Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space

Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space

The aim here is to describe all isomorphism classes of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>SU</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1&l...

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Bibliographic Details
Main Authors:	Indranil Biswas, Francois-Xavier Machu
Format:	Article
Language:	English
Published:	MDPI AG 2024-11-01
Series:	Mathematics
Subjects:	projective space equivariant holomorphic Hermitian vector bundle unitary group
Online Access:	https://www.mdpi.com/2227-7390/12/23/3757
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Summary:	The aim here is to describe all isomorphism classes of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>SU</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-equivariant Hermitian holomorphic vector bundles on the complex projective space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi><msup><mrow><mi mathvariant="double-struck">P</mi></mrow><mi>n</mi></msup></mrow></semantics></math></inline-formula>. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mspace width="0.166667em"></mspace><mo>⊂</mo><mspace width="0.166667em"></mspace><mi>SU</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> is the isotropy subgroup of a chosen point <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mspace width="0.166667em"></mspace><mo>∈</mo><mspace width="0.166667em"></mspace><mi mathvariant="double-struck">C</mi><msup><mrow><mi mathvariant="double-struck">P</mi></mrow><mi>n</mi></msup></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ρ</mi><mspace width="0.166667em"></mspace><mo>:</mo><mspace width="0.166667em"></mspace><mi>G</mi><mspace width="0.166667em"></mspace><mo>⟶</mo><mspace width="0.166667em"></mspace><mi>GL</mi><mo>(</mo><mi>V</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a unitary representation, we obtain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>SU</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-equivariant holomorphic Hermitian vector bundles on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi><msup><mrow><mi mathvariant="double-struck">P</mi></mrow><mi>n</mi></msup></mrow></semantics></math></inline-formula>. Next, given any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mspace width="0.166667em"></mspace><mo>∈</mo><mspace width="0.166667em"></mspace><mi>End</mi><mrow><mo>(</mo><msub><mi>V</mi><mi>ρ</mi></msub><mo>)</mo></mrow><mo>⊗</mo><msup><mrow><mo>(</mo><msubsup><mi>T</mi><msub><mi>z</mi><mn>0</mn></msub><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msubsup><mi mathvariant="double-struck">C</mi><msup><mrow><mi mathvariant="double-struck">P</mi></mrow><mi>n</mi></msup><mo>)</mo></mrow><mo>∗</mo></msup></mrow></semantics></math></inline-formula> satisfying certain conditions, a new structure of an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>SU</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-equivariant holomorphic Hermitian vector bundle on this underlying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mo>∞</mo></msup></semantics></math></inline-formula> holomorphic Hermitian bundle is obtained. It is shown that all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>SU</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-equivariant holomorphic Hermitian vector bundles on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi><msup><mrow><mi mathvariant="double-struck">P</mi></mrow><mi>n</mi></msup></mrow></semantics></math></inline-formula> arise this way.
ISSN:	2227-7390