§1. Marked Riemann surfaces and their canonical differentials.

(a) At several points in the more detailed study of Riemann surfaces, the explicit topological properties of surfaces play an important role; and it is convenient to have these properties established from the beginning of the discussion, to avoid the necessity of inserting topological digressions later. Since the universal covering space of a connected orientable surface of genus g > 0 is a cell, the fundamental group carries essentially all the topological properties of the surface; so it is also convenient to introduce from the beginning and to use systematically henceforth the representation of a Riemann surface in terms of its universal covering space.

Let M be a compact Riemann surface of genus g > 0 , and let [??] be its universal covering space. The topological space M inherits from M in an obvious manner a complex analytic structure, hence [??] is itself a Riemann surface; and [??] is topologically quite trivial, being homeomorphic to an open disc. The covering transformations form a group Γ of complex analytic homeomorphisms T: [??] —> [??]; and the Riemann surface M can be identified with the quotient space [??]/Γ. It will be assumed that the reader is familiar with the topological properties of covering spaces, so that no further details need be given here.

Select a base point po [member of] M and also a base point zo [member of] [??] lying over po. Having made these selections, there is a canonical isomorphism between the covering transformation group Γ and the fundamental group π1. (M,po) of the surface M based at po; this is the isomorphism which associates to any transformation T [member of] Γ the class of loops in π1(M,po) represented by the image in M of any path from zo to Tzo in [??]. Again the details will be omitted, since they can be supplied in a quite straightforward manner by anyone familiar with the topological properties of covering spaces; but it should be noted that this isomorphism does depend on the choice of the base point zo, since the selection of another base point in [??] lying over po alters the isomorphism by an inner automorphism of the group π1 (M,po).

Now for the more detailed properties, recall that topologically M is just a sphere with g handles; hence M can be dissected into a connected contractible set by cutting along 2g paths, as indicated in the accompanying diagram.

Each loop αi or βi lifts to a unique path [??]i or [??]i in [??] beginning at the base point zo; the path [??]i runs from zo to Aizo, where Ai [member of] Γ is the transformation associated to the homoltopy class of αi in π1 (M,po) under the isomorphism introduced above, and the path [??]i. runs from zo to Bizo, where Bi [member of] Γ is associated to βi in the corresponding manner. The complement [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is simply connected, hence lifts homeomorphically to a number of disjoint open subsets of M which are permuted by the action of any element of the covering transformation group. It follows readily, upon tracing out the boundary of this complement in the preceding diagram, that one of these liftings has the form indicated in the following diagram.

In this diagram the elements Ci [member of] Γ axe the commutators

(1) Ci = Ai Bi A-1i B-1i (i = 1, ..., g);

and the domain Δ [subset] [??] as indicated is hameamorphic to the complement [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The point set closure [??] of this domain is a polygonally shaped subset of [??] with boundary consisting of the closed curve

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As usual in the discussion of the groupoid of paths, a product [??]•[??] of paths is the path obtained by traversing first [??] and then [??]]; this multiplication is noncommutative, but is defined only when the end point of [??] coincides with the initial point of [??], hence the notation used in (2) should lead to no confusion. As is well known, the group Γ is generated by the 2g elements A1, ..., Ag, B1, ..., Bg; and these generators are subject to the single relation

(3) C1C2 ... Cg = 1

where again Ci denotes the commutator (l). The fundamental group π1(M,po) is of course correspondingly generated by the loops α1, ..., αg, β1, ..., βg subject to the corresponding relation; that this relation does hold is obvious from (2), although it does require some more work to show that all other relations axe consequences of (3).

A selection of base points po [member of] M, zo [member of] [??] and a set of cuts α1, ..., αg, β1, ..., βg of the above canonical form will be called a marking of the Riemann surface M. A marked Riemann surface thus has a specified base point, a fixed isomorphism between the covering transformation group of its universal covering space and the fundamental group at its chosen base point, a canonical set of generators for that covering transformation group and hence for the fundamental group, and a canonical dissection of the universal covering space into polygonally shaped subsets ΓΔ = {TΔ|T [member of] Γ). The set Δ will be called the standard fundamental domain for the action of the covering transformation group Γ on the universal covering space M; and the notation introduced above for the canonical generators of Γ will be used consistently in the sequel.

There are of course a vast number of possible markings for any given Riemann surface. As noted above, the choice of another base point in [??] has merely the effect of altering the isomorphism between Γ and π1(M,po) by an inner automorphism, hence of altering the canonical generators of π1 by an inner automorphism of Γ while of course leaving the canonical generators of π1 ((M,po)) unchanged, and of replacing the standard fundamental domain Δ by a suitable translate of Δ under the action of Γ. An arbitrary orientation preserving homeomorphism of the topological space M clearly transforms any marking into another marking, modulo choices of base points in [??]; and it is easy to see that conversely any two markings of M can be transformed into one another by some orientation preserving homeomorphism of the underlying topological space, again modulo choices of base points in [??], since the corresponding standard fundamental domains in [??] are evidently homeomorphic under an orientation preserving homeomorphism commuting with Γ. Thus, holding the base points po [member of] M, zo [member of] [??] fixed for simplicity, all possible markings arise from a given marking by applying suitable orientation preserving homeomorphisms of M to itself leaving po fixed. Note that any such homeomorphism determines in turn an automorphism of the fundamental group π1, (M,po), taking the canonical set of generators corresponding to one marking into the canonical set of generators corresponding to another marking. A homeomorphism which is homotopic to the identity (through homeomorphisms leaving the base point fixed) clearly yields the identity automorphism of the fundamental group, so that the canonical sets of generators of the fundamental group π1, (M,po) associated to two markings so related actually coincide; two markings so related will be called equivalent, and the reader should be warned that in much of the literature only these equivalence classes of markings are really considered. The more detailed investigation of these questions is an interesting subject in its own right, but must be left aside at present.

The paths α1, ..., αg, β1, ..., βg can be viewed as singular cycles on M, and as such represent a basis for the singular homology group H1(M,Z) ; thus a marked Riemann surface also has a canonical set of generators for its first homology group. The intersection properties of these one-cycles can be read off immediately from the diagram on page 2. Note that the choice of another marking on the surface, by the application of a homeamorphism of M to itself, determines an automorphism of the homology group H1(M,Z); this automorphism can also be determined directly from the corresponding automorphism of π1(M,Z), recalling that H1(M,Z) is just the abelianization of π1(M,Z). It should be mentioned that there are nontrivial automorphisms of π1 (M,Z) which induce trivial automorphisms of H1(M,Z); there is a real and important distinction between properties of the surface which depend on homological and those which depend on homotopical properties, as will become evident later.

(b) The Abelian differentials on M are the holomorphic differential forms of type (1,0) j they form a g-dimensional complex vector space Γ(M, O1,0). Note that any Abelian differential ω [member of] Γ (M, O1,0 can be viewed as a Γ-invariant holomorphic differential form of type (1,0) on the universal covering surface [??]; this form will also be denoted by ω, a notational convention that really leads to no confusion since M can be identified with the quotient space [??]/Γ. Since [??] is simply connected and since any such Abelian differential ω is closed, there must exist a holomorphic function w on [??] such that ω = dw; such a function is called an Abelian integral for the Riemann surface. Note that the function w is determined uniquely up to an additive constant.

For a Riemann surface with a specified base point zo [member of] [??], the associated Abelian integral can be normalized so that w(zo) = 0, and can thus be viewed as determined uniquely by the Abelian differential ω; indeed the Abelian integral is then given explicitly by the integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note further that since the Abelian differential ω is Γ-invariant, the Abelian integral w(z) has the property that d[w(Tz) - w(z)] = ω(Tz) - ω(z) = 0 for any T [member of] Γ, hence that w(Tz) = w(z) - ω(T) for some constant ω(T) [member of] C depending on T [member of] Γ. It then follows readily that ω(ST) = ω(S) + ω(T) for any two elements S, T [member of] Γ; hence the set of these constants can be viewed as an element ω [member of] Hom (Γ,C) , which will be called the period class of the Abelian differential ω [member of] Γ(M, [??]1,0). This terminology is suggested by the observation that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any T [member of] Γ. Note finally that an Abelian differential is determined uniquely by its period class; for if ω1, ω2 [member of] Γ(M, [??]1,0) have the same period class, their difference ω1 - ω2 = d[w1 - w2] is the derivative of a holomorphic Γ-invariant function on M, hence that difference must vanish since the only holomorphic functions on the compact Riemann surface [??]/Γ are constants.

Now select a marking for the Riemann surface M and a basis ω1, ..., ωg for the Abelian differentials on M. The period classes of these Abelian differentials are determined by their values ωi(Aj), ωi(Bj) on the canonical generators for the covering transformation group Γ; these values can be grouped together to form the associated g × 2g period matrix (Ω', Ω") where Ω'= {ωi(Aj)}, Ω" = (ωi (Bj)} are g × g matrices.

Theorem 1. The period matrix (Ω', Ω") of a basis for the Abelian differentials on a marked Riemann surface M satisfies the Conditions

(i) Ω' • tΩ" - Ω" • tΩ' = 0 , (Riemann's equality), and

(ii) iΩ' • t-[??]" - iΩ" • t-[??]' is positive definite Hermitian, (Riemann's inequality).

Proof. Although this was proved in the earlier lecture notes (Theorem 17), it is perhaps worthwhile repeating that proof to show how the intersection matrix of the canonical basis for the one-cycles on a marked Riemann surface can be calculated and to serve as a model for several quite similar later calculations. The essential point in deriving Riemann's equality is that ωi [conjunction] [and] ωj = 0; viewing these as forms on [??] and integrating over the standard fundamental domain Δ, and recalling that the boundary of Δ has the form given in (2), it follows that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which establishes ([??]). The essential point in deriving Riemann's inequality is that for any Abelian differential ω(z) = f(z)dz in a local coordinate system z = x + iy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

hence that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with equality holding only when ω = 0. In particular, putting ω(z) = [g.summation over (i = 1)] tiωi(z) for arbitrary complex constants ti, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with equality holding only when t1 = ... = tg = 0, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

thus the matrix is P = {Pij} positive definite Hermitian. To determine this matrix explicitly, it follows as above that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which establishes (ii) and concludes the proof.