The Serre duality theorem for Reimann surfaces

In this note we shall fill a gap in the original proof of the Main Theorem in the previous paper ^4], and sharpen the Theorem by showing that the condition (a) alone is sufficient to derive the conclusion of the theorem. This sharpened theorem enables us to solve a problem posed by K. Kodaira in [2]. The authors are grateful to Professor Mudumbai S. Narasimhan for calling our attention to the gap, to him, to Takahiro Kawai and to other friends for discussion about this. § 1. Vanishing of Cohomology In the proof of theorem 1 in Q4], we made too easy use of Serre's duality theorem. Let V be a connected paracompact complex analytic manifold of dimension n and E a holomorphic vector bundle over V. Let us denote by Cp>q=-Cp'q(V, E) the space of E -valued differential forms of type (p, q) and of class C°°, with the topology as given in Q5]. (Cp>q stands for Ap*9 in [5].) The space K%~pt*~q of £*-valued currents of type (n—p, n — q) with compact supports is isomorphic to the topological dual of Cp>9, and the transpose of the sequence

In this chapter we apply the techniques we have learned earlier to study curves. But in fact, except for the proof of the Riemann-Roch theorem (1.3), which uses Serre duality, we use very little of the fancy methods of schemes and cohomology. So if a reader is willing to accept the statement of the Riemann-Roch theorem, he can read this chapter at a much earlier stage of his study of algebraic geometry. That may not be a bad idea, pedagogically, because in that way he will see some applications of the general theory, and in particular will gain some respect for the significance of the Riemann-Roch theorem. In contrast, the proof of the Riemann-Roch theorem is not very enlightening.KeywordsElliptic CurveComplete IntersectionElliptic CurfHyperelliptic CurveQuadric SurfaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Contents Introduction § 1. Fuchsian groups and their sequential generators § 2. Geometry of Fuchsian groups § 3. Free Fuchsian groups of rank two § 4. Spaces of Fricke-Klein-Teichmüller type § 5. Moduli of Riemann surfaces § 6. The space of holomorphic morphisms of a Riemann surface § 7. Lifting Fuchsian groups to § 8. Topological classification of Arf functions and of pairs of Arf functions § 9. Topological classification of independent sets of Arf functions on compact surfaces § 10. The moduli space of spinor bundles of rank one § 11. Super Fuchsian groups, super Riemann surfaces, and their topological types § 12. Moduli of super Riemann surfaces § 13. super Fuchsian groups, super Riemann surfaces, and their topological invariants § 14. Moduli of super Riemann surfaces § 15. Superholomorphic morphisms of super Riemann surfacesBibliography

Let p be a prime number, p > 2. A closed Riemann surface which can be realized as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. If the p-gonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic p-gonal Riemann surface. Accola showed that if the genus is greater than (p − 1)2 the p-gonal morphism is unique. Using the characterization of p-gonality by means of Fuchsian groups we show that there exists a uniparametric family of cyclic p-gonal Riemann surfaces of genus (p − 1)2 which admit two p-gonal morphisms. In this work we show that these uniparametric families are connected spaces and that each of them is the Riemann sphere without three points. We study the Hurwitz space of pairs (X, f), where X is a Riemann surface in one of the above families and f is a p-gonal morphism, and we obtain that each of these Hurwitz spaces is a Riemann sphere without four points.

We prove the Serre duality theorem for the noncommutative projective scheme proj A \;{\operatorname {proj }}\;A when A A is a graded noetherian PI ring or a graded noetherian AS-Gorenstein ring.

A closed Riemann surface $X$ which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering will be called a trigonal morphism. A trigonal Riemann surface $X$ is called real trigonal if there is an anticonformal involution (symmetry) $\sigma$ of $X$ commuting with the trigonal morphism. If the trigonal morphism is a cyclic regular covering the Riemann surface is called real cyclic trigonal. The species of the symmetry $\sigma $ is the number of connected components of the fixed point set $\mathrm{Fix}(\sigma)$ and the orientability of the Klein surface $X/\langle\sigma\rangle$. We characterize real trigonality by means of Fuchsian and NEC groups. Using this approach we obtain all possible species for the symmetry of real cyclic trigonal and real non-cyclic trigonal Riemann surfaces.

A closed Riemann surface X which can be realised as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism.A p-gonal Riemann surface is called real p-gonal if there is an anticonformal involution (symmetry) σ of X commuting with the p-gonal morphism.If the p-gonal morphism is a cyclic regular covering the Riemann surface is called real cyclic p-gonal, otherwise it is called real generic p-gonal.The species of the symmetry σ is the number of connected components of the fixed point set Fix (σ) and the orientability of the Klein surface X/ σ .In this paper we find the species for the possible symmetries of real cyclic p-gonal Riemann surfaces by means of Fuchsian and NEC groups.

Recently, Baker and Norine (Advances in Mathematics, 215(2): 766-788, 2007) found new analogies between graphs and Riemann surfaces by developing a Riemann-Roch machinery on a finite graph $G$. In this paper, we develop a general Riemann-Roch theory for sub-lattices of the root lattice $A_n$ analogue to the work of Baker and Norine, and establish connections between the Riemann-Roch theory and the Voronoi diagrams of lattices under certain simplicial distance functions. In this way, we obtain a geometric proof of the Riemann-Roch theorem for graphs and generalise the result to other sub-lattices of $A_n$. In particular, we provide a new geometric approach for the study of the Laplacian of graphs. We also discuss some problems on classification of lattices with a Riemann-Roch formula as well as some related algorithmic issues.

We use Andreev-Thurston’s theorem on the existence of circle packings to construct approximating solutions to the Beltrami equations on Riemann surfaces. The convergence of the approximating solutions on compact subsets will be shown. This gives a constructive proof of the existence theorem for Beltrami equations.

The articles in these two volumes arose from papers given at the 1991 International Symposium on Geometric Group Theory, and they represent some of the latest thinking in this area. This first volume contains contributions from many of the world's leading figures in this field, and their contributions demonstrate the many interesting facets of geometrical group theory. For anyone whose interest lies in the interplay between groups and geometry, these books will be an essential addition to their library

If the coupling constants in QFT are promoted to functions of space-time, the dependence of the path integral on these couplings is highly constrained by conformal symmetry. We begin the present note by showing that this idea leads to a new proof of Zamolodchikov's theorem. We then review how this simple observation also leads to a derivation of the a-theorem. We exemplify the general procedure in some interacting theories in four space-time dimensions. We concentrate on Banks-Zaks and weakly relevant flows, which can be controlled by ordinary and conformal perturbation theories, respectively. We compute explicitly the dependence of the path integral on the coupling constants and extract the change in the a-anomaly (this agrees with more conventional computations of the same quantity). We also discuss some general properties of the sum rule found in arXiv:1107.3987 and study it in several examples.

The first section of this chapter gives a short account of uniformisation theory for Riemann surfaces, in which the main result is the extended Riemann mapping theorem. We describe the classification of cocompact Fuchsian groups, and the local and global properties of quotients of the hyperbolic plane by such groups. We discuss the inclusion relations between these groups, classified by Singerman, giving particular attention to triangle groups. These groups turn out to be very important, not only for a better understanding of maps and hypermaps, but also because certain automorphic functions related to them act as a source for all Belyĭ functions. These ideas are illustrated with Klein’s quartic curve, a classic example of a Riemann surface of genus 3. In the second section we use triangle groups to prove Grothendieck’s important observation, that even a purely topological definition of a dessin as a graph embedded in a compact oriented surface leads to a unique conformal structure on this surface, so that the dessin corresponds to a Belyĭ function on the resulting Riemann surface. This shows that our previous definitions of dessins (via Belyĭ functions, embedded graphs or permutations) are in fact equivalent. Grothendieck attributed this result to Malgoire and Voisin, and in the meantime there have been many proofs, such as that given by Voevodsky and Shabat, but here we use ideas of Singerman from pre-dessins times, involving maps and triangle groups. The chapter includes an appendix explaining group presentations.

In this paper we return to consider more fully some questions which were touched on in (6). This was the first of a series of papers dealing with the spectral theory of the Laplace–Beltrami operator on a Riemann surface. In the later parts of that series we dealt fully with the case when the fundamental group is finitely generated. In this paper we return to the study of more general questions. In many ways this work continues that of W. Roelcke (7). We shall develop in section 2 the spectral theory for the Laplace operator Δ operating on functions on G\D where D is the unit disc and G is an arbitrary Fuchsian group. The main object is to find generalized eigenfunction expansions which are valid in the sense of absolute convergence rather than L2 convergence. The most convenient form of spectral theory for our purposes appears to be a version due to Gårding for Carleman operators which is presented in ((4), Ch. xvii, §§ 8,9). As a first application we use the results in section 3 to solve a problem of W. K. Hayman. Then we turn to Fuchsian groups G with exponent of convergence δ(G) < ½. Using the results of (6) we first obtain sufficient information on the action of G on the principal circle of G (section 4) and finally determine the spectral resolution in section 5.

Let G be a Fuchsian group containing two torsion free subgroups defining isomorphic Riemann surfaces. Then these surface subgroups K and αKα−1 are conjugate in PSL(2,R), but in general the conjugating element α cannot be taken in G or a finite index Fuchsian extension of G. We will show that in the case of a normal inclusion in a triangle group G these α can be chosen in some triangle group extending G. It turns out that the method leading to this result allows also to answer the question of how many different regular dessins of the same type can exist on a given quasiplatonic Riemann surface.

If $\Gamma$ is a finitely generated Fuchsian group such that its derived subgroup $\Gamma'$ is co-compact and torsion free, then $S={\mathbb H}^{2}/\Gamma'$ is a closed Riemann surface of genus $g \geq 2$ admitting the abelian group $A=\Gamma/\Gamma'$ as a group of conformal automorphisms. We say that $A$ is a homology group of $S$. A natural question is if $S$ admits unique homology groups or not, in other words, is there are different Fuchsian groups $\Gamma_{1}$ and $\Gamma_{2}$ with $\Gamma_{1}'=\Gamma'_{2}$? It is known that if $\Gamma_{1}$ and $\Gamma_{2}$ are both of the same signature $(0;k,\ldots,k)$, for some $k \geq 2$, then the equality $\Gamma_{1}'=\Gamma_{2}'$ ensures that $\Gamma_{1}=\Gamma_{2}$. Generalizing this, we observe that if $\Gamma_{j}$ has signature $(0;k_{j},\ldots,k_{j})$ and $\Gamma_{1}'=\Gamma'_{2}$, then $\Gamma_{1}=\Gamma_{2}$. We also provide examples of surfaces $S$ with different homology groups. A description of the normalizer in ${\rm Aut}(S)$ of each homology group $A$ is also obtained.