Enrique Becerra | Searcharxiv

arXiv2026

Regular Bundles on Orbifolds: A Short Proof of Presentability

Let $\X$ be a Hausdorff second-countable smooth orbifold without boundary, possibly noncompact and ineffective. Suppose $\dim\X\leq n$ and $|G_x|\leq B$ for integers $n\geq0$ and $B\geq1$, where $G_x$ is the full stabilizer at $x$. We construct a smooth Hermitian bundle of explicit rank $R(n,B)$ whose fibre at $x$ is a positive multiple of $\C[G_x]$. Adams operations cancel the Bott obstructions on the boundary spheres of a locally finite good triangulation, and the connectivity of Stiefel manifolds gives actual bundles representing the resulting virtual classes. Smoothing preserves all stabilizer representations. Unitary frames give $\X\simeq[M/U(R(n,B))]$ for a smooth manifold $M$ with a proper locally free action; $M$ is compact exactly when $|\X|$ is compact. Thus every compact smooth orbifold is presentable.

math.AT↗

arXiv2024

Revisiting the Classical McKay Correspondence, Derived Equivalences and the Spectrum of Kleinian Surface Singularities: A Look Through the Mirror

In this article, we revisit the classical McKay correspondence via homological mirror symmetry. Specifically, we demonstrate how this correspondence can be articulated as a derived equivalence between the category of vanishing cycles associated with a Kleinian surface singularity and the category of perfect complexes on the corresponding quotient orbifold. We further illustrate how this equivalence allows for the interpretation of the spectrum of a Kleinian surface singularity solely in terms of the representation-theoretic data of the associated binary polyhedral group.

math.AG↗