The singularities of the modulus space

Abstract

The results collected in the following theorem are contained explicitly in the literature ([l; 2; 8; 3] and references there) or are known to workers in the field and readily deduced from the literature. THEOREM A. (i) the set of conformai equivalence classes of compact Riemann surfaces (henceforth: classes) of fixed genus g^2 can be endowed with the structure of a normal analytic space M 9 of complex dimension 3g--3. This structure is derived from the structure o f a complex analytic manifold, the Torelli space X 9 , by identification under the action of a properly discontinuous group of analytic automorphisms, the reduced mapping class (Siegel modular) group G 9 . In this description X 9 appears as a branched covering of M 9 . P<EzA 9 , the branch locus, if and only if it is a fixed point of a nontrivial finite subgroup G 9 (P) of G 9 , in which case P lies over a class (an H-class) in M 9 containing a surface S (H-surface) admitting at least one nontrivial conformai automorphism and G(P) is isomorphic with H(S), the automorphism group of S.