Compact Riemann Surface Of Genus Research Articles

Die Automorphismengruppe der universellen Familie kompakter Riemannscher Fl�chen vom Geschlecht g?3

In this article it is proven that every automorphism of the universal family of compact Riemann surfaces of genus g⩾3 is induced by an oriented homeomorphism of a fixed Riemann surface.

A modular group and Riemann surfaces of genus 2

Let T(S) be the Teichmiiller space of Compact Riemann surfaces of genus 2 and let M(S) be the corresponding Teichmiiller modular group. In this paper, we determine a fundamental region R(S) for M(S) acting on T(S). R(S) is constructed as a subspace of T(S), as it is described in [8-10]. We make use of and extend the results in [12] to fully describe the action of M(S) on the parameters determining T(S). Our construction also depends strongly on a theorem of Bers [3]. In Section I, we recall some basic definitions, and determine our notation. In Section II, we describe the moduli space for a torus with a hole, the action of the corresponding modular group on this space, and determine a fundamental domain for this modular group. In Section III we describe the space T(S) and in Section IV we discuss the modular group and study its action on T(S). In Section V we state the theorem of Bers in preparation for the actual construction in Section V1. Finally, in Section VII we use the constructed fundamental domain to prove a conjecture of Bets in [2] for the case of Riemann surfaces of genus 2.

On the density of strebel differentials

w 1. Statement of the Main Result Let X be a compact Riemann surface of genus g > 2 and denote by O the sheaf of germs of holomorphic differential 1-forms on X. Let q~H~ ~| be a nonzero holomorphic quadratic form on X, i.e. a form which can be expressed in a local coordinate z as f(z)dz 2, with f holomorphic. Denote by Xq the normalization of the complex curve defined in the total space of the cotangent bundle of X by the equation y2 =q(x), where x~X and yeT*. The surface Xq is the "Riemann surface of q~", i.e. under the projection 7r: (x,y)~-~x, the surface ~'q is a double cover of X ramified at the odd zeroes of q, and there is a canonical form __ 2 % ~ H~ O) such that 7r* q - ~oq.

Some degenerations of a compact Riemann surface of genus 4

A number of degenerations of a compact hyperelliptic Riemann surface of genus 4 are studied, using theta function techniques. 1. Let (5, T, A) be a hyperelliptic Riemann surface of genus 4 with a canonical homology basis. Then S has a representation as a two-sheeted cover of the sphere with ten branch points. We can arrange for the surface to have branch points over 0 , 1 , «>, 1/Xj, 1/X2> * * > IA7» where its real branch points other than 0 and 1 (if any) are all greater than 1 and in ascending order (see [2]). We obtain a concrete representation of (S, T, A) which we henceforth assume is that illustrated in Figure 1.

A remark on degeneration of a compact Riemann surface of genus 2

A compact Riemann surface of genus 2, whose period matrix (π ij ) is arbitrary, is degenerated, thus removing a restriction on the degeneration in a previous paper by the author.

On Schoeneberg's theorem

Let Sbe a compact Riemann surface of genus g ≥ 2 and σ an automorphism (conformal self-homeomorphism) of S of order n. Let S* = S/ « σ« have genus g*. In [5], Schoeneberg gave a sufficient condition that a fixed point P ∈ S of σ should be a Weierstrass point of S, i.e., that Sshould support a function that has a pole of order less than or equal to g at P and is elsewhere regular.

Handle removal on a compact Riemann surface of genus 2

A degeneration of a compact, two-sheeted, Riemann surface in genus 2 is studied. Two branch points coalesce when the period matrix for the surface is degenerated. A Siegel modular transformation is applied to give the degenerating period matrix in the form of a corner matrix.

Degeneration of a compact Riemann surface of genus 2

A problem in the degeneration of a compact, two-sheeted, Riemann surface of genus 2 is studied, using theta function techniques. The three moveable branch points coalesce to the fourth branch point on the first limiting surface while the triangles formed by these points are all essentially similar. Applying a conformal map, we see that these points represent the finite branch points on the second of the limiting surfaces.

On Higher Order Weierstrass Points

Let S be a compact Riemann surface of genus g > 2, let X be a holomorphic line bundle of positive Chern class, and suppose that X admits holomorphic sections which have no common zero. Let F(S, x) = the space of holomorphic sections of x, and let ry(X) = the complex dimension of F(S, x). Let %n = ... X A) X* n times. Define W.(X) to be the set of points p of S for which there exists an element a of F(S, \n) such that the order of c at p is not less than y(Xn) Let W(X) be the union of the W,(X). Our main result was conjectured by Lipman Bers for the case X = a, the canonical bundle:

The Teichmüller spaces are distinct

The Teichmüller space T ( g , n ) T(g,n) of a compact Riemann surface of genus g with n punctures is a complex manifold of dim = 3 g − 3 + n \dim = 3g - 3 + n . In any given dimension, there are a finite number of these Teichmüller spaces and it is natural to ask if all of these are distinct (up to biholomorphic equivalence). We have shown here that with the exception of two special cases in dimensions 1 and 3 all of these spaces are distinct, that is, T ( g , n ) T(g,n) is not biholomorphically equivalent to T ( g ′ , n ′ ) T(g’,n’) unless g ′ = g g’ = g and n ′ = n n’ = n .

Modulus Space is Simply-Connected

Modulus space of genus g consists of conformal equivalence classes of compact Riemann surfaces of genus g. It is the quotient space of the real Teichmuller space of marked Riemann surfaces by the Teichmuller modular group. The result follows using generators and some relations of this group and the fact that it acts discontinuously on Teichmuller space.

A Remark on Mahler's Compactness Theorem

We prove that if G is a semisimple Lie group without compact factors, then for all open sets UCG containing the unipotent elements of G and for all C>O, the set of discrete subgroups rCG such that (a) rn=U[{e}, (b) G/r compact and measure (G/r) 5 C, is compact. As an application, for any genus g and e>0, the set of compact Riemann surfaces of genus g all of whose closed geodesics in the Poincare metric have length 2 , is itself compact. Consider the following general problem: let G be a locally compact topological group and let 9a = { the set of discrete subgroups r C G[. We would like to put a good topology on WaG and we would like to find fairly big subsets of 9JG that turn out to be compact. Mahler studied the case G = Rn, G/r compact, i.e., r is lattice (cf. Cassels [1, Chapter 5]). In this case, the group of automorphisms of G, GL(n, R), acts transitively on the set of lattices, so that the subset WGCEG of lattices can be identified as a homogeneous space under GL(n, R); in fact: 9a _ GL (n, R) / GL (n, Z). So there is only one natural topology on Og and Mahler's theorem states that for all e and K: {rC R (1) if y E r, IIPYII < e= }=O is compact. (2) volume (Rn/r) K K (Cassels [1, p. 137 ].) Chabauty [2] has investigated generalizations of Mahler's theorem to general G and subgroups r such that measure (G/r) < + X'.1 We topologize 9JG by taking as a basis for the open sets the following: Received by the editors April 29, 1970. AMS 1970 subject classifications. Primary 22E40.

Theta constants of two kinds on a compact Riemann surface of genus 2

Theta constants of two kinds on a compact Riemann surface of genus 2

On the Schottky Relation and Its Generalization to Arbitrary Genus

If S is a compact Riemann surface of genus g>2 together with a canonical homology basis (F, A) r = at, *.., y, A = al .**, 3g and if de1, **,9 denotes the basis of abelian differentials of first kind on S dual to (F, A); i.e., 4 there must be these relations. Another way of looking at the problem is the following: The totality of g x g matrices which are symmetric and have positive definite imaginary part is called the Siegel upper half plane of degree g, denoted by ?Rg. Not all elements of g9 are Riemann matrices for some (S, r, A). As a matter of fact for g > 4 the elements of 9g which are Riemannmat rices for (S, F, A) form a set of positive codimension. The problem we are considering is to determine this set. The first person to make a break-through in this direction was F. Schottky [12]. In the case g = 4, Schottky showed that for the set in question in e, the associated even theta constants satisfy a special relation of the form 1/ r, ?+ Vr2 + V r3 = 0 where ri is a product of 8 theta constants. Rationalizing this expression we have an explicit homogeneous polynomial in the Riemann theta constants which of course gives the one relation for g = 4 among the 10 periods. Subsequently, Schottky and Jung in a joint note [13] indicated a way of re-deriving the genus 4 result and generalizing it to arbitrary g. Their idea was to establish certain relations between the Riemann theta constants and what was referred to in [10] as the Schottky theta constants; however, to our knowledge they never establish these relations. These relations were established for the case g = 2 by Rauch and the author in [10, 11] and the relations for g = 2 were seen to be a consequence of the vanishing

THETA CONSTANTS, MODULI, AND COMPACT RIEMANN SURFACES

One of the classical tools for the study of moduli of compact Riemann surfaces is the Riemann theta function. Preliminary results were announced earlier(1) which established relations between two kinds of theta constants on a compact Riemann surface of genus 2. In this note we generalize the results there obtained. The main theorem is as follows:A sufficient condition for [Formula: see text] to be independent of [Formula: see text] is that [Formula: see text] for the 2(g-2)(2(g-1) - 1) characteristics [Formula: see text] where [Formula: see text] ranges over all odd g - 1 characteristics.

Weierstrass points and analytic submanifolds of Teichmueller space

When Rauch originally proved this theorem, he mentioned that it was not altogether clear that the singularity in the exceptional cases of the theorem is genuine and not a product of the method of proof. The present author's proof seemed to lend supporting evidence to the fact that the singularity is indeed genuine. The purpose of this paper is to point out that for g >4 and r =3 the exceptional case does not occur. We begin with a brief summary of the pertinent results from the theory of compact Riemann surfaces. In what follows, S will always denote a compact Riemann surface of genus g. There are exactly g orders ni, 0? n, 4, possessing a Weierstrass point whose Weierstrass sequence begins with n = 3.

Special divisors on compact Riemann surfaces

Introduction. In a previous paper [1] the author obtained some results concerning the distribution of special divisors on compact Riemann surfaces of genus g. The reader is referred to [1 ] for definitions and notation. It was shown in [1, p. 886] that the product structure with STXSr and Tg(S) as factors may be endowed with a structure of complex analytic manifold in such a way that the resulting space, Wr,r, is an analytic fibre space over the base manifold Tg (S) with fibre S, X S, over S(T)ET9(S). We then proved [1, Theorem 2] that if ?, co are completely distinct equivalent special divisors of degree r on S0, then considering the triple (D, co, So(T)) as a point in Wr,r, there is a g codimensional submanifold of W, containing the point (D, co, So(T)) (each point of which has projections onto pairs of equivalent, special divisors on the surface So(T), the base point under the fibre) which projects onto a X codimensional submanifold of Tg(S). Bounds were obtained for X, and from these bounds it followed that a special divisor of degree r < (g+ 1)/2 is always special in the sense of moduli. Finally we showed that if g is odd, a special divisor of degree (g+1)/2 is also special in the sense of moduli. Hence our results for special divisors were [1, Theorem 5 ] that if g is even (odd), a special divisor of degree less than (g+2)/2 ((g+3)/2) is special in the sense of moduli. As a particular example of the method, we computed the dimension of the sublocus of Tg(S) possessing Weierstrass points whose Weierstrass sequences begin with a fixed r <g. It is the purpose of this note to indicate that the techniques used in [1 ] can be employed to yield a general theorem from which the results of [1] emerge as corollaries. Furthermore, in our present treatment, we shall obtain directly that a special divisor of degree less than (g+2)/2 is special in the sense of moduli, eliminating the necessity of Theorem 4 in [1].

Space of Unitary Vector Bundles on a Compact Riemann Surface

Let X be a compact Riemann surface of genus g _ 2. A holomorphic vector bundle on X is said to be unitary if it arises from a unitary representation of the fundamental group of X. We prove in this paper that on the space of isomorphic classes of unitary vector bundles on X of a given rank, there is a natural structure of a normal projective variety (Theorem 8.1). We recall that Mumford has proved that, on the space of (isomorphic classes) stable vector bundles on X of a given rank and degree, there is a natural structure of a non-singular quasi-projective variety (cf. [7]); further, it was proved in [9] that a vector bundle on X of degree zero is stable if and only if it is associated to an irreducible unitary representation of the fundamental group of X. Thus our result shows the existence of a canonical compactification (as an algebraic variety) of the space of stable bundles on X of a given rank and degree zero. We shall now give a brief outline of the proof. It consists in a refinement of the proof of Mumford for the existence of a natural structure of a quasiprojective variety on the space of stable bundles of a given rank and degree (loc. cit.). Let us fix a very ample invertible sheaf OX(1) on X; then if m is a positive integer which is sufficiently large, we have H'(V(m)) 0 0 and H0( V(m)) generates V(m) for any Ve Or,, where Or, stands for the category of unitary vector bundles on X of rank r. Then the rank of H0(V(m)) is the same whatever be V e OR9; let this be p. The Hilbert polynomial of V(m), is also the same whatever be V e OR,; let this be P. Let Q = Quot(E/P) be the scheme in the sense of Grothendieck; E being the free coherent sheaf of rank p on X (cf. [4]). Let R be the open subscheme of Q consisting of the points which represent quotients of E which are locally free, and whose sections can be canonically identified with H0(E). Thus one has a family of vector bundles {Fq}qeR on X such that every Fq can be canonically considered as a quotient vector bundle of the trivial bundle E on X of rank p. The linear group G = Aut E acts on Q, and R is a G-invariant subscheme; further given V e Or there is a q e R such that Fq V, and the set of such points q con-

On the theory of automorphic functions and the problem of moduli

The subject of this presentation is the problem of moduli for certain special types of algebraic-geometrical objects and the connection of this problem with the theory of automorphic functions for certain arithmetically defined discontinuous groups acting on bounded domains in complex Euclidean space. To begin with, the earliest such situation of non trivial content was the theory of elliptic modular functions, connected with the theory of moduli of elliptic curves. This situation was subsequently generalized to the theory of Siegel's modular functions connected with the moduli of normally polarized Abelian varieties. The theory of moduli of algebraic curves or of compact Riemann surfaces of genus n, which has received independent development from the analytical point of view by Teichmuller [9], Ahlfors [l], and Bers [4], may also be extracted to a large degree from the theory of moduli of normally polarized Abelian varieties of dimension n, from among which, by certain algebraic-geometrical criteria due to Matsusaka [6], may be distinguished those which are the canonically polarized Jacobian varieties of curves of genus n. The essential point is that the Jacobian varieties form a Zariski-open subset of an algebraic subset of the normally polarized Abelian varieties [2]. (Of course here we should be careful to point out the distinction between results of a more algebraic nature on the theory of moduli, and the results discussed here which are of a transcendental nature, linking such a theory of moduli with the theory of automorphic functions of some discontinuous group.) The fact that we speak only of the moduli of polarized Abelian varieties comes of course from the well-known phenomenon that if, in the space of nX2n period matrices of complex, w-dimensional tori, we identify those points which correspond to complex analytically isomorphic tori, what we get is a space which is not even Hausdorff, not to speak of being a complex space or algebraic variety. A polarization for a complex torus T is a class consisting of all divisors D

Gaps at Weierstrass points for the modular group

Let 5 be a compact Riemann surface of genus g^2, h: S-+S an automorphism of order N, and H the cyclic group of order N generated by h. One has a representation of H by letting it act on the g complex-dimensional space of abelian differentials of the first kind on 5 by h: g. In [3] the following was proved: (I) Suppose P = h(P) is a fixed point for h with gap sequence 7i> • • • > 7* and that h rotates at P by e, i.e., if z is a local parameter at P, z{P) = 0, then h(z) = ez+ • • • , e = 1. Then, with respect to a suitable basis for Ai, h is represented by the diagonal matrix (h) = d iag (e?i, €*», • • • , €**)• A corollary of this is (II) If P = h(P) is not a Weierstrass point then h has at most four fixed points. Thus if h has more than four fixed points all its fixed points are Weierstrass points. Let r be the inhomogeneous modular group, T(N) the principal congruence subgroup of level N>2, S(N) the compactified fundamental domain for T(N) which is a Riemann surface of genus g(N) = l+iV(iV--6)/24lLltf ( l~ l / £ 2 ) where the product is over primes dividing N. For details see Chapter 1 of [2]. T/T(N) is a group of automorphisms of S(N) whose fixed points are at three kinds of points. Firstly, parabolic points (cusps), equivalent under T/T(N) to 00 which is fixed under the cyclic group of order N generated by (the coset of) T: r—»r + l. Secondly, elliptic points of order 2, equivalent to i = V~-l which is fixed under the cyclic group of order 2 generated by S: r—> — 1/r. Thirdly, elliptic points of order 3, equivalent to p = e which is fixed under the cyclic group of order 3 gener1 Supported by NSF G-18929. The author wishes to acknowledge his gratitude to Professor H. E. Rauch for suggesting this area of research and helping to see it through and also to Professor D. J. Newman for several enlightening discussions.