Moduli of Riemann Surfaces, Real Algebraic Curves, and Their by S. M. Natanzon

By S. M. Natanzon

The gap of all Riemann surfaces (the so-called moduli area) performs a massive position in algebraic geometry and its functions to quantum box thought. the current booklet is dedicated to the examine of topological homes of this area and of comparable moduli areas, reminiscent of the gap of actual algebraic curves, the distance of mappings, and likewise superanalogs of most of these areas. The booklet can be utilized through researchers and graduate scholars operating in algebraic geometry, topology, and mathematical physics.

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1. Let C\,C2,Cz € SL(2,R) be such that {J(C i),J{C 2), J(Cs)} is a sequential set of type (0, k, m). Then C1C2C3 = 1 if and only if 1,0 < a < (3. ) Ai(l - X2)af3\ 1 2 —0) ' (A2 —1) (<* —X2P) ) This matrix has the characteristic equation x 2 — (-А 2/З + a — Xi0 + AiA2a )x + Ai(Ai —1)(A2 - 1)ot(3 = 0 whose roots have the same sign coinciding with that of —A2/?

The set AJVS = {(z |01) . . , 0Ar)< E d1lJV) |Im z“ > 0} 11. SUPER-FUCHSIAN GROUPS AND SUPER-RIEMANN SURFACES 49 is called the upper N -super half-plane. In this section we shall discuss the 1-super half-plane As = A1S. Its automorphism group Aut(A5) consists of transformations A = A[a, 6, c, d, a | £, S] of the form az + b A(*l«) = ( cz -I- d (ad —bc)(e + Sz) (cz + d)2 0, and yfA de­ notes the element in Lo(M) uniquely determined by the properties (V A )2 = A and (VA)i > 0 (see [6], [7]).

1, the analytic structure on T introduced in Section 4 induces an analytic structure on the moduli space M . This is exactly the analytic structure we use in our study of the topology of M . 2 ([28]). The moduli space of Riemann surfaces of type (g, k, m) is diffeomorphic to M69+3fc+2m-6/ Modff>fc>m, where Modff>fc>m is a discrete group. 6. The space o f holom orphic morphism s o f R iem ann surfaces 1 1. A holomorphic morphism of degree d of Riemann surfaces is a triple (P, /, P ), where P and P are Riemann surfaces and / : P —►P is a holomor­ phic mapping of degree d such that /( P ) = P .

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