## Yang-Mills connections over compact strongly pseudoconvex CR manifolds.(English)Zbl 0815.32008

The author extends the (Hermitian-Einstein) Yang-Mills theory (known for holomorphic vector bundles over compact Kähler manifolds) to the real odd-dimensional case. He bases on the ideas of Tanaka, who gave an affine connection and defined the notions of a holomorphic vector bundle and its canonical connection over a strongly pseudoconvex CR manifold. The author shows similarities between Tanaka’s connection in the CR case and the Hermitian-Einstein connection in the Kähler case. A suitably chosen connection is called the Yang-Mills-Tanaka connection by the author, and he asks if every holomorphic vector bundle over a CR manifold admits a unique Yang-Mills-Tanaka connection. To make the first step to a solution, he considers the moduli space of Yang-Mills-Tanaka connections $$D$$ of a complex vector bundle $$E$$ on a compact normal CR manifold. He shows that the tangent space of the moduli space at $$[D]$$ is isomorphic to some cohomology group. As an application he shows a unique existence theorem for the Yang-Mills equation for some vector bundle over a compact CR manifold being a $$U(1)$$-bundle over a compact Kähler manifold. He also shows that the moduli space of anti self-dual connections of $$P^ 2(\mathbb{C})$$ induces the moduli space of Yang-Mills-Tanaka connections of the sphere $$S^ 5$$. Finally he examines the holomorphic tangent bundle of a compact normal CR manifold and he gives conditions for the Tanaka’s connection to be a Yang-Mills-Tanaka connection.

### MSC:

 32V99 CR manifolds 32T99 Pseudoconvex domains 32Q20 Kähler-Einstein manifolds 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 32G13 Complex-analytic moduli problems 32J27 Compact Kähler manifolds: generalizations, classification
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### References:

 [1] Akahori, T.: A new approach to the local embedding theorem of CR structures, the local embedding theorem for n., Mem. Am. Math. Soc.366 (1987) · Zbl 0628.32025 [2] Atiyah, M.F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. Ser. A308, 524–615 (1982) · Zbl 0509.14014 [3] Boutet de Monvel, L.: Integration des equations de Cauchy-Riemann induites formelles. Seminaire Goulaonic-Lions-Schwartz (1974–1975) [4] Chern, S.S., Hamilton, R.S.: On Riemannian metrics adapted to three dimensional contact manifolds In: Hirzebruch, T. et al. (eds.) Arbeitstagung, Bonn 1984 (Lect. Notes Math.,1111, vol. 279–305) Berlin Heidelberg New York: Springer (1985) [5] Donaldson, S.: Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc.50, 1–26 (1985) · Zbl 0547.53019 [6] Jerison, D., Lee, J.M.: The Yamabe problem on CR manifold. J. Differ. Geom.25, 167–197 (1987) · Zbl 0661.32026 [7] Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Tokyo: Iwanami 1987 · Zbl 0708.53002 [8] Kuranishi, M.: Strongly pseudoconvex CR structures over small balls. I. Ann. Math.115, 451–500 (1982); II116, 1–64 (1982); III116, 249–330 (1982) · Zbl 0505.32018 [9] Morrow, J., Kodaira, K.: Complex Manifolds. New York: Holt, Rinehart and Winston 1971 · Zbl 0325.32001 [10] Stanton, N.: The heat equation in several complex variables. Bull. Am. Math. Soc.11, 65–84 (1984) · Zbl 0543.58023 [11] Tanaka, N.: A differential geometric study on strongly pseudoconvex manifolds. (Lect. Notes Math., Kyoto Univ. Kyoto: vol. 9) 1975 · Zbl 0331.53025 [12] Tanno, S.: Variational problems on contact Riemannian manifolds. Trans. Am. Math. Soc.314, 349–379 (1989) · Zbl 0677.53043 [13] Uhlenbeck, K., Yau, S.T.: On the existence of hermitian Yang-Mills connects in stable vector bundles. Commun. Pure Appl. Math.39, S257-S293 (1986) · Zbl 0615.58045 [14] Webster, S.M.: Pseudo-hermitian structure on a real hypersurface. J. Differ. Geom.13, 25–41 (1978) · Zbl 0379.53016
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