Metastable representative obstructions and enumeration formula

Determine, for a fixed dimension m and a rank r satisfying m/2 ≤ r < m, whether Chern classes are the only obstruction to a reduced complex K-theory class h on complex projective space CP^m admitting a rank-r representative, whether the cardinality of Vec^h_r(CP^m) always divides that of Vec^0_r(CP^m), and whether a closed formula exists for the cardinality of Vec^h_r(CP^m) in terms of m, r, and the low-degree Chern classes of h.

Background

The paper computes the enumeration of rank-n complex vector bundles on CP{n+2} representing a prescribed K-theory class when n is odd. The resulting 24-periodic formulas show that existence is controlled by the top two Chern classes and that, when representatives exist, the number of representatives is related to divisibility properties of the first and second Chern classes.

Motivated by these computations and by analogous results for stably trivial bundles and corank-one bundles, the authors propose a broader problem for arbitrary complex projective spaces CPm and ranks r in the metastable range m/2 ≤ r < m. The question asks whether the observed structural features persist uniformly: namely, whether Chern classes completely determine existence, whether nontrivial stable classes yield counts dividing the stably trivial count, and whether the count admits a formula depending only on the dimension, rank, and small Chern classes.

References

Fix a dimension $m$ and a rank $r$ in the {\em metastable range} (that is, for $\frac{m}{2}\leq r<m$). Let $h: {m} \to BU$ be a reduced $K$-theory class. Are Chern classes the only obstruction to $h$ admitting a rank $r$ representative? Is the size of $Vech_r(m)$ always a divisor of the size of $Vec0_r(m)$? Is there a closed formula for the size of $Vech_r(m)$ in terms of $m$, $r$, and small Chern classes of $h$?

— Enumerating corank 2 complex vector bundles on odd complex projective spaces  (2608.23488 - Hu et al., 24 Aug 2026) in Question 1, Section 1 (Introduction), labeled \Cref{conj:bound}

By p. 152, if $n=2$, \Cref{lem:exists-n-odd-c1-even} is also true for $c_1(h)$ odd: every rank $2$ bundle on $3$ with odd first Chern class extends over $4$. The proof makes use of some explicit computations in twisted symplectic $K$-theory, and it is not clear how to adapt this argument to even $n=4$.

— Enumerating corank 2 complex vector bundles on odd complex projective spaces  (2608.23488 - Hu et al., 24 Aug 2026) in Remark following Lemma \Cref{lem:exists-n-odd-c1-even}, Section 3 (Preliminary results)