Isometric Embeddings of Hyperkähler T^*ℂℙ^n

Abstract

We construct explicit U(n)-equivariant isometric embeddings T^*ℂℙⁿ⁻¹ ↪ ℝ^{n⁴+2n²−1}, n≥ 2, with respect to Calabi's hyperkähler metric on T^*ℂℙⁿ⁻¹. To achieve this, we first construct a diffeomorphism from T^*ℂℙⁿ⁻¹ onto the manifold of rank-1 projections in M_n(ℂ), such that the pullback of half of the trace-class norm is a Kähler potential for the hyperkähler metric. We then use the Hilbert space structure of M_n(ℂ) to produce the isometric embeddings into Euclidean space.

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BibTeX RIS

BibTeX

@misc{lackman2026isometric,
  title = {{Isometric Embeddings of Hyperkähler $\mathrmT^*\mathbbCP^n$}},
  author = {Joshua Lackman},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02089},
  eprint = {2611.02089},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02089},
}

RIS

TY  - UNPB
TI  - Isometric Embeddings of Hyperkähler T^*ℂℙ^n
AU  - Joshua Lackman
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.02089
UR  - https://arxiv.org/abs/2611.02089
AB  - We construct explicit U(n)-equivariant isometric embeddings T^*ℂℙ^n−1 ↪ ℝ^n^4+2n^2−1, n≥ 2, with respect to Calabi's hyperkähler metric on T^*ℂℙ^n−1. To achieve this, we first construct a diffeomorphism from T^*ℂℙ^n−1 onto the manifold of rank-1 projections in M_n(ℂ), such that the pullback of half of the trace-class norm is a Kähler potential for the hyperkähler metric. We then use the Hilbert space structure of M_n(ℂ) to produce the isometric embeddings into Euclidean space.
ER  -

Isometric Embeddings of Hyperkähler T^*ℂℙⁿ

Abstract

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