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 Poincar¶eMetric 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 Di®erentials with Pole Logarithmic Singularities 1. Second Main Theorem for Jet Di®erentials with Logarithmic Pole Singularities 2. Complements of Three Plane Curves 1 Chapter 6. Special 2-Jet Di®erentials 1. Statement of Result and Outline of the Method 2. Construction of Special 2-Jet Di®erentials 3. Schwarz Lemma for Special Jet Di®erentials 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-Gri±ths 2. McQuillan's Proof Introduction The main purpose of this article is to present some of the important tech- niques which have been introduced over the years for solving hyperbolicity prob- lems 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 gen- eral 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 alter- native 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 in¯nite number of rational points. It led Lang and Vojta to conjecture that a compact nonsingular algebraic variety de¯ned over Q which is hyperbolic has only a ¯nite number of rational points. For the noncompact case the conjecture is that a nonsingular a±ne algebraic variety de¯ned over Q has a ¯nite 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 ± ¸ 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 ± 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 de¯ned by the homogeneous polynomial f(z0; ¢ ¢ ¢ ; zn) = 0 of degree ±. Let X be the complex ± hypersurface in Pn+1 de¯ned by zn+1 = f(z0; ¢ ¢ ¢ ; zn). Consider the projection map ¼ : X ! Pn de¯ned by [z0; ¢ ¢ ¢ ; zn; zn+1] ! [z0; ¢ ¢ ¢ ; zn]. The map ¼ makes X a branched cover of ± sheets over Pn with cyclic branching of order ± 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 ¼ ± 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 satis¯ed ± when the de¯ning 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 (Detlo®-Schumacher-Wong [DSW93] and [DSW94]). The proof of Conjecture (0.1) becomes more di±cult when the number of components of H becomes smaller. The most di±cult 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 de¯ned 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 di®erent 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 di®erent 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 de¯nite. From the very beginning of the theory of hyperbolicity and the value distri- bution theory of Nevanlinna some of the powerful basic tools are the negative curvature argument, the logarithmic derivative lemma, and the Wronskian. In the ¯rst 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.