The case of dimension two was proved by Andreotti-Frankel [-7] and the case of dimension three by Mabuchi [18] using the result of Kobayashi-Ochiai [12]. Our method of proof uses harmonic maps and the characterization of the complex projective space obtained by Kobayashi-Ochiai [15]. According to the result of Kobayashi-Ochiai the complex projective space of dimension n is characterized by the fact that its first Chern class equals 2c1 (F) for some 2> n+ 1 and some positive holomorphic line bundle F over it. Since by the result of Bishop-Goldberg [-2] the second Betti number of a compact Kiihler manifold M of positive bisectional curvature is 1, for the Main Theorem it suffices to show that ca (M) is 2 times a generator of H2 (M, 7Z.) for some 2> 1+ dim M. This can be done by proving that a generator of the free part of H2 (M, Z) can be represented by a rational curve, because the tangent bundle of M restricted to the rational curve splits into a direct sum of holomorphic line bundles over the rational curve according to the result of Grothendieck [-11]. The existence of the rational curve is obtained in the following way. According to the result of Sacks-Uhlenbeck [22] and its improved formulation by Meeks-Yau [-19], the infimum of the energies of maps from S 2 to M representing the generator of rcz (M) can be achieved by a sum of stable harmonic maps f~ from S 2 to M (1< i< m). The key step in our proof is to show that each f~ is either holomorphic or conjugate holomorphic. The known methods of proving the complex-analyticity of a harmonic map use the formula for the Laplacian of the energy function [23, 25, 26] or a variation of it [24]. Here we use

Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

No abstract uploaded!

Introduction. The classical Schwarz-Pick lemma states that any holomorphic map of the unit disk into itself decreases the Poincare metric. Later Ahlfors generalized this lemma to holomorphic mappings between two Riemann surfaces where curvatures of these Riemann surfaces were used in a very explicit way. More recently, Chern initiated the study of holomorphic mappings between higher-dimensional complex manifold by generalizing the Ahlfors lemma to these spaces. Then this lemma was further extended by Kobayashi, Griffiths, Wu and others. It plays a very important role in their theory. In this note, we shall prove the following generalization of the Schwarz lemma.

This work is intended as both an introductory text and a reference book for those interested in studying several complex variables in the context of partial differential equations. Significant progress has been made in the study of Cauchy-Riemann and tangential Cauchy-Riemann operators; this progress greatly influenced the development of PDEs and several complex variables.