Algebraic Surfaces and 4-manifolds: Some Conjectures and Speculations

Introduction. Since the time of Riemann, there has been a close interplay between the study of the geometry of complex algebraic curves (or, equivalently, compact Riemann surfaces) and the topology of 2-manifolds. These connections arise from the uniformization theorem, which asserts that every simply connected Riemann surface is conformally equivalent to either the Riemann sphere, the plane, or the interior of the unit disk. Prom this it follows that every compact Riemann surface has a conformally equivalent metric of constant curvature. A key idea in the proof of this result is the Dirichlet problem: Find a harmonic function on a Jordan region R in the plane with given boundary values. Any such harmonic function minimizes the functional