assumption n+1 Let V be any (k+1)-dimensional linear subspace of C . Let aj be the unit 2 normal vector of Dj and let |prV (aj)| be the square norm of the orthogonal projection of aj onto V . For every V let JV be the subset of {1, ··· , q} so that

30 aj belongs to the orthogonal complement of V . Since D1, ··· ,Dq are in general position, the number of elements in JV cannot be more than n − k. Thus for any (k +2)-dimensional linear subspace W of Cn+1 containing V the expression

2 |prW (aj)| 2 2 µ |prV (aj)| log ( 2 ) |prV (aj )| is ﬁnite for j ∈ {1, ··· , q} − JV and

2 |prW (aj)| mV,W := min max < ∞ J 1≤j≤q,j²J 2 2 µ |prV (aj)| log ( 2 ) |prV (aj )| with J ranges over the collection of all subsets of n − k elements in {1, ··· , q}. By continuity there exists some positive number mk with mV,W ≤ mk for all V and W . When the situation is applied to the case of V being spanned by f(z), f 0(z), ···, f (k)(z) and W being spanned by f(z), f 0(z), ···, f (k+1)(z), we conclude that for every z there exists a subset Jz,k of n−k elements in {1, ··· , q} such that Φjk(z) ≤ mk for j ∈ {1, ··· , q} − Jk,z. Hence

q X X 1 n−k Φjk ≥ Φjk ≥ (n − k)Πj∈Jk,z Φjk j=1 j∈Jk,z

Φjk 1 Φjk 1 n−k q n−k = mk(n − k)Πj∈Jk,z ( ) ≥ mk(n − k)Πj=1( ) . mk mk Thus Xq nX−1 nX−1 Φjk 1 q n−k (3.3.1.1) ΦjkΩk ≥ mk(n − k)Πj=1( ) Ωk mk j=1 k=0 k=0

q Φjk 1 n−k n−1 n−k N ≥ mΠk=0 (Πj=1( ) Ωk) mk where m is equal to q times the minimum of mk(n − k) for 0 ≤ k ≤ n − 1 and Pn−1 n N = k=0 (n − k) = 2 (n + 1). The inequality (3.3.1.1) together with (3.2.1.3) Pq would give us the defect relation j=1 δ(f, Dj) ≤ n + 1 after applying Green’s Theorem if we can show that Ir(Ωk) and Ir(Ω0) are bounded by a constant multiple of each other for 0 ≤ k ≤ n − 1. By applying Green’s Theorem to (3.2.1.2), we conclude that

Ir(Ωk−1) − 2Ir(Ωk) + Ir(Ωk+1) ≤ constant · (log Ir(Ωk) + log r) k.

We use the notation G ≈ 0 to denote nX−1 |G| ≤ constant · (log Ir(Ωk) + log r) k. k=0

31 Then Ir(Ωk−1) − 2Ir(Ωk) + Ir(Ωk+1) ≈ 0 and from Ω−1 = Ωn = 0 we conclude 1 that k+1 Ir(Ωk) − Ir(Ω0) ≈ 0 by induction on 1 ≤ k ≤ n − 1. This implies that 1 I (Ω ) ≤ (1 + ²)I (Ω ) k k + 1 r k r 0 and 1 (1 − ²)I (Ω ) ≤ I (Ω ) k. r 0 k + 1 r k This ends the proof of the defect relation by negative curvature argument. Ahlfors [A41] gave defect relations not only for the map f but also for the ¡n+1¢ associated curves Fk. The sum of the defects for Fk is no more than k+1 (see also Theorem 5.13 on p.206 of [Wu70]). Ahlfor’s work on the defect relations is a very beautiful theory and a very impressive tour de force. 4. Meromorphic Connections of Low Pole Order The reason that defect relations hold for hyperplanes is that the second fundamental form of the hyperplanes with respect to the standard connection for the tangent bundle is zero. The same negative curvature argument goes through for general hypersurfaces if there is a meromorphic connection of low pole order for the tangent bundle with respect to which the hypersurface has zero second fundamental form. Unfortunately such a meromorphic connection of low pole order for the tangent bundle exists only in very special cases, for example, when the hypersurface is defined by a polynomial with high degree but very few nonzero terms. Such a case could be handled already by Cartan’s Second Main Theorem with truncated counting functions (§3 of Chapter 2). However, the concept of meromorphic connections of low pole order for the tangent bundle is closely related to holomorphic special 2-jet differentials which have been successfully used to get the hyperbolicity of the complement of a generic nonsingular plane curve of high degree [SY94]. Also meromorphic connections of low pole order for the tangent bundle were used to construct hyperbolic surfaces by Nadel [Na89]. In this section we discuss the use of meromorphic connections of low pole order for the tangent bundle. The relations of meromorphic connections of low pole order for the tangent bundle with holomorphic special 2-jet differentials will be discussed in §5 of Chapter 6. A meromorphic connection for the tangent bundle is a connection for the tangent bundle which, with respect to holomorphic local coordinates z1, ··· , zn, α is given by local meromorphic functions Γβγ (1 ≤ α, β, γ ≤ n) with the transformation rule so that for any local holomorphic function f the meromorphic α β γ (2,0)-form D(df) = (∂β∂γ f + Γβγ ∂αf)dz dz is independent of the coordinate chart chosen. Let θL denote the curvature of a Hermitian holomorphic line bundle L. An example of defect relations which can be obtained by using such meromorphic connections is the following ([Si87], [Si90]).

32 Theorem (3.4.1). Let M be a compact complex manifold of complex dimension 2. Let L and F be Hermitian holomorphic line bundles over M and s (respectively t) be a nontrivial global holomorphic section of L (respectively α F ). Assume that L is positive. Let Γβγ be a meromorphic connection of the α tangent bundle of M such that tΓβγ is holomorphic. Assume that the zero-set Zs of s is nonsingular and locally tDα∂βs is a linear combination of ∂αs, ∂βs, and s with smooth coeﬃcients. Let f : C → M be a holomorphic map such that −1 df ∧ tDdf from C to the anticanonical line bundle KM of M is not identically zero. Then the defect of f for the divisor Zs does not exceed any real number

σ such that σθL ≥ θF − θKM .

Remark (3.4.2). The condition that tDα∂βs is a linear combination of ∂αs, ∂βs, and s with smooth coefficients is about the vanishing of the second fun- α damental form of Zs with respect to the meromorphic connection Γβγ of the tangent bundle. Sketch of the Proof of Theorem (3.4.1). Let τ be a holomorphic section of some positive line bundle over M so that there exists a meromorphic connection of the line bundle L whose product with τ is holomorphic. Unlike t whose choice is constrained by the meromorphic connection satisfying the condition of the vanishing of the second fundamental form of Zs, there are different ways of choosing the meromorphic connection for L and the holomorphic section τ. We can get the estimate for different choices of τ and then add up the estimates to eliminate the use of τ in the final estimate. To make the presentation of the argument less encumbered by notations, we just use t as the holomorphic section used to make different meromorphic connections holomorphic. Actually in some places some factors t can be replaced by τ so that such factors can be removed in the final estimate. For notational simplicity we also use the same notation D for covariant differentiation√ with different connections. We use the −1 ¯ ∗ 1 Poincaré-Lelongformula for 2π ∂∂ log f ( |s|2(log |s|2)2 ) to get the estimate of 2 ∗ |Ds| Ir(f ( |s|2(log |s|2)2 )). We get

|tDs|2 (3.4.1.1) I (f ∗( )) = O((log r + I (f ∗θ )) k. r |s|2(log |s|2)2 r L

As in §1, to handle the term from the numerator |f ∗(tDs)|2 in

|tDs|2 I (f ∗( )), r |s|2(log |s|2)2 we use the Poincar´e-Lelongformula for √ −1 1 ∂∂¯log 2π |f ∗(tDs)|2(1 + (log |f ∗(tDs)|2)2

33 to get an estimate

Now use (|f ∗(tD(tDs))|2)λ |f ∗(tD(tDs))|2 = ( )λ |f ∗s|2(log |f ∗s|2)2 |f ∗(tDs)|2(1 + (log |f ∗(tDs)|2)2 (|f ∗(tDs)|2(1 + (log |f ∗(tDs)|2)2)λ |f ∗s|2(log |f ∗s|2)2 to get µ ¶ (|f ∗(tD(tDs))|2)λ (3.4.1.2) I ( )µ = O(r4 + I (f ∗θ )8µ) k r |f ∗s|2(log |f ∗s|2)2 r L for suitable λ > 1 and µ > 0 with a suitable sequence of λ approaching 1. The main idea in the use of the connection is to solve the following equations

∗ 1 2 D(f s) = (df) ((D1s) ◦ f) + (df) ((D2s) ◦ f), α 1 2 (Ddf) ((Dαs) ◦ f) = (Ddf) ((D1s) ◦ f) + (Ddf) ((D2s) ◦ f) by Cramer’s rule for the unknowns (D1s) ◦ f and (D2s) ◦ f to get

2 ∗ 2 ∗ 2 2 α 2 2 (3.4.1.3) |df ∧ Ddf| |f Ds| ≤ 2(|f Ds| |Ddf| + |(Ddf) ((Dαs) ◦ f)| |df| )

|df∧tDdf|2λ (|f ∗(tD(tDs))|2)λ in order to get an estimate of | f ∗s|2(log |f ∗s|2)2 in terms of |f ∗s|2(log |f ∗s|2)2 . We will use the condition on the vanishing of the second fundamental form of Zs to get such an estimate. From the condition on the vanishing of the second fundamental form of Zs we have smooth functions Aα, Bα and Cαβ such that

α ∗ α β (3.4.1.4) (Ddf) ((tDαs) ◦ f) = tf DDs − (df) (df) ((tDαDβs) ◦ f)

∗ ∗ ∗ ∗ α α β = f D(tDs) − f (Dt)(f Ds) − (f Ds)(df) ((Aα + Bα) ◦ f) − (df) (df) Cαβs. It follows from (3.4.1.4) that

α ∗ ∗ ∗ 2 (3.4.1.5) |(Ddf) ((tDαs) ◦ f)| ≤ |f D(tDs)| + C(|f Ds| + |f Ds||df| + |df| ).

To handle the factor |Ddf|2 in the ﬁrst term on the right-hand side of (3.4.1.3), √ −1 ¯ 2 we use the formula for 2π ∂∂ log(1 + |df| ) to get the estimate µ ¶ |Ddf|2 ¡ ¢ (3.4.1.6) I ≤ constant · I (f ∗θ ) + r2 k. r 1 + |df|2 r L

34 Putting together (3.4.1.2), (3.4.1.3), (3.4.1.5), and (3.4.1.6) and using H¨older’s inequality, we get ﬁnally the estimate

µµ ¶µ¶ |df ∧ tDdf|2λ ¡ ¢ I ≤ constant · I (f ∗θ )16λµ + r4 k. r |f ∗s|2(log |f ∗s|2)2 r L Here for the ﬁnal estimate we have to use the remark at the beginning of the sketch of the proof concerning τ to remove one factor of t. The defect relation now results from µ ¶ 1 A log f ∗( ) = r |s|2(log |s|2)2 µ µ ¶µ¶ 1 |df ∧ tDdf|2λ A log − A (log |df ∧ tDdf|2λ) µ r |f ∗s|2(log |f ∗s|2)2 r µµ ¶µ¶ 1 |df ∧ tDdf|2λ ≤ (² log r + (1 + δ) log I ) − A (log |df ∧ tDdf|2λ) µ r |f ∗s|2(log |f ∗s|2)2 r ∗ ∗ ∗ ≤ O(log Ir(f θL) + log r) − λIr(f θKM ) + λIr(f F ) k. Q.E.D.

When the defining equation for a hypersurface S in Pn is defined by a polynomial f of high degree and only a small number of nonzero terms, we can write α ∂β∂γ f = Γβγ ∂αf + Cαβf α for rational functions Γβγ and Cαβ whose denominators and numerators have degrees which are low relative to the degree of f, where ∂α means differentiation α with respect to an inhomogeneous coordinate. We can in such a case use Γβγ as the meromorphic connection of low pole order for the tangent bundle of Pn with respect to which the seocnd fundamental form of S vanishes. One very unsatisfactory point of Theorem (3.4.1) is the nondegeneracy con- −1 dition that df ∧ tDdf from C to the anticanonical line bundle KM of M is not identically zero. A more satisfactory condition would be the image of f not contained in some proper subvariety of M.

For Theorem (3.4.1), when θKM > θF we do not need any L and any s to have a negative defect. In that case the vanishing of the second fundamental form becomes a vacuuous condition and any meromorphic connection for the tangent bundle which satisﬁes θKM > θF could be used. Nadel [Na89] applied such an argument and studied carefully the nondegeneracy condition to produce a class of hyperbolic surfaces in P2. 5. The Method of Dual Curves The question of the hyperbolicity of the complement of an irreducible plane curve can be studied by using the method of dual curves. Unfortunately, so

35 far this method can only give results when the irreducible curve has certain singularities. We present the method in this section. ∗ Let C be an irreducible curve in P2 of degree δ. Its dual curve C is defined as follows. Take a regular point x of C and let L(x) be the tangent line to C ∗ ∗ at x. A line in P2 is a point in the dual space P2 of P2. The dual curve C is ∗ the topological closure of the set of all points in P2 defined by L(x) as x ranges over the set of all regular points of C. There is the duality that the curve C is ∗ ∗ ∗ ∗ the dual curve of C . Let δ be the degree of C in P2. We will discuss this duality later in this section. ∗ ∗ A point x of P2 defines a line `x in P2. If C is nonsingular, then the definition of dual curves means that x does not belong to C if and only if `x ∗ ∗ ∗ intersects C precisely at δ points P1(x), ···,Pδ∗ (x). The δ points P1(x), ·· ∗ δ∗ ∗ ·,Pδ∗ (x) define an element Q(x) in the δ -fold symmetric product Sym C of ∗ δ∗ ∗ ∗ C . So we have a map Φ : P2 − C → Sym C . Take a Hermitian metric h on C∗ of constant negative curvature. We have the product metric h˜ on the δ∗-fold ∗ ∗ Cartesian product (C∗)δ of C∗ defined by h∗. Since Symδ C∗ is a quotient ∗ space of (C∗)δ by the symmetric group of δ∗ objects, we can push forward h˜ ∗ by taking the sum on the branches to get a metric hˆ on Symδ C∗. Finally we ˆ can pull back the metric h via Φ to get a metric on P2 − C whose curvature admits a uniformly negative upper bound. The existence of such a metric on P2 − C of curvature with uniform negative upper bound implies that there is no nonconstant holomorphic map from C to P2 − C and P2 − C is hyperbolic. This approach was studied by Carlson-Green [76] and Grauert-Peternell [GP85]. The limitation of this approach is that C and C∗ cannot both be nonsingular. Since C∗ is defined by taking the topological closure of the set of its generic points, a point x of P2 may not be in C and yet the line `(x), though not tangential to C∗ at a regular point of C∗, can still pass through a cusp of C∗ and does not intersect C∗ at δ∗ distinct points. Moreover, when C∗ has singularities, to get a Hermitian metric on it of curvature with uniform negative upper bound, we have to push forward a Hermitian metric on its normalization # C and the resulting metric on P2 − C may be singular. To avoid singularities that affect the conclusion on hyperbolicity, we can consider Hermitian metrics on C# which vanish at least to an appropriate order at the inverse images of singular points of C∗. Such Hermitian metrics on C# can be constructed from holomorphic 1-forms on C# which vanish to an appropriate order at the inverse images of singular points of C∗. Whether the use of such Hermitian metrics # on C can lead to a Hermitian metric on P2 − C useful for the conclusion on hyperbolicity depends on how bad the singular points of C∗ are and how high the genus of the normalization C# of C∗ is. One can draw conclusions about the hyperbolicity of P2 − C by allowing singularities on C in order to reduce the singularities on the dual curve C∗. The best result obtained so far by this approach is the following.

36 Theorem (3.5.1). Let g be the genus of the normalization of the dual curve ∗ ∗ C of the curve C in P2. If C has only ordinary double points and k cusps with k < 2g − 2, then there is no nonconstant holomorphic map from C to P2 − C. We now discuss the duality between C and C∗ and look at one simple example to get some idea why C and C∗ cannot be both nonsingular and why it is not possible to get the desired metric on C∗ when C is nonsingular. Let F (ξ0, ξ1, ξ2) = 0 deﬁne a curve C in P2, where [ξ0, ξ1, ξ2] is the homogeneous ∗ coordinate of P2. The dual curve C is deﬁned as the image of C under the map

ΨC which sends [ξ0, ξ1, ξ2] to [Fξ0 (ξ0, ξ1, ξ2), Fξ1 (ξ0, ξ1, ξ2), Fξ2 (ξ0, ξ1, ξ2)]. At points where Fξ0 (ξ0, ξ1, ξ2), Fξ1 (ξ0, ξ1, ξ2), Fξ2 (ξ0, ξ1, ξ2) are all zero, the map ΨC is not defined and we may have only the definition of ΨC (C − AC ) for some ∗ finite set AC in C. The dual curve C is then defined as the topological closure of ΨC (C − AC ). ∗ Let C be defined by G(η0, η1, η2) = 0. On C we have

ξ0Fξ0 (ξ0, ξ1, ξ2) + ξ1Fξ1 (ξ0, ξ1, ξ2) + ξ2Fξ2 (ξ0, ξ1, ξ2) = δ F (ξ0, ξ1, ξ2) = 0, where δ is the degree of F . From the deﬁntion of C∗ it is clear that the curves C and C∗ have the same normalization Γ. If t is the local parameter of Γ and we have the parametrization [ξ0(t), ξ1(t), ξ2(t)] of C and the parametrization ∗ [η0(t), η1(t), η2(t)] of C by Γ. Then

(3.5.1.1) ξ0(t)η0(t) + ξ1(t)η1(t) + ξ2(t)η2(t) = 0.

On the other hand, we have F (ξ0(t), ξ1(t), ξ2(t)) = 0 and diﬀerentiation with respect to t yields

0 0 ξ0(t)Fξ0 (ξ0(t), ξ1(t), ξ2(t)) + ξ1(t)Fξ1 (ξ0(t), ξ1(t), ξ2(t)) 0 +ξ2(t)Fξ2 (ξ0(t), ξ1(t), ξ2(t)) = 0. Thus we have also

0 0 0 (3.5.1.2) ξ0(t)η0(t) + ξ1(t)η1(t) + ξ2(t)η2(t) = 0 from the deﬁnition of the map Ψ. The equations (3.5.1.1) and (3.5.1.2) deter- mine the dual curve [η0(t), η1(t), η2(t)]. We now see where the duality comes from. From the diﬀerentiation of (3.5.1.1) with respect to t and by using (3.5.1.2) we obtain

0 0 0 (3.5.1.3) ξ0(t)η0(t) + ξ1(t)η1(t) + ξ2(t)η2(t) = 0. Thus from (3.5.1.1) and (3.5.1.3) we also conclude that C is the dual curve of C∗.

We consider the simple example of the Fermat curve C deﬁned by F (ξ0, ξ1, ξ2) = p p p p−1 ξ0 + ξ1 + ξ2 . Then ηj = p ξj and G(η0, η1, η2) is given by G(η0, η1, η2) =

37 p/(p−1) p/(p−1) p/(p−1) 1 1 p η0 +η1 +η2 . Let q be defined by p + q = 1. Then q = p−1 . Thus q q q G(η0, η1, η2) = η0 +η1 +η2. When q is not an integer, to get a polynomial equa- ∗ q q q tion for the dual curve C of C we have to multiply G(η0, η1, η2) = η0 + η1 + η2 by expressions obtained by the action on it of the Galois group of the minimum q q q normal extension field of C(η0, η1, η2) containing η0, η1, η2. To see whether it is possible to get the desired metric on C∗ by pushing forward metrics on C# defined by holomorphic 1-forms on C# with zeroes at the inverse images of the singular points of C∗, we look at holomorphic 1-forms on C∗. So far the construction of holomorphic 1-forms on C∗ is concerned, we q q q can just consider its defining equation η0 + η1 + η2 = 0 without first changing it to a polynomial equation even when q is not an integer. In inhomgeneous coordinates x = η1 and y = η2 the dual curve C∗ is xq +yq +1 = 0 and the holo- η0 η0 dx dy morphic 1-forms are constructed by using qyq−1 = − qxq−1 and multiplication by dx dy polynomials of low degree in x and y. When q is not an integer qyq−1 = − qxq−1 is not well-defined, but we are only interested in the absolute value of that form dx dy in order to construct a metric. So we consider | yq−1 | = | xq−1 |. This form has no pole at any finite point, because we know that xq + yq + 1 = 0 and x and y cannot be both zero. However, dx has a pole of order two at infinity. So to dx dy remove the pole of | yq−1 | = | xq−1 | at infinity we need q at least 3. When q is q 3 at least 3, we know that p = q−1 is between 1 and 2 and the original Fermat curve C is highly singular. I would like to remark that Grauert [G89] considered a generalization of the technique of dual curves by looking at, instead of tangent lines to the curve, higher-degree curves which osculate to the curve to high order. He used the generalization to study the hyperbolicity of the complements of three quadrics by using metrics from jet differentials rather than from 1-forms.

Chapter 4. Brody’s Reparametrization and Maps of Finite Order A very powerful tool in the study of hyperbolicity problems is the reparametrization of Brody [Br78]. We will discuss it in this Chapter and its use in the recent result of Detloﬀ-Schumacher-Wong [DSW93] on the hyperbolicity of the complement of three quadrics in P2. 1. Brody’s Reparametrization Suppose X is a compact complex manifold with a Hermitian metric. Let ∆r be the open disk of radius r centered at the origin in C. Suppose f is a holomorphic map from an open neighborhood of the topological closure of ∆r in C to X. Let ft(z) = f(tz). We denote by |dft(z)|X,p the pointwise norm of df with respect to the Hermitian metric of X and the Poincar´emetric of ∆r. We denote by |dft(z)|X,e the pointwise norm of df with respect to the Hermitian

38 metric of X and the Euclidean metric of ∆r. We use r2 − |z|2 r2 − |z|2 |df (z)| = |df (z)| = t |df(tz)| . t X,p r t X,e r X,e

|df(0)|X,e. When t = 0, the value of supz∈∆r |dft(z)|X,p is 0. Hence there exists some 0 < t ≤ 1 such that supz∈∆r |dft(z)|X,p = |df(0)|X,e. Let T be the M¨obiustransformation mapping ∆r to itself so that T maps the origin to zt. Let g = ft ◦ Tr. Then |dg|X,p assumes its maximum |df(0)|X,e on ∆r at the origin. We have r r |dg(z)| ≤ |dg(z)| ≤ |df(0)| X,e r2 − |z|2 X,p r2 − |z|2 X,e and |dg(0)|X,e = |df(0)|X,e. Note that the image of g is contained in the image of f.

Theorem (4.1.1) (Brody’s Reparametrization). Suppose rn is a sequence of positive numbers monotonically increasing to ∞ and suppose Fn is a holomorphic map from ∆rn to X with 0 < a ≤ |dFn(0)|X,e ≤ b < ∞. Then there exist 0 < tn ≤ 1 and a M¨obiustransformation Tn of ∆rn such that the func- b tion gn(z) = f(tTn(z)) satisﬁes |dgn(0)|X,e ≥ a and sup |dgn(z)|X,e ≤ 1−κ2 for |z| ≤ κrn and 0 < κ < 1. As a consequence, a subsequence of gn converges to holomorphic map G from C to X with |dG|X,e bounded by b on C. Moreover, the image of G belongs to the subset of X consisting of all points P such that P is the limit of a sequence of points Pn with Pn in the image of Fn. There are a number of immediate consequences of Brody’s reparametization. Corollary (4.1.2). Suppose we have a holomorphic family of compact complex manifolds Mt parametrized by t in the open unit disk ∆. If Mt is hyperbolic for t = 0, then there exists some positive number ² such that Mt is hyperbolic for |t| < ².

Proof. If for a sequence tn → 0 there exists some nonconstant holomorphic map fn from C to Mtn , then by using a Hermitian metric on the manifold ∪t∈∆Mt and recaling the variable in C separately for each n we can assume that |dfn(0)|Mtn ,e = 1 and Brody’s reparametrization theorem would give us a nonconstant holomorphic map f from C to M0 as the limit of gn := (fn)tn ◦ T |∆ with |dg | uniformly bounded and |dg (0)| = 1, where 0 < n rn n Mtn ,e n Mtn ,e

39 tn ≤ 1 and rn monotonically increases to inﬁnity and Tn is an automorphism of the disk ∆rn of radius rn. Q.E.D. Corollary (4.1.3). Suppose M is a compact complex manifold with a Hermi- tian metric and Y is a complex hypersurface in M. If there exists a nonconstant holomorphic map f from C to M − Y , then there exists a nonconstant holomorphic map g from C to M with |dg|M,e uniformly bounded whose image is either contained in M − Y or contained in Y. Proof. By Brody’s reparametrization we get a nonconstant holomorphic map g from C to M with |dg|M,e uniformly bounded, which is the limit of gn := ftn ◦ Tn|∆rn , where 0 < tn ≤ 1 and rn monotonically increases to inﬁnity and

Tn is an automorphism of the disk ∆rn of radius rn. Suppose x = g(z) belongs to Y for some z ∈ C. Locally at x the complex hypersurface Y is deﬁned by a local holomorphic function h. Then for some open neighborhood U of z in C the holomorphic function h ◦ gn is nowhere zero on U and approaches uniformly on U to h ◦ g. By Rouch´e’stheorem h ◦ g must be identically zero on U. Hence the image of g is contained in Y . Q.E.D.

Remark (4.1.4). When the target manifold is Pn, the limit nonconstant holomorphic map f : C → Pn obtained by Brody’s reparametrization can be represented by [f0, ··· , fn] where f0, ··· , fn are entire functions of order at most two without common zeroes. The reason is as follows. From the uniform 2 boundedness of |df| on C it follows that T (r, f, ωM ) = O(r ) and by Lemma (2.1.6) f can be represented by entire functions f0, ··· , fn of order at most two without common zeroes. Remark (4.1.5). When M is a torus with Cn as universal cover, we can give M the Hermitian metric from the standard Euclidean metric on Cn and lift any holomorphic map f : C → M with supC |df| < ∞ to a holomorphic map n F : C → C with supC |dF | < ∞ which means that every component of F is a polynomial of degree at most one. This implies right away the following two statements of M. Green [Gr75]. (i) If A is an abelian variety and X is a subvariety of A which does not contain any translate of an abelian subvariety of A, then A is hyperbolic. The reason is that for any nonconstant holomorphic map f from C to A, we can regard f as a nonconstant holomorphic map from C to A and by Brody’s reparametrization get a nonconstant map g from C to X whose components as a map to the univeral cover of A are polynomials of degree at most one, contradicting the nonexistence in X of any translate of an abelian subvariety of A. (ii) If A is an abelian variety and X is an ample hypersurface of A which does not contain any translate of an abelian subvariety of A, then A−X is hyperbolic. The reason is as follows. Again for any nonconstant holomorphic map f from C to A − X, we can regard f as a nonconstant holomorphic map from C to A and by Brody’s reparametrization get a nonconstant map g from C to A whose

40 components as a map to the univeral cover of A are polynomials of degree at most one. By the nonexistence in X of any translate of an abelian subvariety of A, the image of g cannot be contained completely in X and is therefore disjoint from X. Since the image of g is the translate of some abelian subvariety of A, the ampleness of X forces the image of g to intersect X, which is a contradiction. 2. Complements of Three Qradric Curves By using Brody’s reparametrization (in particular, Remarks (4.1.3) and (4.1.4)) and the explicit computation of the characteristic function of maps de- ﬁned by the exponentials of quadratic polynomials, Detloﬀ-Schumacher-Wong [DSW93] proved the following result on the hyperbolicity of the complement of three quadrics in P2 (given in Theorem (4.2.1)).

Let Γj (j = 1, 2, 3) be three nonsingular quadrics in P2 in normal crossing 0 defined by the vanishing of polynomials Pj of degree 2. For two lines L and L 0 we say that LL is in the linear system spanned by Γ1 and Γ2 if the product of the two linear forms defining L and L0 is a linear combination of the two homogeneous quadratic polynomials defining Γ1 and Γ2. We use Lj (1 ≤ j ≤ 12) to denote distinct lines in P2. Let L1L2 and L3L4 be in the linear system spanned by Γ1 and Γ2, L5L6 and L7L8 be in the linear system spanned by Γ1 and Γ3, and L9L10 and L11L12 be in the linear system spanned by Γ2 and Γ3. Assume the following two conditions. (4.2.0.1) If there is a line tangent to two of the three quadrics, then the two points of tangency does not belong to the third quadric. (4.2.0.2) Any two of the three quadrics intersect at four distinct points which give rise to six lines. The three sets of six lines each give eigtheen distinct lines. Any point of intersection of two quadrics belongs to exactly three of the these eighteen lines and any point of P2 which does not belong to two quadrics belong to at most two of the eighteen lines.

Theorem (4.2.1). There does not exist a nonconstant entire curve f : C → P2 whose image misses Γj (j = 1, 2, 3). The computation of the characteristic function of maps defined by the exponentials of quadratic polynomials is based on the following formula of Ahlfors. λ λ Lemma (4.2.2). Let f : C → Pn be defined by [exp(a0z ), ··· , exp(anz )] with aj ∈ C for 0 ≤ j ≤ n and λ > 0. Let c be the circumference of the convex c λ polygon E spanned by {a0, ··· , an} in C. Then T (r, f) = 2π r + O(1). For this formula Shiffman [Shi79] gave the following very elegant simple proof based on Crofton’s formula. Let pθ be the orthogonal projection of C onto the line through the origin making an angle θ with the real axis. By Crofton’s formula Z π c = length(pθE)dθ θ=0

41 Z π −iθ −iθ = ( sup Re e aν − inf Re e aν )dθ θ=0 0≤ν≤n 0≤ν≤n Z 2π −iθ = ( sup Re e aν )dθ. θ=0 0≤ν≤n Hence

Z 2π 1 iθ λ c λ T (r, f) = log( max Re(aν (re )) )dθ + O(1) = r + O(1). 2π θ=0 0≤ν≤n 2π

λ Remark (4.2.3). Lemma (4.2.2) holds also when each component exp(ajz ) of f is replace by exp Pj(z) with Pj(z) being a polynomial whose highest degree λ term is ajz provided that not all a0, ··· , an are equal. Proof of Theorem (4.2.1). Suppose the contrary. We are going to derive a contradiction. By Corollary (4.1.3) and Remark (4.1.4), we can assume without loss of generality that there exists a nonconstant holomorphic map f : C → P2 − Γ1 ∪ Γ2 ∪ Γ3 such that the three homogeneous components of f are entire functions on C of order at most 2 without common zeroes.

Let Φ = [P1,P2,P3]: P2 → P2. Then T (Φ ◦ f, r) = 2T (f, r) + O(1). We 2 can write Φ ◦ f = [g0, g1, g2] with gj = exp(αjξ + βjξ + γj), αj, βj, γj ∈ C. We can assume without loss of generality that not all αj (j = 1, 2, 3) are equal, 2 2 otherwise we can factor out exp(αjξ ) and compose the map with ξ → ξ . If β1 = β2 = β3, then Φ ◦ f would be constant. Assume that the image of Φ ◦ f does not lie in a hyperplane of P2. By Ahlfors’ formula (Lemma (4.2.2) and Remark (4.2.3)),

T (r, [P ◦ f : P ◦ f]) 2|α − α | lim i j = i j , r→∞ r2 2π T (r, [P ◦ f : P ◦ f : P ◦ f]) |α − α | + |α − α | + |α − α | lim 1 2 3 = 1 2 2 3 2 3 . r→∞ r2 2π Let a and b be complex numbers so that the linear combination of the defining functions for Γ1 and Γ2 with coefficients a and b is the product of the defining functions of L1 and L2. By the Second Main Theorem we have

T (r, [P1 ◦ f, P2 ◦ f]) ≤ N(r, [P1 ◦ f, P2 ◦ f], {z0 = 0})

+N(r, [P1 ◦ f, P2 ◦ f], {z1 = 0}) + N(r, [P1 ◦ f, P2 ◦ f], {az0 + bz1 = 0}) + O(log r)

= N(r, [P1 ◦ f, P2 ◦ f], {az0 + bz1 = 0}) + O(log r).

= N(r, f, L1) + N(r, f, L2) + O(log r).

On the other hand, by the First Main Theorem we have

N(r, [P1 ◦ f, P2 ◦ f], {az0 + bz1 = 0}) ≤ T (r, [P1 ◦ f, P2 ◦ f]) + O(1).

42 Hence

T (r, [P1 ◦ f, P2 ◦ f]) = N(r, f, L1) + N(r, f, L2) + O(log r).

Adding up the six possible choices, we get

X12 N(r, f, Li) = 2(T (r, [P1 ◦ f, P2 ◦ f]) i=1

+T (r, [P2 ◦ f, P3 ◦ f]) + T (r, [P1 ◦ f, P3 ◦ f]) + O(log r).

By the Second Main Theorem we have

X12 (12 − (2 + 1))T (r, f) ≤ N(r, f, Li) + O(log r) i=1 which means that 9 T (r, Φ ◦ f) ≤ 2(T (r, [P ◦ f, P ◦ f]) + T (r, [P ◦ f, P ◦ f]) 2 1 2 2 3 +T (r, [P1 ◦ f, P3 ◦ f])) + O(log r).

After we divide both sides by r2 and taking the limit r → ∞, we get |α − α | + |α − α | + |α − α | |α − α | + |α − α | + |α − α | 9 1 2 2 3 1 3 ≤ 8 1 2 2 3 1 3 2π 2π which is a contradiction. So we conclude that the image of Φ ◦ f must lie in a hyperplane of P2, which one can easily see to contradict (4.2.0.1) or (4.2.0.2). Q.E.D.

Chapter 5. Jet Differentials with Pole Logarithmic Singularities The earliest use of jets and jet differentials for hyperbolicity problems and value distribution theory was probably by A. Bloch in his work on what is known now as Bloch’s theorem [Bl26]. In this Chapter we work only with 1- jet differentials (i.e., sections of symmetric powers of the cotangent bundle). The degree of a 1-jet differential refers to the power of the symmetric product of the cotangent bundle of which it is a section. Later in Chapters 6 and 7 we will use k-jet differentials for k greater than 1. We use the terminology that jet differentials are functions on jets. For example, a 1-jet is a tangent vector and a 1-jet differential of degree 1 is a 1-form. In this Chapter we discuss the use of 1-jet differentials with logarithmic pole singularities. Such logarithmic pole singularities are in wide use in algebraic geometry (e.g. see Deligne [D70], Iitaka [I76], and Sakai [Sa80]). They are used in value distribution theory and hyperbolicity problems by a number of authors (Noguchi [No77], Lu [Lu91], Detloff-Schumacher-Wong [DSW94]). However, all uses of logarithmic

43 pole singularities are so far only for 1-jet differentials. In general the use of special k-jet differentials with logarithmic pole singularities for k > 1 should be studied and it has so far not been done (see §2 and §3 of Chapter 6 for the definition and the use of special k-jet differentials). 1. Second Main Theorem for Jet Differentials with Logarithmic Pole Singularities

Let M be a compact complex manifold. Let Dj (1 ≤ j ≤ q) be nonsingular hypersurfaces of M in normal crossing. Suppose ω is a meromorphic differential on M with at most logarithmic pole singularities along Dj. This means that when ∪jDj is definedP by Πjzj = 0 for some local coordinate dz1 ν1 system z1, ··· , zn, ω is of the form aν ,···,ν (z1, ··· , zn)( ) ··· ν1+···+νn=k 1 n z1 dzn νn ( ) for some local holomorphic functions aν ,···,ν (z1, ··· , zn). Let λj be a zn 1 n nonnegative integer such that the pole order of ω along Dj is no more than λj λj in the sense that locally Πjzj ω is a holomorphic 1-jet differential of degree k. Let S be an ample divisor on which ω vanishes identically. Let LS be the line bundle associated with S and let θS be the positive definite curvature form of some Hermitian metric along the fibers of LS. Theorem (5.1.1) (Second Main Theorem for Jet Differentials with Logarithmic Pole Singularities). Let f : C → M be a holomorphic map ∗ such that f ω is not identicallyP zero and the image of f is not contained in ∪jDj. Then T (r, f, θS) ≤ j λjN(r, f, Dj) + o(T (r, f, θS) + log r) k.

Proof. Let sj be the canonical section of the line bundle associated to Dj. Let t be the canonical section of LS. Let ζ be the coordinate of C. Since, when ∪jDj is deﬁned by Πjzj = 0 for some local coordinate system z1, ··· , zn, we have √ µ ¶ −1 |ω|2 ∂∂¯log f ∗( )/|dζ|2k = 2π |t|2 √ 2λj 2 −1 ¯ ∗ 1 (Πj|zj| )|ω| 2k ∂∂ log(f ( 2λ 2 )/|dζ| ) 2π Πj|zj| j |t|

λj Πj zj ω and since t2 is a holomorphic 1-jet diﬀerential of degree k with value in the −1 line bundle LS , it follows that √ µ ¶ −1 |ω|2 X ∂∂¯log f ∗( )/|dζ|2k ≥ f ∗(θ − λ Zs ). 2π |t|2 S j j j

From √ −1 ¯ 1 1 Dsj ∧ Dsj ∂∂ log 2 2 = 2 (θDj − Zsj) + 2 2 2 2π (log |sj| ) log |sj| |sj| (log |sj| )

44 we obtain √ µ µ ¶ ¶ µ −1 |ω|2 X ∂∂¯log f ∗ /|dζ|2k ≥ f ∗ θ − λ Zs 2π |t|2Π (log |s |2)2 S j j j j j µ ¶ ¶ X 1 Ds ∧ Ds + (θ − Zs ) + j j . log |s |2 Dj j |s |2(log |s |2)2 j j j j By Green’s Theorem Ã ! µ µ 2 ¶¶1/k k ∗ |ω| 2 (5.1.1.1) log Ar f 2 2 2 /|dζ| 2 |t| Πj(log |sj| )

µ µ 2 ¶ ¶ 1 ∗ |ω| 2k − log f 2 2 2 /|dζ| (0) 2 |t| Πj(log |sj| ) µ µ µ 2 ¶ ¶¶ 1 ∗ |ω| 2k ≥ Ar log f 2 2 2 /|dζ| 2 |t| Πj(log |sj| ) µ 2 ¶ 1 ∗ |ω| 2k − log f ( 2 2 2 )/|dζ| (0) 2 |t| Πj(log |sj| ) √ µ 2 ¶ −1 ¯ ∗ |ω| 2k = Ir( ∂∂ log f ( 2 2 2 )/|dζ| 2π |t| Πj(log |sj| ) µ ³ X ∗ ≥ Ir f θS − λjZsj j    ¶ X 1 ´ X Ds ∧ Ds + (θ − Zs ) + I f ∗  j j  . log |s |2 Dj j r |s |2(log |s |2)2 j j j j j

2 P Dsj ∧Dsj |ω| We now compare 2 2 2 and 2 2 2 on M. By the inequality j |sj | (log |sj | ) |t| Πj (log |sj | ) for arithmetic and geometric means there exists a positive constant C such that

2 ¡ X Ds ∧ Ds ¢k |ω| j j ≥ Ck |s |2(log |s |2)2 |t|2Π (log |s |2)2 j j j j j on M. Thus Ã ! µ 2 ¶1/k ∗ |ω| 2 (5.1.1.2) log Ar f ( 2 2 2 ) /|dζ| |t| Πj(log |sj| )

  X Ds ∧ Ds ≤ − log C + log A f ∗ j j  r |s |2(log |s |2)2 j j j

45   X Ds ∧ Ds ≤ − log C + ² log r + (1 + δ) log I f ∗ j j  k. r |s |2(log |s |2)2 j j j Combining (5.1.1.1) and (5.1.1.2), we get X T (r, f, θS) ≤ λjN(r, f, Dj) + o(T (r, f, θS) + log r). j Q.E.D. Corollary (5.1.2) (Defect Relation). Suppose the holomorphic line bundle associated to Dj has a positive deﬁnite curvature form θj (1 ≤ j ≤ q). Let αj be positive numbers such that αjθS ≥ θj on M. Let f : C → M be a holomorphic map such that f ∗ω is not identically zero and the image of f is not contained q Pq Pq in ∪j=1Dj. Then j=1 αjλjδ(f, Dj) ≤ j=1 αjλj − 1. In particular, if λj = λ Pq 1 and αj = α for 1 ≤ j ≤ q, then j=1 δ(f, Dj) ≤ q − αλ . q Corollary (5.1.3) [Lu91]. If f : C → M − ∪j=1Dj is a holomorphic map, then f ∗ω is identically zero.

We now specialize to the case of the complex projective space Pn. For q ≥ 2n − 1 we are going to construct a nontrivial meromorphic 1-jet diﬀerential of high degree with at most logarithmic poles along q nonsingular hypersurfaces in normal crossing and vanishing on some ample divisor.

Theorem (5.1.4). Let q ≥ 2n − 1 and let D1, ··· ,Dq be nonsingular hypersurfaces of degree δ in Pn in normal crossing. Let `, p, and m be positive integers and s be a nonnegative integer such that p < δ and ` > 2s + p and ¡s+n−1¢¡p+n¢¡q+m−2¢ ¡mδ+p+n−1¢ ¡mδ+p−`+n−1¢ ¡m+s−n−1¢ (5.1.4.1) n−1 n q−1 > ( n − n ) n−1 . Then there exists a nontrivial meromorphic 1-jet differential of degree m + s q with at most logarithmic poles along ∪j=1Dj of pole order at most m which vanishes on a hypersurface of degree at least ` − 2s − p. Remark (5.1.5). For fixed p and `, in the inequality (5.1.4.1) the dominant term in m for left-hand side is sn−1pnmq−1 and for the right-hand side is `δn−1m2n−2. So when q ≥ 2n − 1, we can choose s, p, and ` so that sn−1pn > `δn−1 and p < δ and ` > 2s + p to guarantee that the inequality (5.1.4.1) holds. Proof of Theorem (5.1.4). The proof will be done by using the pigeon hole principle. Let Fj(z1, ··· , zn) be a polynomial of degree δ in the coordinates n Fj z1, ··· , zn of the affine open subset C of Pn which defines Dj. Let ωjk = d( ) Fk for 1 ≤ j < k < q. Then ωjk is a meromorphic 1-jet differential on Pn with logarithmic poles along Dj and Dk. Let θj = (Πν6=j,j+1Fν )(FjdFj+1 −Fj+1dFj). Consider X λ1 λn µ1 µn ν1 νq−1 (5.1.5.1) Ω = αλ1···λnµ1···µnν1···νq−1 z1 ··· zn dz1 ··· dzn θ1 ··· θq−1

46 with αλ1···λnµ1···µnν1···νq−1 ∈ C, where the summation ranges over λ1 +···+λn ≤ p, µ1 + ··· + µn = s, ν1 + ··· + νq−1 = m. When {Dj} is in normal crossing, if p < δ and not all αλ1···λnµ1···µnν1···νq−1 are zero, then Ω is not identically zero, otherwise, when we restrict to Dj for a suitable j, some polynomial in z1, ··· , zn of degree < δ would be divisible by the polynomial Fj of degree δ. Let G be a polynomial of degree ` in z1, ··· , zn whose zero-set intersects ∪jDj normally. We will use the pigeon hole principle to show that, when the inequality (5.1.4.1) holds, we can choose {αλ1···λnµ1···µnν1···νq−1 } so that Ω is divisible by G. Then q −m ω = (Πν=1Fν ) Ω is a meromorphic 1-jet diﬀerential on Pn of degree m which q is holomorphic except for logarithmic pole singularities along ∪j=1Dj and which vanishes at the inﬁnity hyperplane of order ≥ ` − 2s − p.

The number of unknowns {αλ1···λnµ1···µnν1···νq−1 } (λ1 + ··· + λn ≤ p, µ1 + ¡s+n−1¢¡p+n¢¡q+m−2¢ ··· + µn = s, ν1 + ··· + νq−1 = m) is equal to n−1 n q−1 . The β1 βn degree of the coeﬃcient of dz1 ··· dzn (β1 + ··· + βn = m + s) in Ω is a polynomial of degree at most mδ − 1 + p in z1, ··· , zn. Of the vector space V of all polynomials of degree at most N in z1, ··· , zn those elements of the form H(z1, ··· , zn)G(z1, ··· , zn) with H being a polynomial of degree at most N − ` in z1, ··· , zn form a vector subspace W whose dimension is equal to the number ¡N−`+n¢ of coeﬃcients in a polynomial of degree N − ` in z1, ··· , zn, namely n . ¡N+n¢ ¡N−`+n¢ So the quotient space V/W has dimension n − n . An element of V is divisible by G if and only if it is mapped to zero in the quotient space V/W . So the number of constraints for an element of V to be divisible by G is ¡N+n¢ ¡N−`+n¢ at most n − n . The number of constraints for Ω to be divisible by ¡N+n¢ ¡N−`+n¢ ¡m+s−n−1¢ G is ( n − n ) n−1 with N = mδ − 1 + p. When the inequality (5.1.4.1) holds, the number of unknowns exceeds the number of constraints so that we can construct a nontrivial Ω of the form (5.1.5.1) which is divisible by G. Q.E.D.

Theorem (5.1.6). Let q ≥ 2n − 1 and let D1, ··· ,Dq be nonsingular complex hypersurfaces of degree δ in Pn in normal crossing. Let `, p, and m be positive integers and s be a nonnegative integer such that p < δ and ` > 2s + p and the inequality (5.1.4.1) holds. Let ω be a nontrivial meromorphic 1-jet diﬀerential of degree m + s (given in Theorem (5.1.4)) with at most logarithmic poles along q ∪j=1Dj of pole order at most λ which vanishes on a complex hypersurface of degree at least ` − 2s − p. Let f : C → Pn be a holomorphic map such that the q ∗ image of f is not contained in ∪j=1Dj and f ω is not identically zero. Then Pq `−2s−p j=1 δ(f, Dj) ≤ q − δλ . The defect relation in Theorem (5.1.6) is far from optimal and also the nondegeneracy condition of the nonvanishing of f ∗ω is unsatisfactory. However, what we can do here in Theorem (5.1.6) gives for the ﬁrst time defect relations for 2n − 1 hypersurfaces in Pn for a general n. 2. Complements of Three Plane Curves

47 Detloff-Schumacher-Wong [DSW94] used meromorphic 1-jet differentials with only logarithmic pole singularities to show the hyperbolicity of the complement of 3 generic nonsingular plane curves with very mild conditions on their degrees. One new ingredient they introduced is the use of a Borel Lemma type argument after the use of Schwarz lemma for meromorphic 1-jet differentials with only logarithmic pole singularities.

Theorem (5.2.1). Let Cj (1 ≤ j ≤ 3) be three nonsingular curves in P2 such that the degree of C1 is at least 3 and the degrees of C2 and C3 are at least 2. Suppose the three nonsingular curves intersect only in normal crossing and if one curve Cj is a quadric then there does not exist a line which is tangential to Ck for k 6= j and which intersects Cj at precisely the two points of tangency with Ck (k 6= j). Then there is no nonconstant holomorphic map from C to 3 P2 − ∪j=1Cj. The idea of their proof is to use the theorem of Riemann Roch to produce a meromorphic 1-jet diﬀerential which is holomorphic except for logarithmic poles along ∪jCj. For the purpose of producing such a meromorphic 1-jet diﬀerential the theorem of Riemann-Roch needs weaker conditions than the pigeon hole principle. However, if one tries to do it for dimension more than two, one needs to have the vanishing of certain cohomology groups to draw conclusion from the theorem of Riemann-Roch which is less convenient than the construction given in Theorem (5.1.4) using the pigeon hole principle. 3 Proof of Theorem (5.2.1). Let C = ∪j=1Cj and the degree of Cj be bj. Let Ω1(C) be the bundle over X of meromorphic 1-forms which are holomorphic except for logarithmic poles at C. By using the exact sequence 0 → Ω1 → P2 Ω1(C) → Ω1(C)/Ω1 → 0, where Ω1 is the cotangent bundle of P , one P2 P2 2 1 2 1 P3 P computes c1(Ω (C)) − c2(Ω (C)) to be −3( j=1 bj − 4) − 6 + i

48 sheets of Φ. By Remark (4.1.4) the three homogeneous components of Φ ◦ f are given by exp(pj(z)) for some polynomial pj(z) of degree at most 2 (0 ≤ j ≤ 2). The identical vanishing of f ∗ω from Corollary (5.1.3) implies that there exist rational functions Rj(ξ1, ξ2) of ξ1, ξ2 such that

Xk j k−j Ri(ξ1, ξ2)(dξ1) (dξ2) ≡ 0 j=1 when ξj = exp(pj(z) − p0(z)) (j = 1, 2). This means that we have an equation of the type

Xk (5.2.1.1) exp(Pj(z))Qj(z) ≡ 0, j=1 where Pj and Qj are polynomials such that Q1, ··· ,Qk have no common zeroes. Consider the map ψ from C to Pk−1 defined by [exp(P1(z))Q1(z), ···, exp(Pk(z))Qk(z)]. By (5.2.1.1) the image of ψ lies in the hyperplane H := Pk { j=1 ζj = 0}, where [ζ1, ··· , ζk] is the homogeneous coordinates of Pk−1. The algebraic degeneracy of Φ◦f (and therefore of f) comes from applying the defect relation to the map ψ from C to H and the hyperplanes H ∩ {ζj = 0} in H (1 ≤ j ≤ k), because, though exp(Pj(z))Qj(z) may be zero, the number of its zeroes is finite and does not make the defect for H ∩ {ζj = 0} smaller than 1. The condition on Cj (1 ≤ j ≤ 3) rules out the existence of such an algebraic curve. Q.E.D. We can now apply the construction in Theorem (5.1.2) and the argument of the refinement of the Borel lemma argument in the proof of Theorem (5.2.1) to get the following result for the higher dimensional case.

Theorem (5.2.2). For any n ≥ 2 there exists a positive integer δn depending only on n (which can be explicitly computed from inequalities involving n) with the following property. For 2n − 1 generic nonsingular complex hypersurfaces Sj (1 ≤ j ≤ 2n − 1) of degree at least δn in Pn every holomorphic map from 2n−1 C to Pn − ∪j=1 Sj is algebraically degenerate in the sense that its image is contained in a proper subvariety of Pn. Remark (5.2.3). I would like to remark that when the number of hypersurfaces in Pn is increased to 2n + 1 and they are in general position, the hyperbolicity of the complement of their unions is known from the work of Babets [Ba84], Eremenko [E93], and Sodin [ES92]. Their work uses only properties of plurisub- harmonicity functions. They consider the lifting of the holomorphic map from C to Cn+1 and on Cn+1 they consider the maximum of certain plurisubharmonic functions constructed from the homogeneous polynomials deﬁning the hypersurfaces.

49 Chapter 6. Special 2-Jet Diﬀerentials This Chapter discusses the result in my recent joint paper [SY94] with Sai Kee Yeung and some other related research recently done jointly with him. The result of [SY94] (which is given here as Theorem (6.1.1)) gives for the ﬁrst time the conjectured hyperbolicity of the complement of a single generic nonsingular complex curve of high degree in P2 even though the degree needed is far from being optimal. Very likely the method can be extended to the case of a single generic nonsingular complex hypersurface of high degree in Pn for n > 2. However, such a generalization has not yet been done. The generalization of the part involving invariant tensors may not be straightforward. 1. Statement of Result and Outline of the Method

Theorem (6.1.1). There exists a positive integer δ0 satisfying the following. Let C be a generic nonsingular complex curve in P2 such that the degree δ of C is at least δ0. Then there is no nonconstant holomorphic map from C to P2 −C. Here generic means that C belongs to a Zariski open subset in the space of all complex curves of degree δ in P2. Moreover, the positive integer δ0 can be any positive integer which satisﬁes the following seven inequalities for some positive integers p0, s0, m0, p∗, s∗, m∗, p, s, and m.

Xm0 1 (s + 3k + 2)(s + 3k + 1)(p + 2)(p + 1) > 4 0 0 0 0 k=0 1 (s + 3m + 1)((δ − 1)(p + (s + 6m )δ ) − (δ2 − 5δ + 4)). 0 0 0 0 0 0 0 2 0 0 3(p0 + 3s0 + 9m0 + 1) + 2 < δ0. Xm∗ 1 (s + 3k + 2)(s + 3k + 1)(p + 2)(p + 1) > 4 ∗ ∗ ∗ ∗ k=0 1 `(2p + 2(s + 6m )δ + 3)(s + 3m + 1). 2 ∗ ∗ ∗ 0 ∗ ∗ p∗ < δ0 − 2 − 3(s∗ + m∗). 1 1 (p + 2)(p + 1)(s + 3m + 2)(s + 3m + 1) < (δ + 2)(δ + 1). 4 ∗ ∗ 0 0 0 0 2 0 0 Xm 1 (s + 3k + 2)(s + 3k + 1)(p + 2)(p + 1) > 4 k=0 1 `(4p + 2(s + 6m)δ + 3)(s + 3m + 1). 2 0 2p < δ0 − 2 − 3(s + m), where ` = 2m0((3m0 + s0)δ0 + p0) + 2m0(3m0 + s0)(δ0 − 1).

Remark (6.1.2). The simplest way to get an example of choices for p0, s0, m0, p∗, s∗, m∗, p, s, m, and δ0 is as follows. Use the second inequality to deﬁne p0

50 in terms of s0, m0, and δ0 as the largest integer satisfying the second inequality. The choice of s0 is not important and can be done arbitrarily. The left-hand 3 side of the first inequality in term of m0 is of the order m0 and the right-hand 2 side of the first inequality in term of m0 is of the order m0. Consider the terms in the first inequality as functions of δ0 and choose m0 large enough so that the dominant term on the left-hand side of the first inequality as a function of δ0 is of greater than the dominant term on the right-hand side as a function of δ0. Then any sufficiently large δ0 would take care of the first two inequalities. Choose p∗ as the largest integer in terms of δ0, s∗, and m∗ so that the fourth and fifth inequalities are satisfied. In the same way that one handles the first inequality, we use the strategy of dominant terms as functions of δ0 to handle the third inequality so that we can fix the choices for p∗, s∗, and m∗. Again we use the same strategy to fix the choices for p, s, and m by first defining p as the largest integer satisfying the seventh inequality as a function of δ0, s, and m. The main idea of the proof of Theorem (6.1.1) is as follows. Let the generic nonsingular complex curve C be defined by a polynomial f(x, y) of degree δ, where (x, y) is an inhomogeneous coordinate system for the affine part C2 of P2. Let X be the branched cover of P2 with branching of degree δ along C. Any holomorphic map ϕ from C to P2 − C can be lifted up to a holomorphic mapϕ ˜ from C to X. We construct explicitly holomorphic 2-jet differentials on the branched cover X which vanish at the infinity curve of X. The construction imitates the explicit construction of holomorphic 1-forms on compact Riemann surfaces of genus at least two which are plane curves. In the case of a Riemann surface if the plane curve is defined by a polynomial g(x, y), one can construct a holomorphic 1-form dx = − dy and others by multiplying it by polynomials gy gx of low degree. The key point in that construction for holomorphic 1-forms on the Riemann surface is the division by gy which takes away the pole order at δ infinity. In our case X is defined by t = f(x, y) in P3 with inhomogeneous coordinates (x, y, t) for the affine part of P3. We construct special holomorphic 2-jet differentials Φ on X by using expressions involving fx, fy, fxx, fxy, fyy. The key point is to construct an expression which is divisible by fy and by a sufficiently high power of t = f 1/δ to take away the pole order at infinity. In our construction we can also impose the condition that such special holomorphic 2-jet differentials vanish on an ample curve of X. A standard Schwarz lemma argument shows that the pullback to C, via a holomorphic mapϕ ˜ from C to X, of any such special holomorphic 2-jet differential Φ which vanishes on an ample curve of X must be identically zero. Thus the image ofϕ ˜ satisfies a second order differential equation which is defined by the 2-jet differential Φ. If we can construct three such special 2-jet differentials which are independent in an appropriate sense, we can eliminate the first and second-order derivatives of the image ofϕ ˜ from the three second-order differential equations and conclude that ϕ˜ must be constant. However, we are unable to produce three such independent special 2-jet differentials. Our construction of a special holomorphic 2-jet dif-

51 ferential depends on the choice of the affine coordinate system (x, y). By using another affine coordinate system obtained by switching x and y, we prove that we are able to come up with two independent special 2-jet differentials so that the image ofϕ ˜ satisfies a first-order differential equation. We then prove thatϕ ˜ is algebraically degenerate in the sense that the image ofϕ ˜ lies in an algebraic curve of X. Ifϕ ˜ is not algebraically degenerate, by the method of explicit construction of special 2-jet differentials by using different affine coordinate systems of P2 we produce a meromorphic special 2-jet differential on X which, on the Zariski closure of the 1-jet image ofϕ ˜, is holomorphic and not identically zero. To such a special 2-jet differential, we can still apply the Schwarz lemma and get enough independent meromorphic 1-jet differentials vanishing on the 1-jet image ofϕ ˜ to get a contradiction to the algebraic nondegeneracy of ϕ. 2. Construction of Special 2-Jet Differentials Let (z, w) be local holomorphic coordinates. By a special 2-jet differential of degree m we mean an expression of the form

Xm 3Xk+s j 3k+s−j 2 2 m−k akj(z, w)(dz) (dw) (d zdw − dzd w) . k=0 j=0

The special 2-jet differential is holomorphic (respectively meromorphic) if each akj(z, w) is a holomorphic (respectively meromorphic) function of z and w. Let p be a positive integer and s be a nonnegative integer. We are going to construct a 2-jet differential Φ of degree m on X of the form Xm 2(m−k) 2 2 m−k Φ = ωs+3kf (d fdx − d xdf) , k=0 where X ν0 ν1 ν2 ωµ = aν0ν1ν2 (x, y)(df) (fdx) (fdy) ν0+ν1+ν2=µ and aν0ν1ν2 (x, y) is a polynomial in x and y of degree ≤ p. We are going to choose the polynomials aν0ν1ν2 (x, y) so that Φ is divisible by fy. Then we will −N −1 conclude that t fy Φ is a holomorphic 2-jet differential on X when certain inequalities involving p, s, δ, m, and N are satisfied, where tδ = f(x, y). This is done by regarding the coefficients of the polynomials aν0ν1ν2 (x, y) as unknowns and counting the number of linear equations corresponding to divisibility of Φ by fy and solving the linear equations when the number of unknowns exceeds the number of equations. The number of equations is (s + 3m + 1)((δ − 1)(p + 1 2 (s + 6m)δ) − 2 (δ − 5δ + 4)). The number of unknowns is

Xm 1 (s + 3k + 2)(s + 3k + 1)(p + 2)(p + 1). 4 k=0

52 The computation of the number of equations is similar to that given in §1 of Chapter 5. When the number of unknowns exceeds the number of equations, for a generic f we can solve the linear equations and the solutions will be rational functions of the coeﬃcients of f. We summarize the result in the following lemma. Lemma (6.2.1). To be able to construct a 2-jet diﬀerential Φ of the form

Xm 2(m−k) 2 2 m−k ωs+3kf (d fdx − d xdf) , k=0 where X ν0 ν1 ν2 ωµ = aν0ν1ν2 (x, y)(df) (fdx) (fdy) ν0+ν1+ν2=µ and aν0ν1ν2 (x, y) is a polynomial in x and y of degree ≤ p, it suﬃces to have the following inequalities p < δ − 1 and

Xm 1 (s + 3k + 2)(s + 3k + 1)(p + 2)(p + 1) > 4 k=0 1 (s + 3m + 1)((δ − 1)(p + (s + 6m)δ) − (δ2 − 5δ + 4)). 2

Moreover, for a generic f the coefficients of aν0ν1ν2 (x, y) are rational functions of the coefficients of f. When we solve the system of homogeneous linear equations for the coefficients of aν0ν1ν2 (x, y), we choose a square submatrix A with nonzero determinant in the matrix of the coefficients of the system of homogeneous linear equations so that A has maximum size among all square submatrices with nonzero determinants and then we apply Cramer’s rule to those equations whose coefficients are involved in A to solve for the the coefficients of aν0ν1ν2 (x, y). When we do this process, we can regard the coefficients of the system of homogeneous linear equations as functions of the coefficients of f. The square submatrix A has maximum size among all square submatrices whose determinants are not identically zero as functions of the coefficients of f. A sufficient condition for the genericity of f involved in this process is that the point represented by the coefficients of f is outside the zero-set of A when A is regarded as a function of the coefficients of f. We factor the Φ obtained in Lemma (6.1.2) as a product of irreducible factors as a polynomial of the variables x, y, dx, dy, d2xdy − d2ydx and one of the −Ny −1 factors divided by t fy for some Ny would be a holomorphic special 2-jet differential Φ1 on X. By switching the roles of x and y in the construction we −Nx −1 get a holomorphic special 2-jet differential Φ2 on X by division by t fx . The holomorphic special 2-jet differential Φ1 is not equal to a rational function

53 times Φ1 for a generic C. For this step we need the second inequality in The- orem (6.1.1) and we need also the result of Sakai [Sa78] on the nonexistence of nontrivial global holomorphic sections of symmetric powers of the cotangent bundle of X. 3. Schwarz Lemma for Special Jet Differentials Let M be a compact complex manifold of complex dimenison n with an ample line bundle L which is the line bundle associated to a nonsingular very ample divisor D. Let t be the element of Γ(M,L) whose zero-set is precisely D. Let Φ be a holomorphic 2-jet differential on M of the following special form Xm ρ σ τ m−j 2 µ ν ν µ j Φ = (ãjρστ (x)dx dx dx ) (d x dx − dx dx ) , j=0 where x = (xµ) is a local holomorphic coordinate system of M and the convention of summing over repeated indices is used. Assume that Φ vanishes on D so that we can write Xm ρ σ τ m−j 2 µ ν ν µ j Φ = t (ajρστ (x)dx dx dx ) (d x dx − dx dx ) j=0

−1 with ajρστ (x) = t a˜jρστ (x) locally holomorophic. There exists some positive 1 ` 1 ` integer ` such that a basis s1, ··· , sq of Γ(M,ΩM ⊗ L ) generates ΩM ⊗ L at every point of M. Let e−ψ be a smooth metric for L whose curvature form is −ψ 2 α strictly positive. Then e |t| is well-defined. Write sj = sjα(ζ)dx . Let η1 and η2 be two positive numbers with η2 > η1` and η1 +3η2m = 1. We introduce the symbol X ψ(η2−η1`) 2 η1 −1 2η2 Ψ = e ( |sj| ) |t Φ| j so that for any holomorphic map ϕ from an open subset U of C to M the symbol ϕ∗Ψ denotes the scalar function on U which satisfies the following equation X ∗ ψ(η2−η1`) ∗ 2 η1 ∗ −1 2η2 ∗ 2η1+6η2 ϕ (e )( |ϕ sj| ) |ϕ (t Φ)| = (ϕ Ψ)|dz| , j where z is the coordinate of C. In other words, in the definition of the symbol Ψ the norm is taken only with respect to the metric of L but not with respect to the metric of the cotangent bundle and not with respect to metric of the bundle of special 2-jet differentials. So the symbol Ψ does not represent a scalar function on M but represents an expression still involving dxµ and d2xµdxν − d2xν dxµ but no longer involving the line bundle L. Lemma (6.3.1) (Schwarz Lemma). If ϕ is a holomorphic map from C to M, then ϕ∗Φ = 0.

54 Lemma (6.3.1) follows from Lemma (6.3.2) below by using Green’s Theorem, the Calculus Lemma, the concavity of the logarithmic function, and the sub mean value property of subharmonic function as in the proof of Lemma (1.1.1). ∗ Lemma (6.3.2). There exists a positive number ²0 such that ∂z∂z log ϕ Ψ ≥ ∗ ²0ϕ Ψ for every holomorphic map ϕ from an open subset U of C to M, where z is the coordinate of C.

Proof. The symbols Aj and ²j below will denote positive constants. When we apply ∂z∂z log to both sides of X ∗ ψ(ϕ)(η2−η1`) α 2 η1 ϕ Ψ = e ( |sjα(ϕ)ϕz | ) · j Xm ρ σ τ m−j µ ν ν µ j 2η2 ·| (ajρστ (ϕ)ϕzϕz ϕz ) (ϕzzϕz − ϕzzϕz ) | , j=0 we generously drop the result from the last factor of the right-hand side and get X ∗ α 2 (6.3.2.1) ∂z∂z log ϕ Ψ ≥ (η2 − η1`)∂z∂zψ(ϕ) + η1∂z∂z log( |sjα(ϕ)ϕz | ). j

P α 2 1/2 Let |ϕz| denote ( α |ϕz | ) and |ϕzz ∧ ϕz| denote X µ ν ν µ 2 1/2 ( |ϕzzϕz − ϕzzϕz | ) . µ6=ν

We now use

2 (6.3.2.2) ∂z∂zψ(ϕ) ≥ ²1|ϕz| and X α 2 −4 2 2 (6.3.2.3) ∂z∂z log( |sjα(ϕ)ϕz | ) ≥ ²3|ϕz| |ϕzz ∧ ϕz| − A2|ϕz| . j

Choose a positive number ²4 such that ²4 is less than the minimum of η1 and (η2 − η1`)²1/A2. Putting the inequalities (6.3.2.1), (6.3.2.2), and (6.3.2.3) together, we have

∗ 2 −4 2 ∂z∂ log ϕ Ψ ≥ ²5|ϕz| + ²3²4|ϕz| |ϕzz ∧ ϕz| .

Finally we get an upper bound for ϕ∗Ψ. For the ﬁrst factor of ϕ∗Ψ, we have a ψ(ϕ)(η2−η1`) ∗ bound e ≤ A3. For the other two factors of ϕ Ψ, we have X Xm α 2 η1 ρ σ τ m−j µ ν ν µ j 2η2 ( |sjα(ϕ)ϕz | ) | (ajρστ (ϕ)ϕzϕz ϕz ) (ϕzzϕz − ϕzzϕz ) | j j=0

55 Xm 2η1 3(m−j) j 2η2 ≤ A4|ϕz| ( |ϕz| |ϕzz ∧ ϕz| ) j=0 Xm 2η1+6η2(m−j)+4jη2 −2 2jη2 ≤ A5 (|ϕz| )(|ϕz| |ϕzz ∧ ϕz|) j=0 2 −2 2 ≤ A6(|ϕz| + (|ϕz| |ϕzz ∧ ϕz|) ), where the last inequality is obtained by using the geometric and arithmetic means inequality and the identity 2η1+6η2(m−j)+4jη2+2jη2 = 2(η1+3η2m) = 2. The lemma now follows with A3A6²0 less than the minimum of ²5 and ²3²4. Q.E.D.

In §2 we have constructed holomorphic special 2-jet diﬀerentials Φ1 and Φ2 on X. We apply the Schwarz Lemma (Lemma (6.3.1)) to conclude that the pullbacks of Φj (j = 1, 2) by a holomorphic map from C to X are identically zero. Ny Let the meromorphic 1-jet diﬀerential ω on P2 be the resultant of t fyΦ1 and Nx 2 2 t fxΦ2 as polynomials of d xdy − d ydx (i.e., ω is obtained from eliminating 2 2 Ny Nx d xdy − d ydx from t fyΦ1 and t fxΦ2). Then the pullback to C of ω by any holomorphic map from C to P2 − C must be identically zero. 4. The Proof of Algebraic Degeneracy

Let ϕ : C → P2 − C be a nonconstant holomorphic map, where C is the smooth curve in P2 defined by the polynomial f(x, y) of degree δ in the affine coordinates (x, y) of P2. In this section we are going to prove that ϕ is algebraically degenerate in the sense that the image of ϕ is contained in a complex (algebraic) curve of P2. Without loss of generality we assume that ϕ is not algebraically degenerate and we are going to derive a contradicion. We can lift ϕ to a nonconstant holomorphic mapϕ ˜ : C → X, where π : X → P2 is the branched cover with branching of order δ along C. From §2 we have two independent irreducible holomorphic special 2-jet differentials Φ1 and Φ2 on X which vanish on the infinity curve. By the Schwarz lemma proved in §3, the pullback of Φj byϕ ˜ is identically zero on C for j = 1, 2. Each Φj corresponds 2 2 to a meromorphic special 2-jet differential Fj(x, y, dx, dy, d xdy − dxd y) for 2 2 j = 1, 2. The pullback of Fj(x, y, dx, dy, d xdy − dxd y) by ϕ is identically zero on C for j = 1, 2. Let h(x, y, dx, dy) be the resultant obtained by eliminating the 2 2 2 2 variable d xdy −dxd y from the two polynomials Fj(x, y, dx, dy, d xdy −dxd y) (j = 1, 2). Then the pullback of h(x, y, dx, dy) by ϕ is identically zero on C. We factor h(x, y, dx, dy) into irreducible factors and the pullback of at least one of the irreducible factors by ϕ is identically zero on C. By replacing h(x, y, dx, dy) by that factor, we can assume that we have a polynomial h(x, y, dx, dy) in x, y, dx, and dy which is not identically zero and is homogeneous in dx and dy such that the pullback of h(x, y, dx, dy) by ϕ is identically zero on C. Let ` be the degree of h(x, y, dx, dy) in x and y and q be the degree of h(x, y, dx, dy) in dx and dy. Since ϕ is assumed to be algebraically nondegenerate, we know that q

56 is positive. Let Vh be the irreducible surface in the projectivization P(TP2 ) of the cotangent bundle TP2 of P2 defined by the zero-set of h(x, y, dx, dy). The image of P(TC) under ϕ is Zariski dense in Vh. We would like to count the degrees involved. The holomorphic special 2-jet differentials Φ1 and Φ2 on X are constructed with a choice of s, m, and p satisfying certain inequalities. Roughly the main inequality is that p2m is of the same order as δ2. Because we would like to use the numbers s, m, and p in another context, we are going to call them s0, m0, and p0 instead of s, m, and p. The degree ` of h(x, y, dx, dy) is no more than 2m0((3m0 + s0)δ + p0) + 2m0(3m0 + s0)(δ − 1) and the homogeneous degree q of h(x, y, dx, dy) is no more than 2m0(3m0 + s0). By using possibly another affine coordinate system (x, y), we can assume without loss of generality that in the expression

q µ ¶ X dy α h(x, y, dx, dy)/dxq = h (x, y) α dx α=0 the polynomial hq(x, y) in x and y is not identically zero. Consider the equation ³ ´α Pq dy α=0 hα(x, y) dx = 0. By diﬀerentiating with respect to x, we obtain

q µ ¶ X dy dy α (∂ h (x, y) + (∂ h (x, y)) ) x α y α dx dx α=0 q µ ¶ X dy α−1 d2ydx − dyd2x + h (x, y)α = 0 α dx dx3 α=0 and we can write 2 2 d ydx − dyd x = Ψh(x, y, dx, dy), where Ã ! q −1 X dy Ψ (x, y, dx, dy) = dx3 − h (x, y)α( )α−1 h α dx α=0 Ã ! q µ ¶ X dy dy α (∂ h (x, y) + (∂ h (x, y)) ) . x α y α dx dx α=0

Deﬁnition (6.4.1). A meromorphic special 2-jet diﬀerential

Φ = Φ(x, y, dx, dy, d2ydx − dyd2x) is said to vanish identically on Vh if

Φ(x, y, dx, dy, Ψh(x, y, dx, dy))

57 vanishes identically on Vh. Φ When Φ is identically zero on Vh, we will also say that dx` vanishes identically on Vh. This is justiﬁable, because dx cannot vanish identically on Vh due to the algebraic nondegeneracy of ϕ. Note that the algebraic nondegeneracy of ϕ also d2fdx−dfd2x d df implies that dx3 = dx ( dx ) is not identically zero on Vh. We introduce this deﬁnition so that if the pullback of Φ by ϕ is identically zero on C, then Φ is identically zero on Vh. Let 2 2 II = fxxdx + 2fxydxdy + fyydy .

Lemma (6.4.2). Let ωα(x, y, fdx, fdy, df) be polynomials of degree at most p in x and y and of homogeneous degree at most s + 3(m − α) in dx and dy Pm 2 α (0 ≤ α ≤ m). Assume that p < δ − 2 − 3(s + m). If α=0 ωα(f II) vanishes identically on Vh, then ωα is identically zero on Vh for 0 ≤ α ≤ m. 2 Proof. Note that ωα and II (and of course f II) are independent of the affine coordinates x and y. We take now an arbitrary affine coordinate sys- Pm 2 α tem (x, y). From the identical vanishing of α=0 ωα(f II) on Vh we con- Pm 2 α d2fdx−dfd2x clude that α=0 ωα(f II) is divisible by fy on Vh. Since dx = ³ ´α d2ydx−dyd2x Pm 2 d2fdx−dfd2x fy dx + II, it follows that α=0 ωα f dx is divisible by 2 2 fy on Vh (in the sense that the the second-order differential d ydx − dyd x is considered as a transcendental indeterminate). Consider the meromorphic Pm 2 2 2 α N ` special 2-jet differential Θ = ( α=0 ωα(f (d fdx − dfd x)) )/(t fyh ) on X, where tδ = f(x, y) and N is the minimum of (s+3(m−α))(δ−1)+2αδ+α(δ−2) for 0 ≤ α ≤ m. Since p < δ −2−3(s+m), though Θ is only meromorphic on X, ∗ yet Θ is holomorphic on π (Vh) ⊂ P(TX ) and vanishes at the curve of X which is above the infinity curve of P2. Since the pullback of the defining function h of Vh by ϕ is identically zero, from an obvious modification of the proof of the Schwarz Pm 2 d2fdx−dfd2x α Lemma in §3 we conclude that the pullback of α=0 ωα(f dx ) by ϕ Pm 2 d2fdx−dfd2x α is identically zero on C and thus α=0 ωα(f dx ) vanishes identically on Vh. Since the affine coordinate system (x, y) used is arbitrary, we can use another affine coordinate system obtained by replacing x by x + λy with λ ∈ C indeterminate. Then µ ¶ Xm d2fdx − dfd2x + λ(d2fdy − dfd2y) α ω f 2 α dx + λdy α=0 vanishes identically on Vh. We differentiate it with respect to λ and conclude that µ ¶ µ ¶ Xm d2fdx − dfd2x + λ(d2fdy − dfd2y) α−1 df(d2xdy − dxd2y) αω f 2 α dx + λdy (dx + λdy)2 α=0

58 vanishes identically on Vh. Set λ = 0 and multiply every term by the same factor µ ¶ µ ¶ d2fdx − dfd2x . df(d2xdy − dxd2y) . dx dx2 It follows that µ ¶ Xm d2fdx − dfd2x α αω f 2 α dx α=0 vanishes identically on Vh. Again this is true for all aﬃne coordinate system (x, y). Now we can repeat this argument with the coordinate x replaced by x+λy for some indeterminate λ. It is the same as applying the above argument with ωα replaced by αωα. Inductively for every positive integer k we have the identical vanishing of µ ¶ Xm d2fdx − dfd2x α αkω f 2 α dx α=0

k on Vh. Since the Vandemonde determinant det(α )1≤α,k≤m is nonzero, it follows that µ ¶ d2fdx − dfd2x α ω f 2 α dx vanishes identically on Vh for 0 ≤ α ≤ m. So we conclude that ωα vanishes identically on Vh for 0 ≤ α ≤ m. Q.E.D. Lemma (6.4.3). Assume that

Xm 1 1 (s+3k +2)(s+3k +1)(p+2)(p+1) > `(2p+2(s+6m)δ +3)(s+3m+1). 4 2 k=0

Then there exist polynomials ωα(x, y, fdx, fdy, df) of degree at most p in x, y and of homogeneous degree at most s + 3(m − α) in dx and dy which are not all Pm 2 α identically zero such that α=0 ωα(f II) is divisible by h, which in particular Pm 2 α implies that α=0 ωα(f II) vanishes identically on Vh. Proof. We prove this by the method of undetermined coeﬃcients. We count the number of unknowns in the choice of ωα(x, y, dx, dy) (0 ≤ α ≤ m) and, as in §2 we end up with

Xm 1 (s + 3k + 2)(s + 3k + 1)(p + 2)(p + 1) 4 k=0 as the total number of unknowns for all possible choices of ωα(x, y, fdx, fdy, df) (0 ≤ α ≤ m). We now compute the number of equations. We consider the vector space W (j, k) of all polynomials g(x, y, dx, dy) of degree at most j in x and y and

59 homogeneous of degree at most k in dx and dy. The dimension of W (j, k) is 1 2 (j + 2)(j + 1)(k + 1). Then the dimension of the quotient space W (j, k) by hW (j − `, k − q) is no more than 1 1 (j + 2)(j + 1)(k + 1) − (j − ` + 2)(j − ` + 1)(k − q + 1) 2 2 1 1 ≤ (j + 2)(j + 1)(k + 1) − (j − ` + 2)(j − ` + 1)(k + 1) 2 2 1 ≤ `(2j + 3)(k + 1), 2 where hW (j − `, k − q) is the image of W (j − `, k − q) → W (j, k) deﬁned by Pm 2 α multiplication by h(x, y, dx, dy). The degree of α=0 ωα(f II) in x and y is at most p + (s + 6m)δ and its degree in dx and dy is at most s + 3m. Thus the 1 total number of equations is at most 2 `(2p + 2(s + 6m)δ + 3)(s + 3m + 1). By assumption

Xm 1 1 (s+3k +2)(s+3k +1)(p+2)(p+1) > `(2p+2(s+6m)δ +3)(s+3m+1). 4 2 k=0 So the number of unknowns is more than the number of equations and we conclude that there exist polynomials ωα(x, y, fdx, fdy, df) of degree at most p in x, y and of homogeneous degree at most s + 3(m − α) in dx and dy which Pm 2 α are not all identically zero such that α=0 ωα(f II) is divisible by h. Q.E.D.

Proposition (6.4.4). Every holomorphic map ϕ from C to P2 − C is algebraically degenerate. Proof. Let us ﬁrst do the proof with the assumption that in Lemma (6.4.2) we have the stronger conclusion that ωα vanishes identically instead of just vanishing identically on Vh. Then we will explain how to handle the diﬃculty which arises without such an assumption. Now we choose m, p and s such that

Xm 1 1 (s + 3k + 2)(s + 3k + 1)(p + 2)(p + 1) > `(2p + 2(s + 6m)δ + 3)(s + 3m + 1) 4 2 k=0 with ` = 2m0((3m0 + s0)δ + p0) + 2m0(3m0 + s0)(δ − 1) and p < δ − 2 − 3(s + m). This is possible, because the dominant term on the left-hand side is p2m3 and the dominant term on the right-hand side is cδ2ms, where c depends on m0, s0, and p0.

By Lemma (6.4.3) we can ﬁnd polynomials ωα(x, y, fdx, fdy, df) of degree at most p in x, y and of homogeneous degree at most s + 3(m − α) in dx and

60 Pm 2 α dy which are not all identically zero such that α=0 ωα(f II) is divisible by Pm 2 α h, which in particular implies that α=0 ωα(f II) vanishes identically on Vh. By Lemma (6.4.2) with the additional assumption about its conclusion we conclude that ωα vanishes identically for 0 ≤ α ≤ m, which is a contradiction. We now explain how to handle the diﬃculty which arises without the additional assumption about the conclusion of Lemma (6.4.2). We ﬁrst make the following two observations. Observation 1. Suppose f(x, y, λ) is a polynomial in x and y with a tran- df scendental parameter λ. Suppose ωj(x, y, dx, dy, f )(j = 1, 2) are two distinct df irreducible polynomials in the 5 variables x, y, dx, dy, f which are homogeneous df in the variables dx, dy, and f . Then for a generic λ ∈ C the common zero-set df 2 of the two polynomials ωj(x, y, dx, dy, f )(j = 1, 2) in the threefold C × P1 2 with coordinates (x, y) ∈ C and [dx, dy] ∈ P1 is of complex dimension one. Observation 2. Suppose g(u, v) is a polynomial in u and v with g(0, 0) 6= 0. If g(u, v) is irreducible as a polynomial in u and v, then g(xy, y) is irreducible as a polynomial in x and y. Geometrically this observation is just a consequence of considering the blowup at the origin of C2 and using the decomposition of a surface into its irreducible components. Algebraically the observation follows from the unique factorization property of the polynomial ring. We have to worry about the following situation. There exists some poly- df nomial ωα(x, y, dx, dy, f ) of degree at most p in x and y and of homogeneous df df degree at most s+3(m−α) in dx, dy, and f such that ωα(x, y, dx, dy, f ) is not df identically zero but is identically zero in Vh. After factoring ωα(x, y, dx, dy, f ) df df as a polynomial of the 5 variables x, y, dx, dy, f , we obtain a polynomial ω(x, y, dx, dy, f ) df df of the 5 variables x, y, dx, dy, f such that ω(x, y, dx, dy, f ) is irreducible as a df polynomial of the 5 variables x, y, dx, dy, f and is not identically zero but is 0 df identically zero on Vh. Let p be the degree of ω(x, y, dx, dy, f ) in x and y.

Assume δ is so large that we have positive integers m∗, p∗, and s∗ such that Xm∗ 1 (6.4.4.1) (s + 3k + 2)(s + 3k + 1)(p + 2)(p + 1) 4 ∗ ∗ ∗ ∗ k=0 1 > `(2p + 2(s + 6m )δ + 3)(s + 3m + 1) 2 ∗ ∗ ∗ ∗ ∗ with ` = 2m0((3m0 + s0)δ + p0) + 2m0(3m0 + s0)(δ − 1) and p∗ < δ − 2 − 3(s∗ + m∗) and 1 1 (6.4.4.2) (p + 2)(p + 1)(s + 3m + 2)(s + 3m + 1) < (δ + 2)(δ + 1). 4 ∗ ∗ 2

61 0 Case 1. p > p∗. Then we can apply the preceding argument with p and m df replaced by p∗ and m∗ and get an irreducible polynomial ω∗(x, y, dx, dy, f ) df in the 5 variables x, y, dx, dy, f with degree no more than p∗ in x and y and df with homogeneous degree no more than s∗ + 3m∗ in dx, dy, and f so that df ω∗(x, y, dx, dy, f ) is not identically zero, but is identically zero in Vh. By con- df df sidering the two 1-jet differentials ω(x, y, dx, dy, f ) and ω∗(x, y, dx, dy, f ) with logarithmic pole singularities along C and considering the pole orders of x, y, dx, dy at the infinity curve and their different degrees in x and y and their irreducibility, we can arrive at a contradiction for a generic f. 0 1 0 0 Case 2. p ≤ p∗. Since the upper bound 4 (p + 2)(p + 1)(s + 3(m − α) + 2)(s + 3(m − α) + 1) for the total number of degrees of freedom in the choice df of coefficients in ω(x, y, dx, dy, f ) is less than the total number of degrees of 1 freedom 2 (δ + 2)(δ + 1) in the choice of coefficients in f, we can use a one- parameter family f(x, y, λ) so that each member of the one-parameter family df has the same ω(x, y, dx, dy, f ). After a coordinate transformation given by an df element of P GL(3, C) we can assume that the degree of ω(x, y, dx, dy, f ) in x and y is positive. By using another affine coordinate x and y we can assume df without loss of generality that ω(0, 0, dx, dy, f ) is not identically zero. df We now repeat the procedure of constructing ωα(x, y, dx, dy, f ) with Xm α ωα(x, y, fdx, fdy, df)II α=0

df divisible by h except that this time we require in addition that ωα(x, y, dx, dy, f ) 0 df is of the form ωα(u, v, dx, dy, f ) with u = xy and v = y for some polynomial 0 df ωα(u, v, dx, dy, f ). In order to be able to apply the method of undetermined coeﬃcients we need to replace p by 2p, because to make the degree in u and v at most p we need the condition that the degree in x and y is at most 2p. Now we get the algebraic nondegeneracy by using Observations 1 and 2 when δ satisﬁes the inequalities (6.4.4.1), (6.4.4.2) and (6.4.4.3) for some m, p, s, m∗, and p∗.

Xm 1 (6.4.4.3) (s + 3k + 2)(s + 3k + 1)(p + 2)(p + 1) 4 k=0 1 > `(4p + 2(s + 6m)δ + 3)(s + 3m + 1) 2 with ` = 2m0((3m0 + s0)δ + p0) + 2m0(3m0 + s0)(δ − 1) and 2p < δ − 2 − 3(s + m). Q.E.D.

62 5. The Deformation Approach There is another approach to the problem of hyperbolicity of the complement of a generic nonsingular plane curve C of high degree. This approach, when implemented successfully, would give hyperbolicity with a degree much closer to the optimal conjectured one of 5. The approach is to consider first a special curve C0 in P2 of high degree defined by an equation with only very few nonzero terms. We can explicitly construct holomorphic special 2-jet differentials on the branched cover XC0 over P2 with branching along C0. Then we try to use those explicitly constructed holomorphic special 2-jet differentials to show that the bundle of special 2-jet differentials over XC0 is positive. If one has such positivity, then on XC for a generic C we can get enough independent holomorphic special 2-jet differentials of sufficiently high degree to get the hyperbolicity of the complement of a generic nonsingular plane curve C of high degree. For this approach the degree required would be the same as the degree of the special curve C0 which can be chosen to be much closer to the optimal conjectured degree. Unfortunately so far the results computed for a number of simple special curves C0 do not turn out to be good enough to implement completely this approach. In this section we discuss some computations done for this approach and the difficulties encountered in attempts to implement it. Let us first look at the construction of holomorphic special 2-jet differentials on XC when C is defined by a polynomial f(x, y) with high degree δ but with only very few nonzero terms. This construction is related to meromorphic connections of low pole order for the tangent bundle discussed in §4 of Chapter 3. Let tδ = f(x, y). We start out with the meromorphic special 2-jet differential 2 2 d fdx − d xdf on P2. Its pole order at the infinity line is of the order at most δ−2 δ−1 δ+4. After we pull it back to XC we can divide it by t , because df = δt dt and d2f = δ(δ −1)tδ−2dt2 +δtδ−1d2t. So we end up with a meromorphic special 2-jet differential on XC of pole order at infinity at most 6. If we can modify d2fdx − d2xdf so that it vanishes on some curve, then we can divide it by the defining equation of that curve and eliminate the pole order of the constructed meromorphic special 2-jet differential on XC at infinity and we get a holomor- 2 2 phic special 2-jet differential on XC . For that purpose we expand d fdx−d xdf and get (6.5.1.1)

2 2 2 2 2 2 (6.5.1.1) d fdx−d xdf = fy(d ydx−d xdy)+(fxxdx +2fxydxdy+fyydy )dx.

The first term on the right-hand side is already divisible by fy. If we can modify the second term on the right-hand side to make it divisible by fy, then we can get a holomorphic special 2-jet differential on XC . So we want fxx, fxy, and fyy to be divisible by fy after modification. The key observation is that fxdx = df −fydy means that we can modify fxdx by a term df which is divisible δ−1 by t so that we end up with a term divisible by fy. So we should try to write fxx, fxy, fyy as linear combinations of fx and fy. We can also throw in f in the

63 linear combination because f is divisible by tδ. This leads us to consider the equations

fxx = a1fx + b1fy + c1f,

fxy = a2fx + b2fy + c2f,

fyy = a3fx + b3fy + c3f, where aj, bj, cj are rational functions. Note that this set of equations is the same used in §4 of Chapter 3 for the construction of meromorphic connection k Γi,j of low pole order for the tangent bundle. Let g be the common denominator of aj, bj, cj. From (6.5.1.1) we get

2 2 2 2 2 d fdx − d xdf = fy(d ydx − d xdy) + ((a1fx + b1fy + c1f)dx 2 +2(a2fx + b2fy + c2f)dxdy + (a3fx + b3fy + c3f)dy )dx.

Using fxdx = df − fydy, we get

2 2 2 2 2 d fdx − d xdf = fy(d ydx − d xdy) + (a1(df − fydy) + b1fydx + c1fdx)dx 2 +2(a2(df − fydy) + b2fydx + c2fdx)dxdy + (a3(df − fydy) + b3fydx + c3fdx)dy .

Moving to the left-hand side all terms on the right-hand side which do not contain the factor fy, we get

2 2 2 2 2 2 d fdx − d xdf − df(a1dx + 2a2dxdy + a3dy ) − fdx(c1dx + 2c2dxdy + c3dy ) ¡ 2 2 2 = fy (d ydx − d xdy) + (−a1dy + b1dx)dx 2¢ +2(−a2dy + b2dx)dxdy + (−a3dy + b3dx)dy .

Note that this meromorphic special 2-jet diﬀerential is the same as

2 α 2 α α β γ β γ α d fdx − (d x + Γβγ dx dx )df − f(Cβγ dx dx )dx ,

α α when x = x and when one uses the language of the connection Γβγ for the 2 α α β γ tangent bundle in §4 of Chapter 3. The expression d x + Γβγ dx dx there is simply the second-order covariant diﬀerential of xα.

Let g be the common denominator of a1, a2, a3, c1, c2, c3. We end up with the holomorphic special 2-jet diﬀerential

2−δ −1 ¡ 2 2 2 2 t fy g d fdx − d xdf − df(a1dx + 2a2dxdy + a3dy ) 2 2 ¢ −fdx(c1dx + 2c2dxdy + c3dy ) which vanishes at inﬁnity if the degree of g is less than δ − 5, the degree of gaj is less than δ − 8, and the degree of gcj is less than δ − 9 for 1 ≤ j ≤ 3.

64 We consider the special curve of degree 3n (with n suﬃciently large) given by the equation f(x, y) = x3n + y3n + xnyn + 1 = 0 in aﬃne coordinates x, y. We have 3(3n − 1)x2n + (n − 1)yn f = f , xx x(yn + 3x2n) x nyn−1 f = f , xy yn + 3x2n x 2nxyn−2 3n − 1 f = − f + f . yy yn + 3x2n x y y

3n 2 n For t = f we have the holomorphic special 2-jet diﬀerential Φ˜ 1 = x y(y + 2n 3x )Φ1 with

1 2nxyn−2 Φ = {(dxd2y − d2xdy) + (dy)3+ 1 t3n−2 yn + 3x2n 3(3n − 1)x2n + (n − 1)yn 1 1 ( dx(dy)2 − dy(dx)2)} x(yn + 3x2n) y x on the branched cover over P2 branched along the curve. Interchanging x and y, 2 n 2n we get from Φ1 the holomorphic special 2-jet diﬀerential Φ˜ 2 = y x(x +3y )Φ2 with 1 2nyxn−2 Φ = {(dyd2x − d2ydx) + (dx)3+ 2 t3n−2 xn + 3y2n 3(3n − 1)y2n + (n − 1)xn 1 1 ( dy(dx)2 − dx(dy)2)}. y(xn + 3y2n) x y

By using the symmetry of the deﬁning equation f(x, y) = x3n + y3n + xnyn + 1 in the three homogeneous coordinates, we get another holomorphic special 2-jet n n diﬀerential Φ˜ 3 = xy(3 + x y )Φ3 with

x3n−5 2nxn+1yn−2 Φ = {(dyd2x − d2ydx) + (dy)3 3 t3n−2 3 + xnyn −3 + 9n − xnyn + 7nxnyn + dy(dx)2 x(3 + xnyn) 3 − 9n + xnyn − 7nxnyn 2nyn+1xn−2 + dx(dy)2 − (dx)3}. y(3 + xnyn) 3 + xnyn

For the deformation we will use the standard method of interpreting holomorphic special 2-jet differentials as holomorphic sections of some line bundle and then study the positivity of the line bundle. For this purpose we describe first the construction of the space over which the line bundle will be defined.

65 For a complex surface M with local coordinates (x, y) we introduce the space SJ2(M) of special 2-jets which is a bundle over M. The space SJ2(M) is similar to the space Jk(Y ) of k-jets for a complex manifold of dimension n which we will use in §1 of Chapter 7. For a point in M we denote the ﬁber of SJ2(M) over P by SJ2(M)P . An element v of Jk(M)P with respect to the coordinate system (x, y) is a triple of complex numbers (a, b, c). When another coordinate system (x0, y0) is used, the element with respect to (x0, y0) is represented by another triple of complex numbers (a0, b0, c0) and the relation between the two triples of complex numbers (a, b, c) and (a0, b0, c0) is as follows. Let f be a holomorphic map to M from an open neighborhood U of 0 in C with d d coordinate ζ with f(0) = P such that ( dζ (x◦f))(0) = a,( dζ (y ◦f))(0) = b, and d2 d d2 d d 0 0 ( dζ2 (x◦f))( dζ (y◦f))(0)−( dζ2 (y◦f))( dζ (x◦f))(0) = c. Then ( dζ (x ◦f))(0) = a , d 0 0 d2 0 d 0 d2 0 d 0 ( dζ (y ◦f))(0) = b , and ( dζ2 (x ◦f))( dζ (y ◦f))(0)−( dζ2 (y ◦f))( dζ (x ◦f))(0) = 0 2 2 c . A coordinate system of SJ2(M)P is given by dx, dy, d xdy − d ydx so that the value of dx, dy, d2xdy −d2ydx at v are respectively a, b, and c. We form the weighted complex projective space P(SJ2(M)P ) of SJ2(M)P as the quotient of C3 − 0 with respect to the action of C∗ given by λ ◦ (a, b, c) = (λa, λb, λ3c) ∗ for λ ∈ C . The space P(SJ2(M)) is the bundle over M whose ﬁber at P is P(SJ2(M)P ).

The fiber P(SJ2(M)P ) is singular at the point defined by the vanishing of the fiber coordinates dx and dy of SJ2(M)P . The singularity is the same as the singularity of C3 − 0/C∗ at the point ξ = η = 0 under the action which sends (λ, (ξ, η, ζ)) ∈ C∗ × C3 to (λξ, λη, λ3ζ). It is the same as the singularity at the origin of C2 under the cyclic group action (²,(ξ, η)) → (²ξ, ²η) with ² being a cubic root of unity. The map (ξ, η) → (σ, τ, ρ, ω) = (ξ3, η3, ξ2η, ξη2) embeds the singularity to the local subvariety in C4 defined by στ = ρω, ρτ = ω2, σω = ρ2. A modification which replaces the single point by a rational curve with self-intersection number −3 resolves the singularity. So the singularity is rather mild.

Define a line bundle LP on P(SJ2(M)P ) as follows. The space P(SJ2(M))P is covered by three open sets defined respectively by the non- 2 2 vanishing of the fiber coordinates dx, dy, d xdy − d y of SJ2(M)P . The tran- 3 sition functions of LP with respect to this covering are the quotients of dx , dy3, d2xdy −d2y so that a weighted homogeneous polynomial P (dx, dy, d2xdy − d2ydx) with total weight k (when the weights of dx and dy are 1 and that of d2xdy−d2ydx is 3) corresponds to a holomorphic section of the kth tensor power of LP . The line bundle L over P(SJ2(M)) is defined from the line bundles LP over P(SJ2(M)P ).

The singular set of P(SJ2(M)) is deﬁned by the vanishing of the ﬁber coordinates dx and dy of SJ2(M). There is a natural map π from the regular part of P(SJ2(M)) to the projectivization P(TX ) of the tangent bundle TX of X which

66 is defined by sending a point with fiber coordinates (dx, dy, d2xdy − d2ydx) to the point with fiber coordinates (dx, dy). We now use this construction for the case when M is the compact complex surface X which is the branched cover of P2 with branching along a nonsingular complex curve C of degree δ in P2. To emphasize the dependence of X on C we denote X by XC . We denote P(SJ2(X)) by PC and L by LC and the singular set of P(SJ2(X)) by YC and π by πC . When C is a special curve C0 which is defined by a polynomial with very few nonzero terms, we have three ˜ holomorphic specical 2-jet differentials Φj (1 ≤ j ≤ 3) of degree 1 on XC0 . We ˜ can regard Φj as a holomorphic section of LC over PC . If the line bundle LC0 is positive on PC0 , then LC will be positive for a generic C and the abundance m of holomorphic sections of LC over PC for a sufficiently large m and a generic C would give us enough independent holomorphic special 2-jet differentials to yield the hyperbolicity of P2 − C. Let Z be the common zeroes of Φ˜ 1, Φ˜ 2, ˜ Φ3. The positivity of LC0 would follow if every complex curve Γ in Z can be 0 deformed to another complex curve Γ in PC0 not lying entirely in Z. 2 2 We eliminate the variable d xdy − d ydx from Φ˜ 1, Φ˜ 2, Φ˜ 3 and get two 1-jet differentials which, up to a factor of a rational function of x and y, are equal to 2 1 Φ + Φ = df(ydx − ²xdy)(ydx − xdy). 1 2 t3n−2(yn + 3x2n)(xn + 3y2n) ² √ 3 where ² = (1+ −3)/2 so that ² = −1. Note that even for this special curve C0 the explicit construction does not give three independent holomorphic special

2-jet differentials on XC0 . A number of attempts with other special curves with more nonzero terms also fail to yield three independent holomorphic special 2- jet differentials. Let S be the subvariety of P(Ω1 ) defined by the vanishing XC0 1 ˜ ˜ of zero-set of df(ydx − ²xdy)(ydx − ² xdy). The common zero-set Z of Φ1, Φ2, ˜ ˜ Φ3 in PC0 is the intersection of the zero-set of Φ1 with the topological closure of π−1(S). In other words, when a point of S is given with fiber coordinates dx C0 and dy, we can get the point in the zero-set of Φ˜ 1 with fiber coordinates dx, dy, d2xdy − d2ydx by using

2nxyn−2 dxd2y − d2xdy = − (dy)3− yn + 3x2n 3(3n − 1)x2n + (n − 1)yn 1 1 ( dx(dy)2 − dy(dx)2) x(yn + 3x2n) y x from the expression for Φ1. Geometrically the set S can be very easily described by using the following surface. For any c ∈ C−0 the submanifold Vc of P(TP2 ) deﬁned by the vanishing of ydx − cxdy, with the natural projection Vc → P2, is biholomorphic to the

67 blowup of the three points [1,0,0], [0,1,0], [0,0,1] in the homogeneous coordinates of P2. For c 6= 1 the intersection of Vc and V1/c is precisely the three rational curves which are the blowups of the three points [1,0,0], [0,1,0], [0,0,1] in the homogeneous coordinates of P2. So by deforming ² to another complex number c it is easy to deform any complex curve inside π−1(S) which is not mapped to C0 C0 in P2. However, the diﬃculty occurs when one tries to deform curves in Z which are mapped to C0 in P2.

On the open subset U of XC0 deﬁned by fx 6= 0, the special 2-jet diﬀerential Φ1 can be written as

1 2 2 3n − 1 2 Φ1 = n−2 (d fdy − df(d y + dy )). t fx y

Since df = 3nt3n−1dt and d2f = 3n(3n − 1)t3n−2dt2 + 3nt3n−1d2t, it follows that df has one more vanishing order of t than d2f and along t = 0 we have

−2n 1 2 t Φ1 = 3n(3n − 1)(dt) dy. fx

So on PC0 above U the zero-set of Φ1 contains the surface T in PC0 above C0 defined by the vanishing of the fiber coordinate dy. The difficulty of this approach is to deform a curve inside T to a curve not contained in Z.

Chapter 7. The Theorem of Bloch 1. The Proof of Bloch, Ochiai, Kawamata, Wong, and Green-Griffiths We now present a proof of Bloch’s Theorem (0.3) whose essence is based on Bloch’s original proof, along the lines of the papers of Ochiai [Oc77], Kawamata [K80], and Wong [Wo80]. Let A be an abelian variety and X be a subvariety in A which is not a translate of an abelian subvariety of A. We are going to prove that there exists no nonconstant holomorphic map f from C to X whose Zariski closure is dense in X. There are two steps in this proof. The first one is an algebraic statement about the map from the projectivization of a jet bundle of X defined by jet differentials from the abelian variety to a complex projective space. The second one is an argument from value distribution theory. Bloch’s original paper [Bl26] already contains arguments corresponding to these two steps. We can assume that X is not ruled by subtori in the sense that the subgroup B of all elements a of A with X + a = X does not have positive dimension, otherwise we can replace A by the quotient of A by the identity component of B.

One of the tools which we use is the space Jk(M) of k-jets for a complex manifold M of complex dimension n, which is a bundle over M. For a point in

68 M we denote the fiber of Jk(M) over P by Jk(M)P . Let zα (1 ≤ α ≤ n) be the local coordinates of M in a neighborhood of P . An element v of Jk(M)P with respect to the coordinate system zα (1 ≤ α ≤ n) is a set of complex numbers (ξjα)1≤j≤k,1≤α≤n. When another coordinate system wα (1 ≤ α ≤ n) is used, the element with respect to wα (1 ≤ α ≤ n) is represented by another set of complex numbers (ηjα)1≤j≤k,1≤α≤n and the relation between the two sets of complex numbers (ξjα)1≤j≤k,1≤α≤n and (ηjα)1≤j≤k,1≤α≤n is as follows. Let f be a holomorphic map to M from an open neighborhood U of 0 in C with coordinate dj ζ such that f(0) = P and ( dζj zα ◦ f)(0) = ξjα (1 ≤ j ≤ k, 1 ≤ α ≤ n). Then dj ( dζj wα◦f)(0) = ηjα (1 ≤ j ≤ k, 1 ≤ α ≤ n). The complex dimension of Jk(M)P j is kn. A coordinate system of Jk(M)P is given by d zα (1 ≤ j ≤ k, 1 ≤ α ≤ n) j so that the value of d zα at v is ξjα. We form the weighted complex projective kn space P(Jk(M)P ) of Jk(M)P as the quotient of C − 0 with respect to the ∗ j ∗ action of C given by λ · (ξjα)1≤j≤k,1≤α≤n = (λ ξjα)1≤j≤k,1≤α≤n for λ ∈ C . The space P(Jk(M)) is the bundle over M whose fiber at P is P(Jk(M)P ). In the case of k = 1 the 1-jet bundle J1(M) is simply the cotangent bundle of M. Let n be the complex dimension of X and let m be the complex dimension of A. Let f : C → X be a holomorphic map such that the Zariski closure of the image of f is X. For ζ ∈ C let Qζ be the point of Jn(X) above f(ζ) defined by the n-jet of the map f at ζ. Since the tangent bundle of A is nm globally trivial, we have Jn(A) = A × C . The inclusion X → A induces an inclusion map Jn(X) → Jn(A) which we compose with the natural projection nm nm nm map Jn(A) = A × C → C to form Ψ : Jn(X) → C . The algebraic statement concerning the jet bundles of X and A is the following. Lemma (7.1.1). There exists some ζ ∈ C such that the rank of the differential dΨ of Ψ at Qζ is equal to the complex dimension of Jn(X). Proof. Suppose the contrary and we are going to derive a contradiction. m Let z1, ··· , zm be the coordinates of the universal cover C of A. There exist

1 ≤ ν1 < ··· < νn ≤ m such that (zν1 , ··· , zνn ) is a coordinate system on some open neighborhood of f(0) in X. Without loss of generality we can assume that Pn νj = j for 1 ≤ j ≤ n. Let dzα = ν=1 ωαν dzν on X. Take ζ0 ∈ C close to the origin. Let Q = Qζ0 . By assumption there exists a nonzero tangent vector T to Jn(X) at Q which is mapped to zero by dπ. The tangent vector T of Jk(X) at ∂j+1 Q is given by (( ∂ζj ∂t zα ◦g)(ζ0, 0))0≤j≤n,1≤α≤n for some holomorphic map g(ζ, t) 2 from a neighborhood of (ζ0, 0) in C to X with g(ζ, 0) = f(ζ). The diﬀerential ∂j+1 dπ of the map π maps T to 0 if and only if ( ∂ζj ∂t zα ◦ g)(ζ0, 0) vanishes for 1 ≤ j ≤ n and 1 ≤ α ≤ m, which for n + 1 ≤ α ≤ m means that at ζ = ζ0 and

69 t = 0 we have Xn j j 0 = ∂t∂ζ (zα ◦ g) = ∂ζ ωαν ∂t(zν ◦ g) ν=1 j µ ¶ Xn X j = (∂j−λ(ω ◦ g))(∂λ∂ (z ◦ g)) λ ζ αν ζ t ν ν=1 λ=0 Xn Xn j j = (∂ζ (ωαν ◦ g))∂t(zν ◦ g) = (∂ζ (ωαν ◦ f))∂t(zν ◦ g), ν=1 ν=1

λ because of the vanishing of (∂t∂ζ (zν ◦ g))(ζ0, 0) for 1 ≤ λ ≤ n and 1 ≤ ν ≤ n.

Since the n complex numbers (∂t(zν ◦ g))(ζ0, 0) (1 ≤ ν ≤ n) are not all zero Pn j due to the nontriviality of T , it follows from the vanishing of ν=1(∂ζ (ωαν ◦ f))(ζ0)(∂t(zν ◦ g))(ζ0, 0) for 1 ≤ j ≤ n, n + 1 ≤ α ≤ m that the rank of the j ((m − n)n) × n matrix ((∂ζ (ωαν ◦ f))(ζ0)) is less than n when 1 ≤ ν ≤ n is regarded as the row index and the double index (j, α) with 1 ≤ j ≤ n and n + 1 ≤ α ≤ m is regarded as the column index. This is true for any arbitrary ζ0 close to 0. The usual Wronskian type argument (§2 of Chapter 2) is one of the tools for proving the linear dependence of a set of n functions. We now apply such an argument simultaneously to (m−n)n sets of n functions to prove that there exist Pn j constants cν (1 ≤ ν ≤ n) not all zero such that ν=1 cν (∂ζ (ωαν ◦ f))(ζ) ≡ 0 for 1 ≤ j ≤ n and n + 1 ≤ α ≤ m. Let 1 ≤ k ≤ n be the smallest integer such j that the rank of the ((m − n)k) × k matrix ((∂ζ (ωαν ◦ f))(ζ)) is less than k for all ζ close to 0 when 1 ≤ ν ≤ k is regarded as the row index and the double index (j, α) with 1 ≤ j ≤ k and n + 1 ≤ α ≤ m is regarded as the column index. We can find meromorphic functions cν (ζ) (1 ≤ ν ≤ k) with ck ≡ 1 Pk j such that ν=1 cν (ζ)(∂ζ (ωαν ◦ f))(ζ) ≡ 0 for 1 ≤ j ≤ k and n + 1 ≤ α ≤ m. Pk j By differentiating ν=1 cν (ζ)(∂ζ (ωαν ◦ f))(ζ) ≡ 0 with respect to ζ once, we Pk j conclude that ν=1(∂ζ cν )(ζ)(∂ζ (ωαν ◦ f))(ζ) ≡ 0 for 1 ≤ j ≤ k − 1 and n + 1 ≤ α ≤ m. By the choice of k we know that the rank of the ((m−n)(k−1))×(k−1) j matrix ((∂ζ (ωαν ◦f))(ζ)) is equal to k−1 for some ζ close to 0 when 1 ≤ ν ≤ k−1 is regarded as the row index and the double index (j, α) with 1 ≤ j ≤ k − 1 and n + 1 ≤ α ≤ m is regarded as the column index. From ck ≡ 1, it follows that (∂ζ cν )(ζ) ≡ 0 for 1 ≤ ν ≤ k − 1. This finishes the proof that there exist Pn j constants cν (1 ≤ ν ≤ n) not all zero such that ν=1 cν (∂ζ (ωαν ◦ f))(ζ) ≡ 0 for 1 ≤ j ≤ n and n + 1 ≤ α ≤ m. Pn Let cα(ζ) = − ν=1 cν (ωαν ◦ f)(ζ)(n + 1 ≤ α ≤ m). It follows from Pn the identical vanishing of ν=1 cν (∂ζ (ωαν ◦ f))(ζ) in ζ that cα(ζ) is a constant function.P Since the Zariski closure of the image of f is dense in X, it follows that n ∂zα cα + cν ωαν is identically zero on X (n + 1 ≤ α ≤ m). Since ωαν = on ν=1 ∂zν

70 X when (z1, ··· , zn) is regarded as the local coordinate system of X, it follows Pn Pn ∂ from the identical vanishing of cα + cν ωαν = cα + ( cν )zα on ν=1 ν=1 ∂zµ X for n + 1 ≤ α ≤ m that the restriction of the function zα to an orbit of Pn ∂ the vector field cν on X is linear in the parameter of the orbit with ν=1 ∂zµ slope −cα. Hence the 1-parameter group action on A defined by the vector field Pm ∂ cµ maps X to itself, contradicting the assumption that X is not ruled µ=1 ∂zµ by subtori. Q.E.D. The simultaneous Wronskian type argument in the proof of Lemma (7.1.1) circumvents the proofs by Kawamata [K] and Wong [W] of Conjecture C on p.84 of Ochiai’s paper [O]. Proof of Bloch’s Theorem (0.3). ˜ Let f : P(Jn(C)) → P(Jn(X)) be induced by f : C → X. Let X be the ˜ Zariski closure of f(P(Jn(C)) in P(Jn(X)). Let N = nm − 1. Let Ψ:˜ X → PN N+1 be induced by Ψ : Jn(X) → C . By Lemma (7.1.1) the dimensions of Ψ(˜ X ) and X are the same. Then any meromorphic function on X is algebraic over the pullback via Ψ˜ of the field of all meromorphic functions on PN . We will consider meromorphic functions on X which are the pullbacks of meromorphic functions on X via the projection map X → X induced by P(Jn(X)) → X.A meromorphic function on PN is the quotient of two jet differentials of the same weight at a point of A. Thus the pullback via Ψ˜ of a meromorphic function on PN is the pullback to X , via the inclusion map X → A, of the quotient of two 2 3 weighted homogeneous polynomials in the differentials dzα, d zα, d zα, ···, etc. on A. Let ϕ1, ··· , ϕ` be meromorphic functions on A such that the field of all meromorphic functions of A is equal to C(ϕ1, ··· , ϕ`). By restricting each ϕj to X we can find weighted polynomials Pjν of the same total weight (1 ≤ j ≤ `, 0 ≤ ν ≤ k ) in the variables dµz (1 ≤ α ≤ m, 1 ≤ µ ≤ n) with the weight of j α P µ kj µ ν d zα being µ such that the pullback of ν=0 Pjν ({d zα})ϕj to X is identically µ µ zero but Pjkj ({d zα}) is not identically zero on X . The fact that Pjkj ({d zα}) µ is not identically zero on X means that the pullback of Pjkj ({d zα}) via f to C ˜ is not identically zero, because X is the Zariski closure of f(P(Jn(C)). We now claim that there is no nonconstant holomorphic map from C to X whose image is Zariski dense in X, which would give a contradiction, finishing the proof of Bloch’s Theorem. The claim follows from the following Schwarz lemma. Lemma (7.1.2) (Schwarz Lemma of Bloch-Ochiai). Let A be an abelian m variety whose universal cover is C with coordinates z1, ··· , zm. Let ϕ1, ··· , ϕ` be meromorphic functions on A such that the field of all meromorphic functions of A is equal to C(ϕ1, ··· , ϕ`). Let Pjν be weighted polynomials of the same µ total weight (1 ≤ j ≤ `, 0 ≤ ν ≤ kj) in the variables d zα (1 ≤ α ≤ m, µ ≥ 1) µ with the weight of d zα being µ. Then there is no nonconstant holomorphic map f from C to A such that for 1 ≤ j ≤ k the pullback of the jet differ- P kj µ ν ential ν=0 Pjν ({d zα})ϕj by f is identically zero on C but the pullback of

71 µ Pjkj ({d zα}) by f is not identically zero on C. Proof. Let T (r, ψ) denote the characteristic function of the map ψ. Since the ﬁeld of all meromorphic functions on A is equal to C(ϕ1, ··· , ϕ`), it follows from P` Lemma (2.1.2) and Lemma (2.1.4) that T (r, f) ≤ C ν=1 T (r, ϕν ◦ f) + O(1). P µ kj d ν From the algebraic dependency given by ν=0 Pjν ({ dζµ zα ◦ f})ϕj ≡ 0 (1 ≤ j ≤ ` ) it follows from Lemma (2.1.4) that

dk T (r, ϕj ◦ f) ≤ C max T (r, zν ◦ f) + O(1) 1≤ν≤m,1≤k≤K dζk for some positive integer K and some positive constant C. We will use C to denote a generic positive constant. Thus

dk (7.1.2.1) T (r, f) ≤ C max T (r, zν ◦ f) + O(1). 1≤ν≤m,1≤k≤K dζk √ P ¯ m 2 Since −1∂∂ ν=1 |zν | is a K¨ahlerform on A, it follows that Ã ! Xm 2 (7.1.2.2) T (r, f) = Ar |zν ◦ f| ) + O(1) ν=1 Ã ! Xm + 2 = Ar (log exp |zν ◦ f|) + O(1). ν=1 The logarithmic derivative lemma gives

dk T (r, z ◦ f) ≤ O(log T (r, exp(z ◦ f))). dζk ν ν Moreover,

¡ + ¢ T (r, exp(zν ◦ f))) = Ar log exp |zν ◦ f| + O(1) ¡ ¢ 1 + 2 2 ≤ Ar log exp |zν ◦ f|) + O(1).

Hence dk (7.1.2.3) max T (r, zν ◦ f) 1≤ν≤N dζk

¡ + 2¢ ≤ O(log max Ar (log exp |zν ◦ f|) + O(1) 1≤ν≤m µX ¶ + 2 ≤ O(log Ar max (log exp |zν ◦ f|) + O(1). 1≤ν≤m

72 Combining (7.1.2.1), (7.1.2.2), and (7.1.2.3), we get T (r, f) ≤ O(log T (r, f)) + O(1). which implies that T (r, f) = O(1), contradicting f being non constant. Q.E.D. The paper of Green-Griffiths [G-G] does not need the step that the dimension of Ψ(˜ X ) is the same as that of X . Instead the Schwarz lemma for jet differentials is used (Proposition (2.5) and Corollary (2.7) in [G-G]). In general such a Schwarz lemma holds only for special jet differentials (see §3 of Chapter 6). For special jet differentials the argument about using power series in con- cluding X + a = X from Jk(X) + a = Jk(X) for all nonegative integer k is not that clear. 2. McQuillan’s Proof Faltings [F91] proved the analog in number theory of both Bloch’s Theorem (0.3) and Conjecture (0.4). His proof is based on Vojta’s idea [Vo92] of the existence of line bundles on the product of copies of a subvariety of an abelian variety other than the obvious ones from the abelian variety if the subvariety is not a translate of an abelian subvariety. McQuillan [McQ93] adapted Faltings’ proof to function theory and gave a new proof of Bloch’s theorem. Usually a result in number theory is much harder to prove than its analog in function theory. However, in the case of Conjecture (0.4), though its analog in number theory (Theorem II on p.571 of Faltings) has been proved, Conjecture (0.4) in function theory is still open. The reason is that the proximity function is given by an integral whereas in number theory the local distance with respect to a place is just a nonnegative number. While it is easy to deal with the maximum of a finite set of nonnegative numbers, the relation between the maximum of a finite number of proximity functions and the maximum of the integrands in them poses difficulties which are unresolved up to this point. We now present McQuillan’s proof of Bloch’s theorem. Let A be an abelian variety and X be a subvariety of A which is not a translate of an abelian subvariety of A. We have to show that there is no nonconstant holomorphic map f from C to A such that the Zariski closure of the image of f is X.

Suppose there is such a map f and we are going to derive√ a contradiction. −1 P α The characteristic function we use is the following. Let ω = 2π α,β hαβdw ∧ β dw , with hαβ being constants, be a K¨ahlerclass which is integral and is the curvature form of a positive line bundle L over A, where wα is the coordinate of the complex Euclidean space which is the universal covering of A. We use the ∗ characteristic function T (r, f) given by T (r, f) = Ir(f ω). By Lemma (1.1.2), 1 X 1 X I (f ∗ω) = A ( h f αf β) − h f αf β(0). r 2 r αβ 2 αβ α,β α,β P α β We will use Ar( α,β hαβf f ) to compute the characteristic function T (r, f).

73 Let Xm (respectively Am) be the product of m copies of X (respectively m m(m−1)/2 A). Consider the map p : X → A deﬁned by (xj)1≤j≤m going to (xj − xk)1≤j

∗ ∗ ` ∗ (7.2.1.1) Ir(F˜ Ω) = Ir(Z(F˜ s/z )) − Ar(log |F˜ s|) −` log r + log |F˜∗s/z`|(0). ∗ The term Ar(log |F˜ s|) can be made negative by rescaling the metric of the line ∗ ∗ ` bundle. We have to control the two terms Ir(F˜ Ω) and log |F˜ s/z |(0) when we choose λ1, ··· , λm to deﬁne F (z) = (f(λ1z), ··· , f(λmz)). Let us ﬁrst compute Xm X I (F˜∗Ω) = d(−² I (f ∗ ω) + I ((f ∗ − f ∗ )ω)). r r λj r λj λk j=1 1≤j

74 To get an upper bound of I ((f ∗ − f ∗ )ω) we use r λj λk Z (λj − λk)z f(ζ)dζ f(λjz) − f(λkz) = √ 2π −1 |ζ|=R (ζ − λjz)(ζ − λkz) to get 2 |λj − λk|r 2 Ar(|fλj − fλk | ) ≤ AR(|f| ). (R − |λj|r)(R − |λk|r) Thus |λ − λ |r (I ((f ∗ − f ∗ )ω) + O(1)) ≤ C j k (I (|f|2) + O(1)). r λj λk R (R − |λj|r)(R − |λk|r)

1 We use T (r + T (r) ) ≤ 2T (r) k (see e.g. p.245, Lemma 2 in §3 of Chapter 9 in 1 [Ne70]). So we choose R = r + T (r) . We now worry about the term log |F˜∗s/z`|(0). This step diﬀers from Section 4.3 of McQuillan’s paper because some simple modiﬁcations are needed to take care of the minor inaccuracies there. Assume without loss of generalization that F (0) = 0. In a local trivialization of the line bundle M(−²)⊗d we can expand ∗ ∗ P∞ F s into homogeneous components F s = µ=` sµ so that s` is not identically 0 0 0 0 zero. Choose nonzero real numbers λ1, ··· , λm such that s`(1 + λ1, ··· , 1 + λm) 1 1 is nonzero. Choose a positive number ηr so that 4 ≤ ηr ≤ 0 3 for T (r) λj T (r) 0 1 ≤ j ≤ m. Let λj = 1 + ηrλj /r for r > 0. Then 1 1 1 R − |λ |r ≥ R − r − η |λ0| ≥ − ≥ j r j T (r) T (r)3 2T (r)

0 0 2 and |λj − λk|r = |λj − λk|ηr ≤ T (r)3 . So

|λ − λ |r (I ((f ∗ − f ∗ )ω) + O(1)) ≤ C j k (I (|f|2) + O(1)) r λj λk R (R − |λj|r)(R − |λk|r) 8 1 ≤ C (T (r + ) + O(1)) T (r) T (r) 8 ≤ C (2T (r) + O(1)) = O(1). T (r)

Similarly ∗ ∗ (Ir((fλj − f )ω) + O(1)) ≤ O(1). Thus Xm X (7.2.1.2) I (F˜∗Ω) = d(−² I (f ∗ ω) + I ((f ∗ − f ∗ )ω)) r r λj r λj λk j=1 1≤j

75 = −²dmT (r) + O(1). We now compute

˜∗ ` 0 0 (F s/z )(0) = s`(λ1, ··· , λm) = s`(1 + ηrλ1/r, ··· , 1 + ηrλm/r).

0 0 Consider the expansion of the function s`(1 + λ1t, ··· , 1 + λmt) as a function of t at t = 0. We get

0 0 p p+1 ` s`(1 + λ1t, ··· , 1 + λmt) = ϕpt + ϕp+1t + ··· + ϕ`t with 0 6= ϕp²C. Then

0 0 |s`(λ1, ··· , λm)| = |s`(1 + ηrλ1/r, ··· , 1 + ηrλm/r)| 1 1 ≥ ϕ ( )p 2 p rT (r)4 for r suﬃciently large. Finally from (7.2.1.1) and (7.2.1.2) we have 1 1 −²dmT (r) + O(1) ≥ −` log r + log ϕ ( )p. 2 p rT (r)4

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