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Mathematics > Differential Geometry
Title: Continuity of HYM connections with respect to metric variations
(Submitted on 25 Mar 2024 (this version), latest version 20 Apr 2024 (v3))
Abstract: We investigate the set of (real Dolbeault classes of) balanced metrics $\Theta$ on a balanced manifold $X$ with respect to which a torsion-free coherent sheaf $\mathcal{E}$ on $X$ is slope stable. We prove that the set of all such $[\Theta] \in H^{n - 1,n - 1}(X,\mathbb{R})$ is a convex cone defined by a finite number of linear inequalities. For this, we prove that the set of all $c_1(\mathcal{S})$ for $\mathcal{S} \subset \mathcal{E}$ saturated and coherent is finite.
When $\mathcal{E}$ is a Hermitian vector bundle, the Kobayashi--Hitchin correspondence provides associated Hermitian Yang--Mills connections, which we show depend continuously on the metric, even around classes with respect to which $\mathcal{E}$ is only semi-stable. In this case, the holomorphic structure induced by the connection is the holomorphic structure of the associated graded object. The method relies on semi-stable perturbation techniques for geometric PDEs with a moment map interpretation and is quite versatile, and we hope that it can be used in other similar problems.
Submission history
From: Rémi Delloque [view email][v1] Mon, 25 Mar 2024 14:38:49 GMT (28kb)
[v2] Tue, 9 Apr 2024 12:24:37 GMT (27kb)
[v3] Sat, 20 Apr 2024 14:32:03 GMT (27kb)
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