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

Santos, João Paulo

doi:10.2140/agt.2020.20.2177

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

João Paulo Santos

Algebraic & Geometric Topology 20 (2020) 2177–2268

DOI: 10.2140/agt.2020.20.2177

Abstract

We study the moduli space $𝔐_{k}^{r} ({\tilde{ℙ}}_{q}^{2})$ of rank $r$ holomorphic bundles with trivial determinant and second Chern class $c_{2} = k$ , over the blowup ${\tilde{ℙ}}_{q}^{2}$ 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 $𝔐_{k}^{r} ({\tilde{ℙ}}_{q}^{2})$ and the degree $k$ component of the bar construction $B (𝔐^{r} ℙ^{2}, {(𝔐^{r} ℙ^{2})}^{q}, {(𝔐^{r} {\tilde{ℙ}}_{1}^{2})}^{q})$ . The space $𝔐_{k}^{r} ({\tilde{ℙ}}_{q}^{2})$ is isomorphic to the moduli space $𝔐 ℐ_{k}^{r} (X_{q})$ of charge $k$ based $SU (r)$ instantons on a connected sum $X_{q}$ of $q$ copies of $\bar{ℙ^{2}}$ and we show that, for $k = 1, 2$ , we have a homotopy equivalence between $𝔐 ℐ_{k}^{r} (X_{q} # X_{s})$ and the degree $k$ component of $B (𝔐 ℐ^{r} (X_{q}), 𝔐 ℐ^{r} (S^{4}), 𝔐 ℐ^{r} (X_{s}))$ . 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.

Keywords

moduli space, holomorphic bundles, instantons, bar construction

Mathematical Subject Classification 2010

Primary: 14D21, 58D27

Secondary: 14J60, 55P48

References

Publication

Received: 6 August 2015

Revised: 19 August 2019

Accepted: 11 November 2019

Published: 4 November 2020

Authors

João Paulo Santos
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos Departamento de Matemática Instituto Superior Técnico Universidade de Lisboa Lisboa Portugal

Volume 20, issue 5 (2020)