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Convexity of moment polytopes of algebraic varieties

By William Liu and Communicated Mohan Ramach


Abstract. 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

Topics: c○2003 American Mathematical Society
Year: 2013
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