Communications in Analysis and Geometry

Volume 12 (2004)

Number 1

Potential Functions and Actions of Tori on Kähler Manifolds

Pages: 281 – 303

DOI: http://dx.doi.org/10.4310/CAG.2004.v12.n1.a13

Authors

D. Burns

V. Guillemin

Abstract

Let M be a Kähler manifold equipped with a free Hamiltonian action of the standard n-torus, T with moment map, Φ : M → ℝn. For λ ∈ ℝn the symplectic quotient Mλ = Φ -1(λ)/T 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 T-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.

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