Related papers: An equivariant description of certain holomorphic …
We study holomorphic integrable systems on the hyperk\"ahler manifold $G\times S_{\text{reg}}$, where $G$ is a complex semisimple Lie group and $S_{\text{reg}}$ is the Slodowy slice determined by a regular…
Consider a Hamiltonian action of a compact Lie group on a symplectic manifold which has the strong Lefschetz property. We establish an equivariant version of the Merkulov-Guillemin $d\delta$-lemma and an improved version of the…
By complexifying a Hamiltonian system one obtains dynamics on a holomorphic symplectic manifold. To invert this construction we present a theory of real forms which not only recovers the original system but also yields different real…
With a nilpotent element in a semisimple Lie algebra g one associates a finitely generated associative algebra W called a W-algebra of finite type. This algebra is obtained from the universal enveloping algebra U(g) by a certain Hamiltonian…
We show that generic symplectic quotients of a Hamiltonian $G$-space $M$ by the action of a compact connected Lie group $G$ are also symplectic quotients of the same manifold $M$ by a compact torus. The torus action in question arises from…
For any Lie group $G$, we construct a $G$-equivariant analogue of symplectic capacities and give examples when $G = \mathbb{T}^k\times\mathbb{R}^{d-k}$, in which case the capacity is an invariant of integrable systems. Then we study the…
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…
Given a symplectic manifold $(M,\omega)$ endowed with a proper Hamiltonian action of a Lie group $G$, we consider the action induced by a Lie subgroup $H$ of $G$. We propose a construction for two compatible Witt-Artin decompositions of the…
In this paper we introduce invariants of semi-free Hamiltonian actions of $S\sp 1$ on compact symplectic manifolds (which satisfy some technical conditions related to positivity) using the space of solutions to certain gauge theoretical…
We prove several results on symplectic varieties with a Hamiltonian action of a reductive group having invariant Lagrangian subvarieties. Our main result states that the images of the moment maps of a Hamiltonian variety and of the…
The symplectic implosion construction of Guillemin, Jeffrey and Sjamaar associates to a Hamiltonian action of a compact group K on a symplectic manifold X its 'imploded cross section'. When X is a complex projective variety and K acts…
An action of a complex reductive group $\mathrm G$ on a smooth projective variety $X$ is regular when all regular unipotent elements in $\mathrm G$ act with finitely many fixed points. Then the complex $\mathrm G$-equivariant cohomology…
We outline the construction of invariants of Hamiltonian group actions on symplectic manifolds. These invariants can be viewed as an equivariant version of Gromov-Witten invariants. They are derived from solutions of a PDE involving the…
We study pseudoholomorphic curves in symplectic quotients as adiabatic limits of solutions of a system of nonlinear first order elliptic partial differential equations in the ambient symplectic manifold. The symplectic manifold carries a…
Let M be a symplectic 4-manifold. A semitoric integrable system on M is a pair of real-valued smooth functions J, H on M for which J generates a Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall introduce new global…
Given a quasi-split reductive group $G$ and a symmetric variety $X$, we introduce a notion of endoscopic varieties for $(G,X)$, and establish the foundational properties of these varieties such as matching of stable semi-simple orbits. To…
Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is…
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…
For each connected complex reductive group G, we find a family of new examples of complex quasi-Hamiltonian G-spaces with G-valued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on…
Starting from generic bilinear Hamiltonians, constructed by covariant vector, bivector or tensor fields, it is possible to derive a general symplectic structure which leads to holonomic and anholonomic formulations of Hamilton equations of…