A result of Jost and Zuo is used to show that for a large class of finite-dimensional hyperkähler quotients, the only L2 harmonic forms lie in the middle dimension, and are of type (k,k) with respect to all complex structures. The argument is extended to some moduli spaces which appear as infinite-dimensional quotients such as Higgs bundle and SU(2) monopole moduli spaces. In the latter case we recover some of the S-duality predictions of Sen.
In this paper we study the geometry of the total space $Y$ of a cotangent bundle to a Kähler manifold $N$ where $N$ is obtained as a Kähler reduction from $\mathbb C^n$. Using the hyperkähler reduction we construct a hyperkähler metric on $Y$ and prove that it coincides with the canonical Feix-Kaledin metric. This metric is in general non-complete. We show that the metric completion $\tilde Y$ of the space $Y$ is equipped with a structure of a stratified hyperkähler space. We give a necessary condition for the Feix-Kaledin metric to be complete using an observation of R.Bielawski. Pick a compl…
We represent algebraic curves via commuting matrix polynomials. This allows us to show that the Hilbert scheme of cohomologically stable twisted rational curves of degree $d$ in ${\Bbb P}^3\backslash {\Bbb P}^1$ is isomorphic to a complexified hyperkähler quotient of an open subset of a vector space by a non-reductive Lie group.
We show the $C^0$ estimate for solutions to Hessian quotient equations on hyperKähler with torsion manifolds without any additional assumption on its hypercomplex structure by a clever use of the cone condition directly.
We study singular hyperkahler quotients of the cotangent bundle of a complex semisimple Lie group as stratified spaces whose strata are hyperkahler. We focus on one particular case where the stratification satisfies the frontier condition and the partial order on the set of strata can be described explicitly by Lie theoretic data.
We use twistor methods to promote Namikawa's universal Poisson deformations of conic affine symplectic singularities to families of hyperkähler structures deforming hyperkähler cone metrics. The metrics we produce are generally incomplete, but for specific classes of hyperkähler cones, even these incomplete metrics have some interest and applications: we study in detail the case of the nilpotent cone of a simple complex Lie algebra, with applications to hyperkähler metrics with symmetries and hyperkähler quotients, and the case of Kleinian singularities, with applications to codimension-4 sing…
We investigate the collapsing geometry of hyperkähler 4-manifolds. As applications we prove two well-known conjectures in the field. (1) Any collapsed limit of unit-diameter hyperkähler metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3-torus by an involution, a singular special Kähler metric on the 2-sphere, or the unit interval. (2) Any complete hyperkähler 4-manifold with finite energy (i.e., gravitational instanton) is asymptotic to a model end at infinity.
