Some Examples of the Twistor Construction

Abstract

A hyperkähler manifold X is a kähler manifold of even dimension with a holomorphic 2-form ω, everywhere nonsingular, and covariant constant. In [2] Calabi constructed such metrics on the cotangent bundle of IPⁿ, with ω equal to the canonical symplectic 2-form ω_can on T*(IPⁿ), and gave an analytic construction of an associated complex manifold of dimension 2n+1. This construction, for n=1, is equivalent to the Atiyah-Hitchin-Singer version of Penrose’s curved twistor space ([1], [4], [6]). Calabi asked for an algebraic or geometric description of this twistor space. On the other hand, it is widely known that the construction is reversible, i.e. there is an inverse reconstruction of a hyperkähler metric from geometric data. In examining the question of the geometric construction of the twistor space for Calabi’s examples, it appeared that a simple enough procedure emerged to enable one to construct new hyperkähler metrics where Calabi’s method no longer worked. The metrics constructed are on cotangent bundles again, and the constant, holomorphic, non-degenerate 2-form ω is the canonical symplectic form. They are local only, i.e. defined on a neighborhood of the zero section in T*(M). To make the necessary calculations feasible we assume M is a generalized flag manifold, i.e., a compact, simply-connected homogeneous kähler manifold.