Holomorphic bundles on the blown-up plane and the bar construction

Abstract

We study the moduli space ${\mathfrak{𝔐}}_k^r\left({\tilde{\mathbb{P}}}_q^2\right)$ of rank $r$ holomorphic bundles with trivial determinant and second Chern class $c_2=k$, over the blowup ${\tilde{\mathbb{P}}}_q^2$ of the projective plane at $q$ points, trivialized on a rational curve. We show that, for $k=1,2$, we have a homotopy equivalence between ${\mathfrak{𝔐}}_k^r\left({\tilde{\mathbb{P}}}_q^2\right)$ and the degree $k$ component of the bar construction $B\left({\mathfrak{𝔐}}^r{\mathbb{P}}^2,{\left({\mathfrak{𝔐}}^r{\mathbb{P}}^2\right)}^q,{\left({\mathfrak{𝔐}}^r{\tilde{\mathbb{P}}}_1^2\right)}^q\right)$. The space ${\mathfrak{𝔐}}_k^r\left({\tilde{\mathbb{P}}}_q^2\right)$ is isomorphic to the moduli space $\mathfrak{𝔐}{\mathcal{ℐ}}_k^r\left(X_q\right)$ of charge $k$ based $SU\left(r\right)$ instantons on a connected sum $X_q$ of $q$ copies of $\overline{{\mathbb{P}}^2}$ and we show that, for $k=1,2$, we have a homotopy equivalence between $\mathfrak{𝔐}{\mathcal{ℐ}}_k^r\left(X_q\#X_s\right)$ and the degree $k$ component of $B\left(\mathfrak{𝔐}{\mathcal{ℐ}}^r\left(X_q\right),\mathfrak{𝔐}{\mathcal{ℐ}}^r\left(S^4\right),\mathfrak{𝔐}{\mathcal{ℐ}}^r\left(X_s\right)\right)$. Analogous results hold in the limit when $k\to \infty$. As an application we obtain upper bounds for the cokernel of the Atiyah–Jones map in homology, in the rank-stable limit.