Teichmüller Theory of Riemann Surfaces

Abstract

For detailed discussions of Teichmüller theory and moduli spaces, see [Abi80], [Gar87], [HM98], [Leh87], [Nag88], [Tro92], etc. Here we give only a brief introduction mostly without proofs. We first review some facts regarding the tensor calculus in the complex plane and Riemann surfaces that are used for the rest of the book. Recall that $dz : = dx + idy, d\bar{z} :=\bar{dz} : dx - idy$ Are the differential forms and $\partial _z : = \frac{1}{2}(\partial _x - i\partial _y ),\partial _{\bar z} : = \frac{1}{2}(\partial _x + i\partial _y )$ Are vector fields, z ϵ C. Thus $dz \wedge d\bar z = - 2idx \wedge dy$ is the multiple of the Euclidean area form. The vector fields $\partial _z ,\partial _{\bar z}$ are thought of as sections of two different complex line bundles T ^1,0 (the holomorphic line bundle) and T ^0,1 (the antiholomorphic line bundle). Since the bundle T ^1,0 is 1-dimensional, there are natural isomorphisms between the tensor product (T ^1,0)^⊗n, the symmetric product Sⁿ(T ^1,0) and the exterior product $\wedge ^n T^{1,0}$. In what follows, we will identify these bundles. The same applies to the bundle T ^1,0 and to the duals: holomorphic and antiholomorphic contangent bundles (T ^1,0)*, (T ^0,1)* whose sheaves of local sections are generated by the forms $dz,d\bar z$