Related papers: Invariant K\"ahler potentials and symplectic reduc…

Let $(M,\omega)$ be a Hamiltonian $G$-space with a momentum map $F:M \to {\frak g}^*$. It is well-known that if $\alpha$ is a regular value of $F$ and $G$ acts freely and properly on the level set $F^{-1}(G\cdot \alpha)$, then the reduced…

When a complex semisimple group $G$ acts holomorphically on a K\"ahler manifold $(X,\omega)$ such that a maximal compact subgroup $K\subset G$ preserves the symplectic form $\omega$, a basic result of symplectic geometry says that the…

Let $M$ be a connected compact quantizable K\"ahler manifold equipped with a Hamiltonian action of a connected compact Lie group $G$. Let $M//G=\phi^{-1}(0)/G=M_0$ be the symplectic quotient at value 0 of the moment map $\phi$. The space…

We consider actions of reductive complex Lie groups $G=K^C$ on K\"ahler manifolds $X$ such that the $K$--action is Hamiltonian and prove then that the closures of the $G$--orbits are complex-analytic in $X$. This is used to characterize…

We have developed N=1 supersymmetric nonlinear realization methods, which realize global symmetry breaking in N=1 supersymmetric theories. The target space of nonlinear sigma models with a linear model origin is a G^C-orbit, which is a…

We prove that when Hodge theory survives on non-compact symplectic manifolds, a compact symplectic Lie group action having fixed points is necessarily Hamiltonian, provided the associated almost complex structure preserves the space of…

We study Hamiltonian actions of compact Lie groups K on Kaehler manifolds which extend to a holomorphic action of the complexified group K^C. For a closed normal subgroup L of K we show that the Kaehlerian reduction with respect to L is a…

A symplectic reductive homogeneous space is a pair $(G/H,\Omega)$, where $G/H$ is a reductive homogeneous $G$-space and $\Omega$ is a $G$-invariant symplectic form on it. The main examples include symplectic Lie groups, symplectic symmetric…

A Theorem due to Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of…

We study meromorphic actions of unipotent complex Lie groups on compact K\"ahler manifolds using moment map techniques. We introduce natural stability conditions and show that sets of semistable points are Zariski-open and admit geometric…

Let $M$ be a compact symplectic manifold on which a compact torus $T$ acts Hamiltonialy with a moment map $\mu$. Suppose there exists a symplectic involution $\theta:M\to M$, such that $\mu\circ\theta=-\mu$. Assuming that 0 is a regular…

Let $G \subset GL(V)$ be a reductive algebraic subgroup acting on the symplectic vector space $W=(V \oplus V^*)^{\oplus m}$, and let $\mu:\ W \rightarrow Lie(G)^*$ be the corresponding moment map. In this article, we use the theory of…

For a connected Lie group G, we show that a complex structure on the total space TG of the tangent bundle of G that is left invariant and has the property that each left translation G-orbit is a totally real submanifold is induced from a…

Let $G/K$ be an irreducible Hermitian symmetric spaces of compact type with the standard homogeneous complex structure. Then the real symplectic manifold $(T^*(G/K),\Omega)$ has the natural complex structure $J^-$. We construct all…

We consider a connected symplectic manifold $M$ acted on by a connected Lie group $G$ in a Hamiltonian fashion. If $G$ is compact, we prove give an Equivalence Theorem for the symplectic manifolds whose squared moment map $\parallel \mu…

We consider compact symplectic manifolds acted on effectively by a compact connected Lie group $K$ in a Hamiltonian fashion. We prove that the squared moment map $||\mu||^2$ is constant if and only if $K$ is semisimple and the manifold is…

In this note, we shall prove geodesic convexity of the space of K\"ahler potentials on an ALE K\"ahler manifold. This extends earlier results in the compact case proved in the fundamental work of X-X. Chen. We further prove the boundedness…

Let G be a compact simple Lie group and the O the minimal nilpotent orbit in g^C. We determine all G-invariant K\"ahler potentials for hyperK\"ahler metrics compatible with the KKS complex symplectic form on O.

The (local) invariant symplectic action functional $\A$ is associated to a Hamiltonian action of a compact connected Lie group $\G$ on a symplectic manifold $(M,\omega)$, endowed with a $\G$-invariant Riemannian metric $<\cdot,\cdot>_M$. It…

In this note we prove the following theorem: Let $G$ be a compact Lie group acting on a compact symplectic manifold $M$ in a Hamiltonian fashion. If $L$ is an $l$-dimensional closed invariant submanifold of $M$, on which the $G$-action is…