Related papers: Singular Reduction of Generalized Complex Manifold…

We consider the Poisson reduced space $(T^*Q)/K$ with respect to a cotangent lifted action. It is assumed that $K$ is a compact Lie group which acts by isometries on the Riemannian manifold $Q$ and that the action on $Q$ is of single…

Let $X$ be a smooth projective variety acted on by a reductive group $G$. Let $L$ be a positive $G$-equivariant line bundle over $X$. We use the Witten deformation of the Dolbeault complex of $L$ to show, that the cohomology of the sheaf of…

We study the action of a real reductive group $G$ on a real submanifold $X$ of a Kahler manifold $Z$. We suppose that the action of a compact connected Lie group $U$ with Lie algebra $\mathfrak{u}$ extends holomorphically to an action of…

We study the partial resolutions of singularities related to Hilbert schemes of points on an affine space. Consider a quotient of a vector space $V$ by an action of a finite group $G$ of linear transforms. Under some additional assumptions,…

This note surveys the well-known structure of G-manifolds and summarizes parts of two papers that have not yet appeared in print: one with joint with J. Bruning and F. W. Kamber, and another with I. Prokhorenkov. In particular, from a given…

Riemannian geometry is a particular case of Hamiltonian mechanics: the orbits of the hamiltonian $H=\frac{1}{2}g^{ij}p_{i}p_{j}$ are the geodesics. Given a symplectic manifold (\Gamma,\omega), a hamiltonian $H:\Gamma\to\mathbb{R}$ and a…

We develop the theory of generalized bi-Hamiltonian reduction. Applying this theory to a suitable loop algebra we recover a generalized Drinfeld-Sokolov reduction. This gives a way to construct new examples of algebraic Frobenius manifolds.

We prove that the Grothendieck-Springer simultaneous resolution viewed as a correspondence between the adjoint quotient of a Lie algebra and its maximal torus is Lagrangian in the sense of shifted symplectic structures. As Hamiltonian…

We develop Lie-Poisson integrators for general Hamiltonian systems on $\mathbf{R}^{3}$ equipped with the rigid body bracket. The method uses symplectic realisation of $\mathbf{R}^{3}$ on $T^{*}\mathbf{R}^{2}$ and application of symplectic…

The purpose of this short note is to prove the uniqueness of Hamiltonian volume minimizing Lagrangian submanifolds which are Hamiltonian isotopic to RP^n in CP^n modulo isometric group actions.

Based on the Euler-Lagrange cohomology groups $H_{EL}^{(2k-1)}({\cal M}^{2n}) (1 \leqslant k\leqslant n)$ on symplectic manifold $({\cal M}^{2n}, \omega)$, their properties and a kind of classification of vector fields on the manifold, we…

We apply the Guillemin-Lerman-Sternberg theorem to reprove a formula of Heckman for the Duistermaat-Heckman measure associated to the coadjoint action of $T$, a maximal torus of a compact semisimple Lie group $G$, on a regular coadjoint…

We consider three 'classical doubles' of any semisimple, connected and simply connected compact Lie group $G$: the cotangent bundle, the Heisenberg double and the internally fused quasi-Poisson double. On each double we identify a pair of…

We study Hamiltonian field theories on the multisymplectic bundle of a principal G-bundle with Hamiltonian densities invariant under a subgroup $H\subset G$. Using the covariant bracket formulation, we reduce the polysymplectic space and…

We discuss the application of the method of the gaugeless Hamiltonian reduction to general relativity. This method is based on explicit resolving the global part of the energy constraint and on identification of one of the metric components…

We introduce a new method to perform reduction of contact manifolds that extends Willett's (math.SG/0104080) and Albert's results. To carry out our reduction procedure all we need is a complete Jacobi map $J$ from a contact manifold $M$ to…

We characterize isometric actions on compact Kaehler manifolds admitting a Lagrangian orbit, describing under which condition the Lagrangian orbit is unique. We furthermore give the complete classification of simple groups acting on the…

The Guillemin-Sternberg conjecture states that "quantisation commutes with reduction" in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved for compact Lie groups $G$ acting on compact…

A symplectic $T^n$ - reduction on a complex Grassmann manifold $G_{n,2}$ for the canonical action of the maximal compact torus depends on the $S_n$ - orbit of a maximal chamber in a hypersimplex $\Delta _{n,2}$. The chamber decomposition of…

The purpose of this note is to show that a connection with closed skewsymmetric torsion and reducible holonomy admits a locally defined Riemannian submersion together with a projected geometry on the base. We reframe known submersion…