Riemann Surfaces and Their Moduli

The purpose of these notes is to give an overview of Riemann surfaces, and their moduli spaces. In general, theorems will be presented without proof, although references will be provided. These notes were originally prepared for the summer 2004 session of the UIUC Graduate String Seminar1. Proofs of nearly all the theorems may be found, unless otherwise noted, in [FK80]. CONTENTS 1. Mathematical review 1 1.1. Complex structure 1 1.2. Conformal structure 2 1.3. Equivalence of conformal and complex structure 2 2. Classification of Riemann surfaces 2 2.1. Uniformization theorem for Riemann surfaces 3 2.2. Automorphisms 3 2.3. The Exceptional Riemann Surfaces 4 3. Moduli of Riemann surfaces 4 3.1. Surfaces with universal cover Ĉ 4 3.2. Surfaces with universal cover C 5 3.3. Surfaces with universal cover U 6 References 6 1. MATHEMATICAL REVIEW 1.1. Complex structure. Let M be a 2m-dimensional manifold, with an atlas {Ui,φi}; S i Ui =M , φi :Ui → Cm. If the functions φi ◦φ−1 j :Cm → Cm are holomorphic on their domains of definition, the atlas is called complex-analytic. Two atlases {Ui,φi} and {Vi,ψi} are said to be compatible if their union is again an atlas. Clearly this defines an equivalence relation on the set of atlases, an equivalence class of which is known as a complex structure. A manifold together with a complex structure is called a complex manifold. Its complex dimension is defined to be m; dimCM = m. A complex manifold of complex dimension 1 is called a Riemann surface. Another way to define a complex structure is to complexify the tangent bundle and introduce an almost-complex structure J. An almost-complex structure onM is a tensor of type (1,1) (a section of the bundle End(TM ⊗C)→M ) which squares to −1. In local coordinates this is JjJ j k = −δk. An almost complex structure J is said to be integrable if the Nijenhuis tensor N(X ,Y ) = [X ,Y ]+ J[JX ,Y ]+ J[X ,JY ]− [JX ,JY ] vanishes for all smooth vector fields X and Y . An integrable almost-complex structure defines a complex structure on a manifold. In one complex dimension, it can be shown that the Nijenhuis tensor vanishes identically, so that every almost-complex structure defines a complex structure. Another useful fact is that any complex structure has a unique associated almost-complex structure.