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Isometric Embeddings and Hyperkähler Geometry of the Cotangent Bundle of Complex Projective Space via the Scheme of Rank-1 Projections

Joshua Lackman

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 $T^*\mathbb{CP}^{n-1}$.

Isometric Embeddings and Hyperkähler Geometry of the Cotangent Bundle of Complex Projective Space via the Scheme of Rank-1 Projections

Abstract

We show that the hyperkahler geometry of can be described algebraically by the affine scheme of rank-1 projections, and that this description simultaneously yields explicit -equivariant isometric embeddings as well as a generalization of the hyperkahler geometry of 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 to the space of idempotents in whose images are rank-1 projective modules. In particular, its space of -points is identified with .

Paper Structure

This paper contains 16 sections, 21 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Overview
  4. Main Operator
  5. Star Rings
  6. Basic Definitions
  7. Examples of $^*$–Rings
  8. Projection Space
  9. Tangent Bundle of Functors
  10. Tangent Bundle of Projection Space
  11. Geometry of Rank–1 Projections
  12. Hyperkähler–Like Geometry of the Covering Space of Projections
  13. The Scheme of Rank–1 Projections
  14. Proof of the Main Identity
  15. Isometric Embeddings
  16. ...and 1 more sections

Key Result

Lemma 3.2

In the context of the previous definition: $q+\varepsilon \mathcal{L}_q(a)$ is an idempotent and writing $qq^*q=rq$ for $r\in Z(\mathcal{A}),$ we have that Furthermore, for all $s\in Z(\mathcal{A}),$

Theorems & Definitions (65)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 1.1.1
  • Definition 1.1.2
  • Example 1.1.3
  • Example 1.1.4
  • Example 1.1.5
  • Example 1.1.6
  • Definition 1.2.1
  • ...and 55 more