Isometric Embeddings and Hyperkähler Geometry of the Cotangent Bundle of Complex Projective Space via the Scheme of Rank-1 Projections
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918411256201216 |
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| author | Lackman, Joshua |
| author_facet | Lackman, Joshua |
| contents | 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 $T^*\mathbb{CP}^{n-1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25632 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Isometric Embeddings and Hyperkähler Geometry of the Cotangent Bundle of Complex Projective Space via the Scheme of Rank-1 Projections Lackman, Joshua Algebraic Geometry Differential Geometry Symplectic Geometry 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 $T^*\mathbb{CP}^{n-1}$. |
| title | Isometric Embeddings and Hyperkähler Geometry of the Cotangent Bundle of Complex Projective Space via the Scheme of Rank-1 Projections |
| topic | Algebraic Geometry Differential Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2603.25632 |