A hyperk\"ahler metric on twisted cotangent bundles of the complex projective space

TL;DR

This paper explicitly constructs a hyperk"ahler metric on twisted cotangent bundles of complex projective spaces, revealing their structure as complex semisimple coadjoint orbits.

Contribution

It provides an explicit local coordinate description of hyperk"ahler metrics on these bundles, linking geometric structures to coadjoint orbits.

Findings

Explicit hyperk"ahler metric construction

Identification with complex semisimple coadjoint orbits

Enhanced understanding of geometric structures on cotangent bundles

Abstract

We construct a hyperk\"ahler metric on twisted cotangent bundles of the complex projective space $\mathbb{CP}^n$ explicitly in terms of local coordinates. Note that the twisted cotangent bundles of $\mathbb{CP}^n$ are holomorphically isomorphic to complex semisimple coadjoint orbits of $\mathrm{SL}_{n+1}(\mathbb{C})$.