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

We show that the hyperkahler geometry of $T^{\ast }{\mathbb{C}\mathbb{P}}^{n-1}$ can be described algebraically by the affine scheme of rank-1 projections, and that this description simultaneously yields explicit $SU(n)$-equivariant isometric embeddings $$T^{\ast }{\mathbb{C}\mathbb{P}}^{n-1}\hookrightarrow {\mathbb{R}}^{(n^2+1{)}^2},$$ as well as a generalization of the hyperkahler geometry of $T^{\ast }{\mathbb{C}\mathbb{P}}^{n-1}$ 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 $T^{\ast }{\mathbb{C}\mathbb{P}}^{n-1}$.