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2020 Holomorphic bundles on the blown-up plane and the bar construction
João Paulo Santos
Algebr. Geom. Topol. 20(5): 2177-2268 (2020). DOI: 10.2140/agt.2020.20.2177

Abstract

We study the moduli space 𝔐 k r ( β„™ Μƒ q 2 ) of rank r holomorphic bundles with trivial determinant and second Chern class c 2 = k , over the blowup β„™ Μƒ 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 ( β„™ Μƒ q 2 ) and the degree k component of the bar construction B ( 𝔐 r β„™ 2 , ( 𝔐 r β„™ 2 ) q , ( 𝔐 r β„™ Μƒ 1 2 ) q ) . The space 𝔐 k r ( β„™ Μƒ 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 Β copies of β„™ 2 Β― 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 β†’ ∞ . As an application we obtain upper bounds for the cokernel of the Atiyah–Jones map in homology, in the rank-stable limit.

Citation

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João Paulo Santos. "Holomorphic bundles on the blown-up plane and the bar construction." Algebr. Geom. Topol. 20 (5) 2177 - 2268, 2020. https://doi.org/10.2140/agt.2020.20.2177

Information

Received: 6 August 2015; Revised: 19 August 2019; Accepted: 11 November 2019; Published: 2020
First available in Project Euclid: 10 November 2020

MathSciNet: MR4171566
Digital Object Identifier: 10.2140/agt.2020.20.2177

Subjects:
Primary: 14D21, 58D27
Secondary: 14J60, 55P48

Rights: Copyright Β© 2020 Mathematical Sciences Publishers

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