Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access March 12, 2022

Polystable bundles and representations of their automorphisms

Nicholas Buchdahl and Georg Schumacher
From the journal Complex Manifolds

Abstract

Using a quasi-linear version of Hodge theory, holomorphic vector bundles in a neighbourhood of a given polystable bundle on a compact Kähler manifold are shown to be (poly)stable if and only if their corresponding classes are (poly)stable in the sense of geometric invariant theory with respect to the linear action of the automorphism group of the bundle on its space of in˝nitesimal deformations.

MSC 2010: 53C07; 14L24; 32G13; 32L10; 32Q15

References

[1] M. F. Atiyah, N. J. Hitchin and I. M. Singer: Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A, 362 (1978), 425–461.10.1098/rspa.1978.0143Search in Google Scholar

[2] S. Bando and Y.-T. Siu: Stable sheaves and Einstein-Hermitian metrics. In: Geometry and analysis on complex manifolds, (World Sci. Publ., River Edge, NJ, 1994), 39–50.Search in Google Scholar

[3] T. A. Brönnle: Deformation constructions of extremal metrics. Ph.D. thesis, Imperial College, London, UK, 2011.Search in Google Scholar

[4] N. P. Buchdahl: Stable 2-bundles on Hirzebruch surfaces, Math. Z. 194 (1987), 143–152.10.1007/BF01168013Search in Google Scholar

[5] N. P. Buchdahl: Hermitian-Einstein connections and stable vector bundles over compact complex surfaces, Math. Ann. 280 (1988), 625–648.10.1007/BF01450081Search in Google Scholar

[6] N. P. Buchdahl: Sequences of stable bundles over compact complex surfaces, J. Geom. Anal. 9 (1999), 391–42810.1007/BF02921982Search in Google Scholar

[7] N. P. Buchdahl and G. Schumacher: An analytic application of Geometric Invariant Theory, J. Geom. Phys. 165 (2021), 104237.10.1016/j.geomphys.2021.104237Search in Google Scholar

[8] N. P. Buchdahl and G. Schumacher: An analytic application of Geometric Invariant Theory II: Coarse moduli spaces. J. Geom. Phys. 175 (2022), 104467.10.1016/j.geomphys.2022.104467Search in Google Scholar

[9] X. Chen and S. Sun: Calabi ˛ow, geodesic rays, and uniqueness of constant scalar curvature Kähler metrics, Ann. of Math. 180 (2014), 407–454.10.4007/annals.2014.180.2.1Search in Google Scholar

[10] S. K. Donaldson: Instantons and geometric invariant theory, Comm. Math. Phys. 93 (1984), 453–460.10.1007/BF01212289Search in Google Scholar

[11] S. K. Donaldson: Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. 50 (1985), 1–26.10.1112/plms/s3-50.1.1Search in Google Scholar

[12] S. K. Donaldson and P. B. Kronheimer: The geometry of four-manifolds, Oxford University Press, New York, 1990.Search in Google Scholar

[13] Y. Fan: Construction of the moduli space of Higgs bundles using analytic methods, arXiv:2004.07182v2 (2020).Search in Google Scholar

[14] O. Forster and K. Knorr: Über die Deformationen von Vektorraumbündeln auf kompakten komplexen Räumen, Math. Ann. 209 (1974), 291–34610.1007/BF01351725Search in Google Scholar

[15] A. Fujiki and G. Schumacher: The moduli space of Hermite-Einstein bundles on a compact Kähler manifold, Proc. Japan Acad. Ser. A Math. Sci. 63 (1987), 69–72.10.3792/pjaa.63.69Search in Google Scholar

[16] M. Garcia-Fernandez and C. Tipler: Deformation of complex structures and the coupled Kähler-Yang-Mills equations, J. London Math. Soc. 89 (2014), 779–796.10.1112/jlms/jdt084Search in Google Scholar

[17] A. Jacob: Existence of approximate Hermitian-Einstein structures on semi-stable bundles, Asian J. Math. 18 (2014), 859–883.10.4310/AJM.2014.v18.n5.a5Search in Google Scholar

[18] A. W. Knapp: Lie Groups Beyond an Introduction, Birkhäuser, Boston, 1996.10.1007/978-1-4757-2453-0Search in Google Scholar

[19] G. Kempf and L. Ness: Lengths of vectors in representation spaces, Lecture Notes in Mathematics Vol. 732, Springer, Berlin-Heidelberg-New York, 1978, 233–244.10.1007/BFb0066647Search in Google Scholar

[20] F. C. Kirwan: Cohomology of quotients in symplectic and algebraic geometry. Princeton University Press, Princeton, NJ, 1984.10.1515/9780691214566Search in Google Scholar

[21] S. Kobayashi: First Chern class and holomorphic tensor ˝elds, Nagoya Math. J. 77 (1980), 5–11.10.1017/S0027763000018602Search in Google Scholar

[22] S. Kobayashi: Curvature and stability of vector bundles, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), 158–162.Search in Google Scholar

[23] S. Kobayashi: Di˙erential geometry of complex vector bundles, Princeton University Press, Princeton, NJ 1987.10.1515/9781400858682Search in Google Scholar

[24] M. Kuranishi: On the locally complete families of complex analytic structures, Ann. of Math. 75 (1962), 536–577.10.2307/1970211Search in Google Scholar

[25] E. Lerman: Gradient ˛ow of the norm squared of a moment map, Enseign. Math. 51 (2005), 117–127.Search in Google Scholar

[26] M. Lübke: Stability of Einstein-Hermitian vector bundles, Manuscripta Math. 42 (1983), 245–257.10.1007/BF01169586Search in Google Scholar

[27] D. Luna: Slices étales, Bull. Soc. Math. France 33 (1973), 81–105.Search in Google Scholar

[28] K. Miyajima: Kuranishi family of vector bundles and algebraic description of the moduli space of Einstein-Hermitian connections, Publ. Res. Inst. Math. Sci. 25 (1989), 301–320.10.2977/prims/1195173613Search in Google Scholar

[29] D. Mumford, J. Fogarty and F. Kirwan: Geometric invariant theory (3rd ed.), Springer, Berlin, 1994.10.1007/978-3-642-57916-5Search in Google Scholar

[30] A. Neeman: The topology of quotient varieties, Ann. of Math. 122 (1985), 419–459.10.2307/1971309Search in Google Scholar

[31] L. Ness: A strati˝cation of the null cone via the moment map. With an appendix by David Mumford. Amer. J. Math. 106 (1984), 1281–1329.10.2307/2374395Search in Google Scholar

[32] G. Székelyhidi: The Kähler-Ricci ˛ow and K-polystability, Am. J. Math. 132 (2010), 1077–1090.10.1353/ajm.0.0128Search in Google Scholar

[33] F. Takemoto: Stable vector bundles on algebraic surfaces, Nagoya Math. J. 47 (1972), 29–48.10.1017/S0027763000014896Search in Google Scholar

[34] R. P. Thomas: Notes on GIT and symplectic reduction for bundles and varieties. In: Surveys in di˙erential geometry 10, Int. Press, Somerville, MA (2006), 221–273.10.4310/SDG.2005.v10.n1.a7Search in Google Scholar

[35] K. K. Uhlenbeck and S.-T. Yau: On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986), S257–S293.10.1002/cpa.3160390714Search in Google Scholar

Received: 2021-07-31
Revised: 2021-12-22
Accepted: 2022-02-02
Published Online: 2022-03-12

© 2022 Nicholas Buchdahl et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Scroll Up Arrow