arXiv subjects

Piotr Kobak

Publications and source records attributed to Piotr Kobak.

3 recordsLinked to original sources

arXiv2000

HyperKähler Potentials in Cohomogeneity Two

A hyperKähler potential is a function rho that is a Kähler potential for each complex structure compatible with the hyperKähler structure. Nilpotent orbits in a complex simple Lie algebra are known to carry hyperKähler metrics admitting such potentials. In this paper, we explicitly calculate the hyperKähler potential when the orbit is of cohomogeneity two. In some cases, we find that this structure lies in a one-parameter family of hyperKähler metrics with Kähler potentials, generalising the Eguchi-Hanson metrics in dimension four.

math.DG↗

arXiv2000

The HyperKähler Geometry Associated to Wolf Spaces

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

math.DG↗

arXiv2000

HyperKähler Potentials via Finite-Dimensional Quotients

It is known that nilpotent orbits in a complex simple Lie algebra admit hyperKähler metrics with a single function that is a global potential for each of the Kähler structures (a hyperKähler potential). In an earlier paper the authors showed that nilpotent orbits in classical Lie algebras can be constructed as finite-dimensional hyperKähler quotient of a flat vector space. This paper uses that quotient construction to compute hyperKähler potentials explicitly for orbits of elements with small Jordan blocks. It is seen that the Kähler potentials of Biquard and Gauduchon for SL(n)-orbits of elements with X^2=0, are in fact hyperKähler potentials.

math.DG↗