Related papers: Hyperkahler implosion and Nahm's equations
We discuss symplectic and hyperk\"ahler implosion and present candidates for the symplectic duals of the universal hyperk\"ahler implosion for various groups.
Let G be compact Lie group. It is shown that the cotangent bundle of the complexification of G admits a hyperkahler structure which is invariant under left and right translations by elements of G. The proof is to realize the cotangent…
The main purpose of the paper is to study hyperkahler structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkahler structures. Motivated by the work of…
The geometry of the universal hyperKaehler implosion for SU(n) is explored. In particular, we show that the universal hyperKaehler implosion naturally contains a hypertoric variety described in terms of quivers. Furthermore, we discuss a…
We present some fundamental facts about a class of generalized K\"ahler structures defined by invariant complex structures on compact Lie groups. The main computational tool is the BH-to-GK spectral sequences that relate the bi-Hermitian…
We review the quiver descriptions of symplectic and hyperk\"ahler implosion in the case of SU(n) actions. We give quiver descriptions of symplectic implosion for other classical groups, and discuss some of the issues involved in obtaining a…
We apply methods from strict quantization of solvable symmetric spaces to obtain universal deformation formulae for actions of a class of solvable Lie groups. We also study compatible co-products by generalizing the notion of smash product…
The aim of the note is to extend the uniformization theorem to compact Kahler spaces X with mild singularities and establish a kind of rigidity of their universal coverings. We assume the fundamental group of X is large, residually finite…
We give an abstract version of the hard Lefschetz theorem, the Lefschetz decomposition and the Hodge-Riemann theorem for compact Kaehler manifolds.
The Riemann normal coordinate expansion method is generalized to a Kahler manifold. The Kahler potential and holomorphic coordinate transformations are used to define a normal coordinate preserving the complex structure. The existence of…
Let $(Z,\omega)$ be a connected Kahler manifold with an holomorphic action of the complex reductive Lie group $U^{\mathbb C}$, where $U$ is a compact connected Lie group acting in a hamiltonian fashion. Let $G$ be a closed compatible Lie…
In this paper we construct compact forms associated with a complex Lie supergroup with Lie superalgebra of classical type.
A complex Lie supergroup can be described as a real Lie supergroup with integrable almost complex structure. The necessary and sufficient conditions on an almost complex structure on a real Lie supergroup for defining a complex Lie…
In this paper we construct smooth, non-radial solutions of the compressible Euler and Navier-Stokes equation that develop an imploding finite time singularity. Our construction is motivated by the works [Merle, Rapha\"{e}l, Rodnianski, and…
We introduce an analogue in hyperkahler geometry of the symplectic implosion, in the case of SU(n) actions. Our space is a stratified hyperkahler space which can be defined in terms of quiver diagrams. It also has a description as a…
We prove a version of the affine Kempf-Ness theorem for non-algebraic symplectic structures and shifted moment maps, and use it to describe hyperkahler quotients of T*G, where G is a complex reductive group.
Looijenga--Lunts and Verbitsky showed that the cohomology of a compact hyper-K\"ahler manifold $X$ admits a natural action by the Lie algebra $\mathfrak{so} (4, b_2(X)-2)$, generalizing the Hard Lefschetz decomposition for compact K\"ahler…
We discuss Nakamaye's Theorem and its recent extension to compact complex manifolds, together with some applications.
We develop a theory of reduction for generalized Kahler and hyper-Kahler structures which uses the generalized Riemannian metric in an essential way, and which is not described with reference solely to a single generalized complex…
Given a Lie algebra $\frak{g}$, the \emph{Nahm algebra} of $\frak{g}$ is the vector space $\frak{g}\times \frak{g}\times \frak{g}$ with the natural commutative, nonassociative algebra structure associated with the Nahm equations $\dot{x} =…