Convexity of Moment Polytopes of Algebraic Varieties

We consider the situation of a compact irreducible subvariety of a smooth compact complex variety equipped with a Kähler form preserved by a torus action. We study the image of that subvariety under the moment map of the Kähler form. 1. Main result The moment map of a Hamiltonian T -symplectic/Kähler manifold has been one of the main interests of study for mathematical as well as physical reasons. In the early 80’s, Atiyah in [A] considered the following situation: Let M be a compact finite dimensional Kähler manifold on which a real torus acts in a Hamiltonian fashion. There is then a natural extension of the real torus action to the complexified torus action by applying the almost complex structure induced by the complex structure to the infinitesimal real torus action and then integrating to obtain the action of the “imaginary part” of the complexified torus. He proved the following: Theorem 1.1 ([A]). Let f be a moment map of M with respect to the torus action. Let Y be an orbit of the complexified torus, and let Z1, · · · , Zp be the components of the set of all fixed points of the torus lying in the closure of Y . Then the image of the closure of Y under the moment map is equal to the convex hull of the images of c1, · · · , cp where ci = f(Zi) for each i. The convex hull in the theorem is sometimes called the moment polytope. One can ask if the above theorem holds for more general subspaces than a single orbit of M . The answer turns out to be positive in the following setting: Suppose thatM is a compact finite dimensional smooth complex variety equipped with a Kähler form ω. Let T be a real torus acting on (M,ω) that preserves ω and the complex structure such that the moment map exists. Denote its Lie algebra by t. Let TC be the complexified torus which also acts on M . Let X be a compact Received by the editors May 25, 2001. 2000 Mathematics Subject Classification. Primary 53D20, 37J15, 58E40; Secondary 22E67, 14M15.