A classification of complex rank 3 vector bundles on complex projective 5-space

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.