Uniformisation

The chapter discusses what is known about the fundamental group and universal cover of compact complex manifold. For algebraic curves, the primary theory is classical: a curve of genus 0 is isomorphic to $\mathbb{P}^{1}$ , by the Riemann mapping theorem, curves of genus 1 are uniformised by $\mathbb{C}$ with the fundamental group a lattice of translations, and curves of genus ≥2 by the upper half-plane, with the covering group a cocompact discrete subgroup of $\mathop{{\mathrm{SL}}}(2,\mathbb{R})$ . Conversely, given a cocompact discrete group acting on any bounded domain (of any dimension), the quotient is a projective algebraic variety, and has pluricanonical embeddings into projective space provided by Poincaré series. In higher dimensions the theory is much more fragmentary. Standard constructions of projective geometry such as complete intersections lead to simply connected varieties. By taking appropriate group quotients of these, one can obtain every finite group as the fundamental group of a compact complex manifold. The final section raises the question (now considered to be a deep and studied under the name of Shafarevich’s conjecture) of whether the universal cover of a complete algebraic variety is holomorphically convex.