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

Santos, João Paulo¹

¹Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

Algebraic and Geometric Topology, Tome 20 (2020) no. 5, pp. 2177-2268

Résumé

We study the moduli space 𝔐kr(ℙ ̃q2) of rank r holomorphic bundles with trivial determinant and second Chern class c2 = k, over the blowup ℙ ̃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(ℙ ̃q2) and the degree k component of the bar construction B(𝔐rℙ2,(𝔐rℙ2)q,(𝔐rℙ ̃12)q). The space 𝔐kr(ℙ ̃q2) is isomorphic to the moduli space 𝔐ℐkr(Xq) of charge k based SU(r) instantons on a connected sum Xq of q copies of ℙ2¯ and we show that, for k = 1,2, we have a homotopy equivalence between 𝔐ℐkr(Xq # Xs) and the degree k component of B(𝔐ℐr (Xq),𝔐ℐr (S4),𝔐ℐr (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.

Détail

DOI : 10.2140/agt.2020.20.2177

Classification : 14D21, 58D27, 14J60, 55P48
Keywords: moduli space, holomorphic bundles, instantons, bar construction

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