This is the first of a series of papers where we relate tangent cones of Hermitian–Yang–Mills connections at a singularity to the complex algebraic geometry of the underlying reflexive sheaf. In this paper we work on the case when the sheaf is locally modeled on the pullback of a holomorphic vector bundle from the projective space, and we shall impose an extra assumption that the graded sheaf determined by the Harder–Narasimhan–Seshadri filtrations of the vector bundle is reflexive. In general, we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of a Harder–Narasimhan–Seshadri filtration of an algebraic tangent cone, which is a certain torsion-free sheaf on the projective space. In this paper we also prove this conjecture when there is an algebraic tangent cone which is locally free and stable.

We study the set of connected components of certain unions of affine Deligne–Lusztig varieties arising from the study of Shimura varieties. We determine the set of connected components for basic $\sigma$-conjugacy classes. As an application, we verify the Axioms in recent work by the first author and Rapoport for certain PEL-type Shimura varieties. We also show that, for any nonbasic $\sigma$-conjugacy class in a residually split group, the set of connected components is “controlled” by the set of straight elements associated to the $\sigma$-conjugacy class, together with the obstruction from the corresponding Levi subgroup. Combined with the second author’s earlier article, this allows one to verify, in the residually split case, the description of the mod-$p$ isogeny classes on Shimura varieties conjectured by Langlands and Rapoport. Along the way, we determine the Picard group of the Witt vector affine Grassmannian first proposed by Bhatt, Scholze, and Zhu, which is of independent interest.

For every compact Kähler manifold $X$ of algebraic dimension $a\left(X\right)=dimX-1$, we prove that $X$ has arbitrarily small deformations to some projective manifolds.