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Hermitian Yang-Mills metrics on reflexive sheaves over asymptotically cylindrical Kähler manifolds. (English) Zbl 1516.53029

Summary: We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical (ACyl) Kähler manifolds: If \(\mathcal E\) is a reflexive sheaf over an ACyl Kähler manifold, which is asymptotic to a \(\mu\)-stable holomorphic vector bundle, then it admits an asymptotically translation-invariant projectively Hermitian Yang-Mills metric (with curvature in \(L^2_{\operatorname{loc}}\) across the singular set). Our proof combines the analytic continuity method of K. Uhlenbeck and S. T. Yau [Commun. Pure Appl. Math. 39, S257–S293 (1986; Zbl 0615.58045)] with the geometric regularization scheme introduced by S. Bando and Y.-T. Siu [in: Geometry and analysis on complex manifolds. Festschrift for Professor S. Kobayashi’s 60th birthday. Singapore: World Scientific. 39–59 (1994; Zbl 0880.32004)] .

MSC:

53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
32L05 Holomorphic bundles and generalizations
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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