Fibering Compact Kähler Manifolds over Projective Algebraic Varieties of General Type

Abstract

Regarding compact Riemann surfaces S, the Uniformization Theorem gives a trichotomy according to the genus of S. Other than the Riemann sphere ℙ1 (of genus 0) and elliptic curves (of genus 1), S is conformally equivalent to the quotient of the unit disc by a torsion-free cocompact Fuchsian group of Möbius transformations, and as such is equipped with a Hermitian metric of constant negative curvature. For n-dimensional compact complex manifolds X this precise trichotomy in terms of the genus is replaced by the rough classification according to the Kodaira dimension K (X) = -∞, 1, 2,…, n, which is the transcendence degree of the field of meromorphic functions arising from pluricanonical sections, i.e., holomorphic sections of positive powers of the canonical line bundle K X . When K (X) = n ≥ 1, X is said to be of general type. They are the analogues of compact Riemann surfaces of genus ≥ 2. In 2 complex dimensions the Enriques-Kodaira classification of compact complex surfaces gives an essentially complete description for X of Kodaira dimension -∞, 0, 1. If X is a compact Kähler surface with K (X) < 2 and with infinite fundamental group, then either some finite unramified covering of X is biholomorphic to a compact complex torus, or X is an elliptic surface over a compact Riemann surface S of genus ≥ 1.