# A blend of methods of recursion theory and topology: A П 0^1 by Kalantari I., Welch L.

By Kalantari I., Welch L.

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By considering only stable holomorphic structures in E or Einstein-Hermitian structures in E we obtain a Hausdorff moduli space. Given an Hermitian structure h in a C °~ complex vector bundle E over a compact KEhler manifold M, the moduli space ~ ( E , h) of irreducible Einstein-Hermitian connections in (E, h) carries a natural K/thler metric while AA(E) may not. In spite of apparent advantages of Ad(E, h) over Ad(E), we consider here 46 mainly the latter since it is much easier to keep track of various holomorphic objects on A~I(E) than those on A74(E, h).

A complex submanifold Sx through x which is transversal to the orbit G(x) in the sense that Tx(,-l(0)) = Tx(S~) • T~(G(x)). The quotient space W = # - I ( O ) / G is called the reduced phase space. If we take S , sufficiently small, then the projection lr : #-1(0) --+ W defines a homeomorphism of Sx onto an open subset ~r(Sx) of W and introduces a local coordinate system in W. This makes W into a (not necessarily Hausdorff) complex manifold. If the action of G on # - 1 ( 0 ) is proper, then W is Hausdorff.

It has possible double poles at s = - k /'or k = 0 , 1 , . - . and at most single poles/'or s = 2 + p - k which can only occur i/. there exists 7r C Ldls¢(F\G ) such that 7r C Ind([ [P®~r) and ~" is a constituent o/. a ® R ~ . The continuation of I~(r, s) is achieved by a similar m e t h o d and gives possible poles at the poles of the Eisenstein series for F. If we now return to our original problem, combining the above with Proposition 1 gives the following result. THEOREM 1. The Kloosterman-Selberg zeta function Z r (% s) has a meromorphic continuation to nil of C.