Hermitian Yang–Mills Connections on Blowups

Abstract

Consider a vector bundle over a Kähler manifold which admits a Hermitian Yang–Mills connection. We show that the pullback bundle on the blowup of the Kähler manifold at a collection of points also admits a Hermitian Yang–Mills connection, for Kähler classes on the blowup which make the exceptional divisors small. Our proof uses gluing techniques, and is hence asymptotically explicit. This recovers, through the Hitchin–Kobayashi correspondence, algebro-geometric results due to Buchdahl and Sibley.

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Acknowledgements

The authors would like to thank Ben Sibley for helpful discussions. The second named author is thankful to the CIRGET who support his postdoctoral position.

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Correspondence to Lars Martin Sektnan.

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Dervan, R., Sektnan, L.M. Hermitian Yang–Mills Connections on Blowups. J Geom Anal 31, 516–542 (2021). https://doi.org/10.1007/s12220-019-00286-0

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Keywords

  • Hermitian Yang–Mills connections
  • Kähler manifolds
  • Blowups

Mathematics Subject Classification

  • Primary 53C07
  • Secondary 53C55