Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space

Biswas, Indranil; Machu, Francois-Xavier

doi:10.3390/math12233757

Open AccessArticle

Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space

by

Indranil Biswas

^1,* and

Francois-Xavier Machu

²

¹

Department of Mathematics, Shiv Nadar University, NH91, Tehsil Dadri, Greater Noida 201314, Uttar Pradesh, India

²

Ecole Supérieure d’Informatique Électronique Automatique (ESIEA), 74 bis Av. Maurice Thorez, 94200 Ivry-sur-Seine, France

^*

Author to whom correspondence should be addressed.

Mathematics 2024, 12(23), 3757; https://doi.org/10.3390/math12233757

Submission received: 9 October 2024 / Revised: 11 November 2024 / Accepted: 27 November 2024 / Published: 28 November 2024

(This article belongs to the Special Issue Advanced Algebraic Geometry and Applications)

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Abstract

The aim here is to describe all isomorphism classes of

SU (n + 1)

-equivariant Hermitian holomorphic vector bundles on the complex projective space

C P^{n}

. If

G \subset SU (n + 1)

is the isotropy subgroup of a chosen point

x_{0} \in C P^{n}

, and

ρ : G ⟶ GL (V)

is a unitary representation, we obtain

SU (n + 1)

-equivariant holomorphic Hermitian vector bundles on

C P^{n}

. Next, given any

v \in End (V_{ρ}) \otimes {(T_{z_{0}}^{0, 1} C P^{n})}^{*}

satisfying certain conditions, a new structure of an

SU (n + 1)

-equivariant holomorphic Hermitian vector bundle on this underlying

C^{\infty}

holomorphic Hermitian bundle is obtained. It is shown that all

SU (n + 1)

-equivariant holomorphic Hermitian vector bundles on

C P^{n}

arise this way.

Keywords: projective space; equivariant holomorphic Hermitian vector bundle; unitary group

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MDPI and ACS Style

Biswas, I.; Machu, F.-X. Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space. Mathematics 2024, 12, 3757. https://doi.org/10.3390/math12233757

AMA Style

Biswas I, Machu F-X. Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space. Mathematics. 2024; 12(23):3757. https://doi.org/10.3390/math12233757

Chicago/Turabian Style

Biswas, Indranil, and Francois-Xavier Machu. 2024. "Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space" Mathematics 12, no. 23: 3757. https://doi.org/10.3390/math12233757

APA Style

Biswas, I., & Machu, F.-X. (2024). Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space. Mathematics, 12(23), 3757. https://doi.org/10.3390/math12233757

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space

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