Pit-Mann Wong

Strong uniqueness polynomials: The complex case

The theory of strong uniqueness polynomials, satisfying the separation condition (first introduced by Fujimoto [H. Fujimoto (2000). On uniqueness of meromorphic functions sharing finite sets. Amer. J. Math., 122, 1175–1203.]), for complex meromorphic functions is quite complete. We construct examples of strong uniqueness polynomials which do not necessary satisfy the separation condition by constructing regular 1-forms of Wronskian type, a method introduced in Ref. [T.T.H. An, J.T.-Y. Wang and P.-M. Wong. Unique range sets and uniqueness polynomials in positive characteristic. Acta Arith. (to appear).] We also use this method to produce a much easier proof in establishing the necessary and sufficient conditions for a polynomial, satisfying the separation condition, to be a strong uniqueness polynomials for meromorphic functions and rational functions.

Picard number, holomorphic sectional curvature, and ampleness

We prove that for a projective manifold with Picard number equal to one, if the manifold admits a Kähler metric whose holomorphic sectional curvature is quasi-negative, then the canonical bundle of the manifold is ample.

On the holomorphic sectional and bisectional curvatures in complex Finsler geometry

The concepts of holomorphic sectional and bisectional curvatures for holomorphic vector bundles in complex Finsler geometry are used to characterize the concept of big vector bundles in algebraic geometry.

The concepts of general position and a second main theorem for non-linear divisors

The recent works of Evertse–Ferretti (Evertse and Ferretti, A generalization of the subspace theorem with polynomials of high degree, Dev. Math. (2008), pp.175–198) and Corvaja–Zannier (Corvaja and Zannier, On a general Theus equation, Ann. Math. 160 (2004), pp. 705–726; Corvaja and Zannier, On integral points on surfaces, Amer. J. Math. 126 (2004), pp. 1033–1055) in diophantine approximations (Ru, A defect relation for holomorphic curves intersecting hypersurfaces, Amer. J. Math. 126 (2004), pp. 215–266; An, A defect relation for non-Archimedean analytic curves in arbitrary projective varieties, Proc. Amer. Math. Soc. 135 (2007), pp. 1255–1261) in complex and p-adic Nevanlinna theory extend the classical subspace theorem and the classical second main theorem to the case of non-linear divisors in general position. These had been long standing problems in diophantine approximation and Nevanlinna theory. However, their results when specialized to the case of hyperplanes are weaker than the classical results. In this article, we refine the concept of general position to the concepts of p-jet general position. These concepts of general position involve jets of order p and coincide with the usual concept of general position for hyperplanes, but are different for hypersurfaces of higher degrees. With the assumption that the hypersurfaces are in n-jet general position, a second main theorem, with ramification term, for non-linear divisors and d-non-degenerate map f : ℂ → ℙ n is obtained. The result when specialized to hyperplanes is precisely the classical result of Ahlfors (The theory of meromorphic curves, Acta Soc. Sci. Fenn. 3 (1941), pp. 1–31) (see also Cartan, Sur les zeros des combinations lineares de p fonctions holomorphes donees, Mathematica 7 (1933), pp. 5–31; Stoll, About the value distribution of holomorphic maps into the projective space, Acta Math. 123 (1969), pp. 83–114; Cowen and Griffiths, Holomorphic curves and metrics of non-negative curvature, J. Anal. Math. 29 (1976), pp. 93–153; for small functions, see Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192 (2004), pp. 225–294). In fact the proof is a modification of Ahlfors proof. There are a number of variations of Ahlfors proof, we choose the ‘approximate negatively curved’ approach used in Cowen and Griffiths (Holomorphic curves and metrics of non-negative curvature, J. Anal. Math. 29 (1976), pp. 93–153), Wong (Defect relation for meromorphic maps on parabolic manifolds and Kobayashi metrics on ℙn omitting hyperplanes, Ph.D. thesis, University of Notre Dame (1976)), Wong (Holomorphic curves in spaces of constant curvature, in Complex Geometry (Proceedings of the Osaka Conference, 1990), Lecture Notes in Pure and Applied Mathematics, Vol. 143, Marcel Dekker, New York, Basel, Hong Kong, 1993, pp. 201–223), Wong (On the second main theorem of Nevanlinna theory, Amer. J. of Math. 111 (1989), pp. 549–583) and Cowen (The Kobayashi metric on P n ∖ (2n + 1) hyperplanes, in Value Distribution Theory, Part A, R.O. Kujala and A.L. Vitter III, eds., Marcel Dekker, New York, 1974, pp. 205–223).

Second Main Theorems in Number Theory and Nevanlinna Theory

The most important theorem in Nevanlinna theory is the Second Main Theorem. In the classical case, it states that for any transcendental holomorphic map f: C → CP1 and any q distinct points a1,..., aq in CP1, the following estimate holds for any ɛ > 0 and for all positive real number r outside a set of finite Lebesgue measure:

Stein manifolds with compact symmetric center

{\Sigma _{1 \leqslant i \leqslant q}}m(f,{a_{i,r}}) \leqslant (2 + \varepsilon )T(f,r).

Geometry of the complex homogeneous Monge-Ampère equation

(1.1)

Unique range sets and uniqueness polynomials in positive characteristic

The concept of an open parabolic Riemann surface is classical. It is well-known that there are many different but equivalent characterizations, each of which can be generalized in some way to complex manifolds of higher dimension. However, as is to be expected, these generalized concepts are in general not equivalent. In this article parabolicity is defined via the existence of an (unbounded) exhaustion satisfying the homogeneous complex Monge-Ampere equation which, in the case of Riemann surfaces is simply the Laplace equation. There are several reasons for choosing this as the definition of parabolic manifolds. First of all it includes affine algebraic manifolds and with this definition, the classical value distribution theory over ℂn can be extended to parabolic manifolds in such a way that the defect relation is still intrinsic (cf. section 1 below). Secondly, this definition is indeed quite intrinsic as it is possible to obtain (under appropriate assumptions on the regularity of the exhaustion function) uniformization theorems for parabolic manifolds. Thirdly, by allowing bounded exhaustions (so that the manifolds under consideration are no longer parabolic but rather, Kobayashi hyperbolic), it is discovered that the Monge-Ampere condition is intrinsically related to the condition of certain complex curves being geodesics(or extremal disc) of the Kobayashi metric (cf. sections 4 and 5).