Hermitian Yang–Mills metrics on reflexive sheaves over asymptotically cylindrical Kähler manifolds

Abstract

We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical Kähler manifolds: If E is a reflexive sheaf over an ACyl Kähler manifold, which is asymptotic to a µ-stable holomorphic vector bundle, then it admits an asymptotically translationinvariant projectively Hermitian Yang-Mills metrics (with curvature in L 2 loc across the singular set). Our proof combines the analytic continuity method of Uhlenbeck and Yau [35] with the geometric regularization scheme introduced by Bando and Siu [3]. 1 metrics on holomorphic vector bundles which are flat at infinity [2]. Ni and Ren [27] proved that a holomorphic vector bundle over a complete non-compact Kähler manifold with a spectral gap admits a PHYM metric if and only if it admits a metric whose failure to be PHYM is in L p for p > 1 (using an argument similar to Donaldson's solution of the Dirichlet problem for the PHYM equation [8]). Ni [26] showed that the same conclusion holds, for example, if the Kähler manifold satisfies a L 2 Sobolev inequality and p ∈ [1, n/2), or if it is non-parabolic (i.e., admits a positive Green's function) and p = 1. Main resultIn this article we concentrate on the asymptotically cylindrical case, and in view of the applications we have in mind we work with reflexive sheaves (not just holomorphic vector bundles).Theorem 1.1. Let V be an asymptotically cylindrical (ACyl) Kähler manifold with asymptotic cross-section D. Let E D be a µ-stable vector bundle over D, and E a reflexive sheaf asymptotic to E D .In this situation there exists an asymptotically translation-invariant Hermitian metric H on E which satisfies the projective Hermitian Yang-Mills (PHYM) equation). Remark 1.3. A PHYM metric H on E is Hermitian Yang-Mills (HYM) if and only if the induced metric h on det E is HYM, that is, iΛF h = tr(iΛF H ) rk Eis constant. Every asymptotically translationinvariant line bundle over an ACyl Kähler manifold has a HYM metric; however, this metric will typically not be asymptotically translation invariant. See Section 2.3 for a detailed discussion.Remark 1.4. The definition of asymptotically cylindrical Kähler manifolds we work with is given in Definition 2.1; it includes being asymptotically fibred.Remark 1.5. The question of the existence of HYM metrics on holomorphic bundles (with trivial determinant) over ACyl Calabi-Yau 3-folds was studied earlier by Sá Earp [29] (using the Yang-Mills heat flow). Our result improves on his in that we consider general ACyl Kähler manifolds and handle reflexive sheaves; moreover, we give a complete proof of the exponential decay to a PHYM metric over D (which is crucial for applications).Remark 1.6. In dimension four, there is prior work on the relation between ASD instantons and holomorphic vector bundles over cylindrical manifolds by Guo [12] and Owens [28]. Examples and applicationsThere are plenty of examples of ACyl Kähler manifolds and reflexive sheaves on them. Given any smooth projective variety Z containing a smooth divisor D and fibred by |D|, V := Z \ D can be given t...

Tangent cones of Hermitian Yang–Mills connections with isolated singularities

“…We have |∇ H⋄ s| 2 1 − ∆|s| 2 .Proof. Since H = H ⋄ e s is PHYM, we have∆|s| 2 + 2|υ(−s)∇ H⋄ s| 2 ≤ −4 K H⋄ , s with υ(−s) = 1 − e − ads ad s ∈ End(gl(E));see, e.g.,[7, Proposition A.6]. The assertion follows using1 − e −x x 1 1 + |x| , K H⋄ L ∞ ≤ c,which is a consequence of (2.3), and the bound on |s| established in Proposition 3.Proof of Proposition 4.2.…”

Hermitian–Yang–Mills Connections on Collapsing Elliptically Fibered K3 Surfaces

“…Consider , defined by From the definition of the complexified gauge action ( 3.1 ), we have This allows one to compute (for instance, see [ 39 , Appendix A]).…”

“…Our first task is to specify . Fix an endomorphism satisfying Using this curvature bound, along with the inequality (see for instance Proposition A.6 in [ 39 ]), we can apply Moser iteration to conclude Here only depends on and . We now set .…”

A note on singular Hermitian Yang-Mills connections

“…The singularity problem can be addressed by the continuity method due to Bando and Siu [1]; we follow the exposition in [8], Section 6. The idea is to take repeated blow ups Z in the interior of Z so that E Z∖Sing(E) extends as a vector bundle Ẽ across the exceptional locus.…”

“…Then the Bando-Siu interior estimate (cf. Appendix C, D in [8]) gives C k loc -estimates on Hǫ over the locally free locus of E, uniform in ǫ. Furthermore, there are uniform L 2 curvature bounds on Hǫ because of the topological energy formula for HYM connections…”