RECORD
A classification of complex rank 3 vector bundles on complex projective 5-space
Algebraic Topology · Submitted 11 Jan 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This shows that Chern classes are incomplete invariants of topological rank $3$ bundles on $\mathbb CP^5$.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | result 0.70 | 0.44 | 0.23 | asserted 1.00 |
| s1 | 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. | result 0.99 | 0.95 | 0.37 | asserted 1.00 |
| s2 | This shows that Chern classes are incomplete invariants of topological rank $3$ bundles on $\mathbb CP^5$. | interpretation 0.95 | 0.86 | 0.40 | asserted 1.00 |
| s3 | 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 $3$. | method 0.55 | 0.95 | 0.05 | asserted 1.00 |
| s4 | 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. | contribution 0.56 | 0.96 | 0.25 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v3
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
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.
Human review
How the literature uses it
1 arXiv paper cite this in 4 sentences
What it cites · 1 sentence without a confident label
Cites as background 1 paper
-
For $r$ strictly between $1$ and $n$, there is no uniform answer (although some patterns have been found when restricting to bundles with all Chern classes zero, see [ref] ).
Metastable complex vector bundles over complex projective spaces · Introduction · 0.83
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.