A classification of complex rank 3 vector bundles on complex projective 5-space
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917738229792768 |
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| author | Opie, Morgan |
| author_facet | Opie, Morgan |
| contents | Given integers $a_1,a_2,a_3$, there is a complex rank $3$ topological bundle on $\mathbb CP^5$ with $i$-th Chern class equal to $a_i$ if and only if $a_1,a_2,a_3$ satisfy the Schwarzenberger condition. Provided that the Schwarzenberger condition is satisfied, we prove that the number of isomorphism classes of rank $3$ bundles $V$ on $\mathbb C P^5$ with $c_i(V)=a_i$ is equal to $3$ if $a_1$ and $a_2$ are both divisible by $3$ and equal to $1$ otherwise.
This shows that Chern classes are incomplete invariants of topological rank $3$ bundles on $\mathbb CP^5$. To address this problem, we produce a universal class in the $tmf$-cohomology of a Thom spectrum related to $BU(3)$, where $tmf$ denotes topological modular forms localized at $3$. From this class and orientation data, we construct a $\mathbb Z/3$-valued invariant of the bundles of interest and prove that our invariant separates distinct bundles with the same Chern classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_04313 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A classification of complex rank 3 vector bundles on complex projective 5-space Opie, Morgan Algebraic Topology Given integers $a_1,a_2,a_3$, there is a complex rank $3$ topological bundle on $\mathbb CP^5$ with $i$-th Chern class equal to $a_i$ if and only if $a_1,a_2,a_3$ satisfy the Schwarzenberger condition. Provided that the Schwarzenberger condition is satisfied, we prove that the number of isomorphism classes of rank $3$ bundles $V$ on $\mathbb C P^5$ with $c_i(V)=a_i$ is equal to $3$ if $a_1$ and $a_2$ are both divisible by $3$ and equal to $1$ otherwise. This shows that Chern classes are incomplete invariants of topological rank $3$ bundles on $\mathbb CP^5$. To address this problem, we produce a universal class in the $tmf$-cohomology of a Thom spectrum related to $BU(3)$, where $tmf$ denotes topological modular forms localized at $3$. From this class and orientation data, we construct a $\mathbb Z/3$-valued invariant of the bundles of interest and prove that our invariant separates distinct bundles with the same Chern classes. |
| title | A classification of complex rank 3 vector bundles on complex projective 5-space |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2301.04313 |