Potential Functions and Actions of Tori on Kähler Manifolds

inherits from M a Kähler structure; and in the first part of this paper we will describe what the Kähler form and Ricci form look like locally on coordinate patches in Mλ. Then in the second part of this paper we will discuss some global implications of these results. This will include (1) A Kählerian proof of the Duistermaat-Heckman theorem. (2) A formula, due to Biquard and Gauduchon, for the Kähler potential on a symplectic quotient. (3) A convexity theorem of Atiyah for moment images of TC-orbits. (4) A formula in terms of moment data for the Kähler metric on a toric variety. (5) A formula for the Kähler form on the symplectic quotient of a Kähler–Einstein manifold. A few comments about each of these items: (1) The usual proof of the Duistermaat-Heckman theorem is global and topological in nature. Our proof in the Kähler case consists of showing that locally on a coordinate patch in Mλ two canonically defined Kähler forms are identical. (2) The formula of Biquard–Gauduchon was used by Calderbank, David and Gauduchon to give an elegant economical proof of the theorem alluded to in item 4. (This theorem is due to one of us, and a much less elegant proof of it was presented in [Gu].) In this paper we use a primitive local form of the Biquard– Gauduchon theorem (see § 4, formula (4.5)) to give an even simpler derivation of this formula.