Moduli of Riemann surfaces, transcendental aspects

These notes are an informal introduction to moduli spaces of compact Riemann surfaces via complex analysis, topology and Hodge Theory. The prerequisites for the first lecture are just basic complex variables, basic Riemann surface theory up to at least the Riemann-Roch formula, and some algebraic topology, especially covering space theory. The first lecture covers moduli in genus 0 and genus 1 as these can be understood using relatively elementary methods, but illustrate many of the points which arise in higher genus. The notes cover more material than was covered in the lectures, and sometimes the order of topics in the notes differs from that in the lectures. We have seen in genus 1 case that M1 is the quotient Γ1/X1 of a contractible complex manifold X1 = H by a discrete group Γ1 = SL2(Z). The action of Γ1 on X1 is said to be virtually free - that is, Γ1 has a finite index subgroup which acts (fixed point) freely on X1. In this section we will generalize this to all g >= 1 - we will sketch a proof that there is a contractible complex manifold Xg, called Teichmueller space, and a group Γg, called the mapping class group, which acts virtually freely on Xg. The moduli space of genus g compact Riemann surfaces is the quotient: Mg = Γg/Xg. This will imply that Mg has the structure of a complex analytic variety with finite quotient singularities. Teichmueller theory is a difficult and technical subject. Because of this, it is only possible to give an overview. In this lecture, we compute the orbifold Picard group of Mg for all g >= 1. Recall that an orbifold line bundle over Mg is a holomorphic line bundle L over Teichmueller space Xg together with an action of the mapping class group Γg on it such that the projection L → Xg is Γg-equivariant. An orbifold section of this line bundle is a holomorphic Γg-equivariant section Xg → L of L. This is easily seen to be equivalent to fixing a level l>= 3 and considering holomorphic line bundles over Mg[l] with an Spg(Z/lZ)-actionsuch that the projection is Spg(Z/lZ)-equivariant. Working on Mg[l] has the advantage that we can talk about algebraic line bundles more easily. An algebraic orbifold line bundle over Mg is an algebraic line bundle over Mg[l] for some l equipped with an action of Spg(Z/lZ) such that the projection to Mg[l] is Spg(Z/lZ) equivariant. A section of such a line bundle is simply an Spg(Z/lZ)-equivariant section defined over Mg[l]. Isomorphism of such orbifold line bundles is defined in the obvious way. Let PicorbMg denote the group of isomorphisms classes of algebraic orbifold line bundles over Mg. Our goal in this lecture is to compute this group. It is first useful to review some facts about the Picard group of a smooth projective variety