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

Abstract

We study the moduli space ${\mathfrak{M}}_k^r({\tilde{\mathbb{P}}}_q^2)$ 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 ${\mathfrak{M}}_k^r({\tilde{\mathbb{P}}}_q^2)$ and the degree $k$ component of the bar construction $\mathrm{B}({\mathfrak{M}}^r{\mathbb{P}}^2,({\mathfrak{M}}^r{\mathbb{P}}^2{)}^q,({\mathfrak{M}}^r{\tilde{\mathbb{P}}}_1^2{)}^q)$. The space ${\mathfrak{M}}_k^r({\tilde{\mathbb{P}}}_q^2)$ is isomorphic to the moduli space $\mathfrak{M}{\mathcal{I}}_k^r(X_q)$ of charge $k$ based $SU(r)$ 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 $\mathfrak{M}{\mathcal{I}}_k^r(X_q\#X_s)$ and the degree $k$ component of $\mathrm{B}(\mathfrak{M}{\mathcal{I}}^r(X_q),\mathfrak{M}{\mathcal{I}}^r(S^4),\mathfrak{M}{\mathcal{I}}^r(X_s))$. 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.

Final version

References in corpus (1)

• Monads for framed torsion-free sheaves on multi-blow-ups of the projective plane

Cited by in corpus (1)

• The Kuranishi map for vector bundles on certain products of curves