Uniqueness and algebraic characterization of analytic tangent cones

Let $B$ be a ball in $\mathbb C^n$ and let $\mathcal E$ be a reflexive sheaf over $B$ with an isolated singularity at $0$. For an admissible Hermitian–Yang–Mills connection $A$ on $\mathcal E$, an analytic tangent cone is a triple $(A_\infty,\Sigma,\mu)$ obtained as a limit under rescaling toward $0$. Its underlying reflexive sheaf on $\mathbb C^n$ is denoted $\mathcal E_\infty$. An algebraic tangent cone is a torsion-free sheaf $\underline{\mathcal E}^{alg}$ on $\mathbb C\mathbb P^{n-1}$ obtained by extending the pullback of $\mathcal E$ from the punctured ball across the exceptional divisor. Let $\pi:\mathbb C^n_*\to\mathbb C\mathbb P^{n-1}$ be the natural projection and let $\psi:\mathbb C^n_*\to\mathbb C^n$ be inclusion. Write $Gr^{HNS}$ for the graded object associated to the Harder–Narasimhan–Seshadri filtration. The analytic tangent cone conjecture. (I) For every admissible Hermitian–Yang–Mills connection $A$ on $\mathcal E$, all analytic tangent cones at $0$ have gauge-equivalent admissible connections $A_\infty$; consequently, $\mathcal E_\infty$ is unique up to isomorphism. (II) There is an algebraic tangent cone $\underline{\mathcal E}^{alg}$ on $\mathbb C\mathbb P^{n-1}$ such that, for every admissible Hermitian–Yang–Mills connection $A$ on $\mathcal E$, the sheaf corresponding to its analytic tangent cones satisfies

$$\mathcal E_\infty\cong\psi_*\pi^*((Gr^{HNS}(\underline{\mathcal E}^{alg}))^{**}).$$

The conjecture seeks uniqueness of analytic tangent cones and their determination by algebraic data, while the surrounding discussion emphasizes that uniqueness is not known a priori because tangent cones may depend on the rescaling subsequence. The conjecture also identifies the analytic limit with the graded Harder–Narasimhan–Seshadri object of an algebraic tangent cone.