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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

We study the moduli space 𝔐kr( ̃q2) of rank r holomorphic bundles with trivial determinant and second Chern class c2 = k, over the blowup ̃q2 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 𝔐kr( ̃q2) and the degree k component of the bar construction B(𝔐r2,(𝔐r2)q,(𝔐r ̃12)q). The space 𝔐kr( ̃q2) is isomorphic to the moduli space 𝔐kr(Xq) of charge k based SU(r) instantons on a connected sum Xq of q copies of 2¯ and we show that, for k = 1,2, we have a homotopy equivalence between 𝔐kr(Xq # Xs) and the degree k component of B(𝔐r (Xq),𝔐r (S4),𝔐r (Xs)). Analogous results hold in the limit when k . As an application we obtain upper bounds for the cokernel of the Atiyah–Jones map in homology, in the rank-stable limit.

moduli space, holomorphic bundles, instantons, bar construction
Mathematical Subject Classification 2010
Primary: 14D21, 58D27
Secondary: 14J60, 55P48
Received: 6 August 2015
Revised: 19 August 2019
Accepted: 11 November 2019
Published: 4 November 2020
João Paulo Santos
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos
Departamento de Matemática
Instituto Superior Técnico
Universidade de Lisboa