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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2024-11-01
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| author | Indranil Biswas Francois-Xavier Machu |
| author_facet | Indranil Biswas Francois-Xavier Machu |
| author_sort | Indranil Biswas |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-59f07283436a4b0497d2b07e097e2025 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-59f07283436a4b0497d2b07e097e20252024-12-13T16:27:39ZengMDPI AGMathematics2227-73902024-11-011223375710.3390/math12233757Equivariant Holomorphic Hermitian Vector Bundles over a Projective SpaceIndranil Biswas0Francois-Xavier Machu1Department of Mathematics, Shiv Nadar University, NH91, Tehsil Dadri, Greater Noida 201314, Uttar Pradesh, IndiaEcole Supérieure d’Informatique Électronique Automatique (ESIEA), 74 bis Av. Maurice Thorez, 94200 Ivry-sur-Seine, FranceThe 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.https://www.mdpi.com/2227-7390/12/23/3757projective spaceequivariant holomorphic Hermitian vector bundleunitary group |
| spellingShingle | Indranil Biswas Francois-Xavier Machu Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space Mathematics projective space equivariant holomorphic Hermitian vector bundle unitary group |
| title | Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space |
| title_full | Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space |
| title_fullStr | Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space |
| title_full_unstemmed | Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space |
| title_short | Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space |
| title_sort | equivariant holomorphic hermitian vector bundles over a projective space |
| topic | projective space equivariant holomorphic Hermitian vector bundle unitary group |
| url | https://www.mdpi.com/2227-7390/12/23/3757 |
| work_keys_str_mv | AT indranilbiswas equivariantholomorphichermitianvectorbundlesoveraprojectivespace AT francoisxaviermachu equivariantholomorphichermitianvectorbundlesoveraprojectivespace |