Invent. math. 126, 65– 84 (1996) K ahlerian potentials and convexity properties of the moment map Peter Heinzner, Alan Huckleberry Fakult at und Institut f ur Mathematik, Ruhr Universit at Bochum, Universit atsstrasse 150, D-44780 Bochum, Germany; e-mail: Heinzner@cplx.ruhr-uni-bochum.de, Huckleberry@cplx.ruhr-uni-bochum.de Oblatum 6-XI-1995 & 27-I-1996 Since Kostant proved his convexity theorems for torus actions on ag varieties ([Ko]), there have been numerous contributions to this subject in the more general symplectic and K ahlerian setting. For example, let K be a compact Lie group acting in a Hamiltonian fashion on a connected compact symplectic manifold X . Then the intersection (X ) + of the image of the moment map  : X → (Lie K ) * with a positive Weyl chamber t * + is convex. Thus (X )= K · (X ) + , where (X ) + is a natural convex section. This was proved by Atiyah and Guillemin–Sternberg ([A], [G-S]) in the abelian case, i.e., where K = T is a compact torus, by Mumford [M]) for X projective algebraic with an integral K ahlerian structure and K not necessarily abelian and in its nal form in the compact symplectic case by Kirwan ([K]). A precise description of (X ) + := (X ) ∩ t * + in the compact K ahlerian framework can be given in terms of the image of the components of the set of T -xed points in X . In the projective case, where the K ahlerian structure comes from a projective embedding, there are explicit connections to the rep- resentations theory of G = K C which are due to Brion and to Mumford–Ness ([B], [N]). The purpose of the present paper is to prove convexity properties of  in the setting of non-compact K ahlerian spaces. Note that, by removing an appro- priate K -invariant subset from X one can easily construct non-convex -images from convex ones. On the other hand Hilgert–Neeb –Plank proved ([H-N-P]) that convexity holds for a proper moment map  : X → (Lie K ) * . Another convexity result has been obtained by Sjamaar for X ane and  the restric- tion of the moment map of a representation ([S2]). Here the representation theoretical approach of Brion is used. Now, let X be an irreducible complex space endowed with a holo- morphic action of a complex reductive group G. Assume that a maximal The research for this work was partially supported by SFB-237 of the DFG.