L 2 Riemann-Roch Theorem for Elliptic Operators

Abstract

1. Let X be a compact riemannian surface (i.e. a compact complex manifold with dimc X = 1) or, in other words, a non-singular complex algebraic curve. Denote by g the genus of X, i.e. a non-negative integer such that X is homeomorphic to the sphere with g handles. In particular, if g = 0 then X is just a riemannian sphere, and if g = 1 then topologically X is a 2-torus (in this case X is called an elliptic curve). Consider a divisor μ on X, i.e. an element of the free abelian group generated by the points in X. So μ is in fact a finite collection of points x 1,…, x k in X with multiplicities p l, …, p k which are arbitrary integers. Let us consider the space \(\mathcal{O}(\mu )\) of meromorphic functions on X which are subordinated to μ.