Morse and semistable stratifications of Kähler spacesby ${\Bbb C}^{\ast}$ -actions

Abstract

We prove that the Morse decomposition in the sense of Kirwan and semistable decomposition in the sense of GIT of a \({\Bbb C}^{\ast}\)-Kahler manifold coincide if the moment map is proper and if the fixed points set \(X^{{\Bbb C}^{\ast}}\) has a finite number of connected components. For general Kahler space with holomorphic action of a complex reductive group G, if every component of the moment map is proper, the two decompositions also coincide if each semistable piece is Zariski open in its topological closure and the moment map square is minimal degenerate Morse function in the sense of Kirwan.