Isometric Embeddings and Hyperk\"{a}hler Geometry of the Cotangent Bundle of Complex Projective Space via the Scheme of Rank-1 Projections

TL;DR

This paper describes the hyperk"ahler geometry of the cotangent bundle of complex projective space algebraically via rank-1 projections, providing explicit embeddings and generalizations to various rings, including noncommutative ones.

Contribution

It introduces an algebraic scheme of rank-1 projections that captures the hyperk"ahler geometry of $T^*\,\mathbb{CP}^{n-1}$ and extends this framework to arbitrary rings, including split-complex and bicomplex numbers.

Findings

Explicit $SU(n)$-equivariant isometric embeddings of $T^*\,\mathbb{CP}^{n-1}$ into Euclidean space.

Generalization of hyperk"ahler geometry to rings with involutions, including noncommutative rings.

Identification of the scheme's $\,\mathbb{C}$-points with the cotangent bundle of complex projective space.

Abstract

We show that the hyperkahler geometry of $T^*\mathbb{CP}^{n-1}$ can be described algebraically by the affine scheme of rank-1 projections, and that this description simultaneously yields explicit $SU(n)$-equivariant isometric embeddings \[ T^*\mathbb{CP}^{n-1} \hookrightarrow \mathbb{R}^{(n^2+1)^2}, \] as well as a generalization of the hyperkahler geometry of $T^*\mathbb{CP}^{n-1}$ to arbitrary commutative rings with involutions (and some noncommutative ones). In particular, we obtain para-hyperkahler and complex hyperkahler manifolds by taking the rings to be the split-complex numbers and bicomplex numbers, respectively. The functor of points of the scheme of rank-1 projections is the functor that maps a commutative ring $\mathcal{R}$ to the space of idempotents in $M_n(\mathcal{R})$ whose images are rank-1 projective modules. In particular, its space of $\mathbb{C}$-points is identified with $T^*\mathbb{CP}^{n-1}$.