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Analytic tangent cones of admissible Hermitian Yang-Mills connections. (English) Zbl 1481.53031

Let us consider a reflexive sheaf \(\mathcal{E}\) on the unit ball of \(\mathbb{C}^n\) with isolated singularity at 0, and an admissible Hermitian-Yang-Mills connection \(A\) on \(\mathcal{E}\) with respect to a Kähler metric \(\omega\). The rescaled sequence of connections \(A_\lambda:=\lambda^*A\) (obtained by the dilatations \(\lambda\in \mathbb{C}^n\)), that are Hermitian-Yang-Mills with respect to a rescaled Kähler metric \(\omega_\lambda\) induced by \(\omega\), converge to a smooth Hermitian-Yang-Mills connection \(A_\infty\) outside a closed complex-analytic subvariety \(\Sigma\) of \(\mathbb{C}^n \setminus \{0\}\) by well-known results in the literature. The set \(\Sigma\) is called the analytic bubbling set. This connection \(A_\infty\) extends to an admissible Hermitian-Yang-Mills connection on \(\mathbb{C}^n\) and defines a reflexive sheaf \(\mathcal{E}_\infty\). The sequence of Yang-Mills energy measures \(\mu_\lambda= \vert F_{A_\lambda}\vert^2 \omega_\lambda^n\) converges weakly (after taking subsequences) to a Radon measure \(\mu\) on \(\mathbb{C}^n\). The triple \((A_\infty,\Sigma, \mu)\) is called by the authors an analytic tangent cone of \(A\) at \(0\). In a nutshell, the goal of this very rich paper is to investigate this analytic tangent cone and to see that this object has a canonical algebraic interpretation. The leitmotiv of the study is to show the uniqueness of the objects obtained at the analytic limit.
Under the above setting and the assumption that \(\mathcal{E}\) is isomorphic to the pull-back of some holomorphic vector bundle \(\underline{\mathcal{E}}\) over \(\mathbb{CP}^{n-1}\), it is proved that \(\mathcal{E}_\infty\) is isomorphic to the double dual of the graded object associated to the Harder-Narasimhan-Seshadri filtration \(Gr(\underline{\mathcal{E}})\) and \(A_\infty\) is gauge equivalent to the natural Hermitian-Yang-Mills connection living on it. Futhermore, the analytic bubbling set \(\Sigma\) is also independent of the choice of subsequences. It agrees with the singular set where the sheaf \(Gr(\underline{\mathcal{E}})\) fails to be locally free, and the measure \(\mu\) is completely determined by \(\underline{\mathcal{E}}\).
The paper generalizes earlier results of the authors [Duke Math. J. 169, No. 14, 2629–2695 (2020; Zbl 1457.14079)] where \(Gr(\underline{\mathcal{E}})\) was supposed to be reflexive. A nice consequence of the paper is that there exists an admissible Hermitian-Yang-Mills connection on a rank-2 reflexive sheaf over \(\mathbb{CP}^3\) such that at all of its singular points, the analytic tangent cones have flat connections but non empty bubbling sets.
Very recently, a more general result (the technical assumptions have been removed) has been proved by the authors in [Invent. Math. 225, No. 1, 73–129 (2021; Zbl 1471.32031)].

MSC:

53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
14H60 Vector bundles on curves and their moduli
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References:

[1] 10.1142/9789814350112_0002 · doi:10.1142/9789814350112_0002
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