M. Brion proved a convexity result for the moment map image of an irreducible subvariety of a compact integral Kähler manifold preserved by the complexification of the Hamiltonian group action. V. Guillemin and R. Sjamaar generalized this result to irreducible subvarieties preserved only by a Borel subgroup. In another direction, L. O’Shea and R. Sjamaar proved a convexity result for the moment...

Moment invariants have been frequently used as features for shape recognition. They are computed based on the information provided by both the shape boundary and its interior region. Although several fast algorithms for computing traditional moment invariants have been proposed, none has ever shown the theoretical results of moment invariants computed based on the shape boundary only. This pape...

A manifold obtained by k simultaneous symplectic blow-ups of CP of equal sizes ǫ (where the size of CP ⊂ CP is one) admits an effective two dimensional torus action if k ≤ 3 and admits an effective circle action if (k−1)ǫ < 1. We show that these bounds are sharp if 1/ǫ is an integer. 1. Toric actions and circle actions in dimension four Hamiltonian torus actions. Let a torus T ∼= (S) act on a c...

We prove that the norm-square of a moment map associated to a linear action of a compact group on an affine variety satisfies a certain gradient inequality. This allows us to bound the gradient flow, even if we do not assume that the moment map is proper. We describe how this inequality can be extended to hyperkähler moment maps in some cases, and use Morse theory with the norm-squares of hyper...

We present a proof due to Duistermaat that the gradient flow of the norm squared of the moment map defines a deformation retract of the appropriate piece of the manifold onto the zero level set of the moment map. Duistermaat’s proof is an adaptation of Lojasiewicz’s argument for analytic functions to functions which are locally analytic.

This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties and the moment map. In particular, we explain the relation between linear precision and the moment map.

Let K be a connected compact Lie group and M a Hamiltonian K-manifold, i.e., a symplectic K-manifold equipped with a moment map μ : M → k := (LieK). A theorem of Kirwan (implicitly in [Ki]) asserts: if M is connected and compact then the level sets of μ are connected. The purpose of this note is to prove such a statement in the category of algebraic varieties. First, we reformulate Kirwan’s the...