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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References
Anderson, L.B., Braun, V., Karp, R.L., Ovrut, B.A.: Numerical Hermitian Yang–Mills connections and vector bundle stability in heterotic theories. J. High Energy Phys. 6, 107 (2010)
Arezzo, C., Pacard, F.: Blowing up and desingularizing constant scalar curvature Kähler manifolds. Acta Math. 196(2), 179–228 (2006)
Arezzo, C., Pacard, F., Singer, M.: Extremal metrics on blowups. Duke Math. J 157(1), 1–51 (2011)
Biquard, O., Rollin, Y.: Smoothing singular constant scalar curvature Kähler surfaces and minimal Lagrangians. Adv. Math. 285, 980–1024 (2015)
Donaldson, S.K.: Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. 50(1), 1–26 (1985)
Donaldson, S.K.: Infinite determinants, stable bundles and curvature. Duke Math. J. 54(1), 231–247 (1987)
Douglas, M.R., Karp, R.L., Lukic, S., Reinbacher, R.: Numerical solution to the Hermitian Yang–Mills equation on the Fermat quintic. J. High Energy Phys. 12, 083 (2007)
Huybrechts, D.: Complex Geometry, Universitext. An Introduction. Springer, Berlin (2005)
Kobayashi, S.: Differential Geometry of Complex Vector Bundles, Publications of the Mathematical Society of Japan. Kanô Memorial Lectures, vol. 15, 5th edn. Princeton University Press, Princeton, NJ (1987)
Lockhart, R.B., McOwen, R.C.: Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12(3), 409–447 (1985)
Lübke, M., Teleman, A.: The Kobayashi–Hitchin correspondence. World Scientific Publishing Co. Inc., River Edge, NJ (1995)
Nicholas, P.: Buchdahl, blowups and gauge fields. Pac. J. Math. 196(1), 69–111 (2000)
Sektnan, L.M.: Poincaré Type Kähler Metrics and Stability on Toric Varieties, Ph.D. Thesis, Imperial College, London (2016)
Seyyedali, R., Székelyhidi, G.: Extremal Metrics on Blowups Along Submanifolds, ArXiv e-prints (2016)
Sibley, B.: Asymptotics of the Yang–Mills flow for holomorphic vector bundles over Kähler manifolds: the canonical structure of the limit. J. Reine Angew. Math. 706, 123–191 (2015)
Székelyhidi, G.: On blowing up extremal Kähler manifolds. Duke Math. J. 161(8), 1411–1453 (2012)
Székelyhidi, G.: An Introduction to Extremal Kähler Metrics. Graduate Studies in Mathematics, vol. 152. American Mathematical Society, Providence, RI (2014)
Taubes, C.H.: The existence of anti-self-dual conformal structures. J. Diff. Geom. 36(1), 163–253 (1992)
Uhlenbeck, K., Yau, S.-T.: On the existence of Hermitian–Yang–Mills connections in stable vector bundles. Comm. Pure Appl. Math. 39, S257–S293 (1986)
Wang, X.: Canonical metrics on stable vector bundles. Comm. Anal. Geom. 13(2), 253–285 (2005)
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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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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DOI: https://doi.org/10.1007/s12220-019-00286-0