Laurenţiu G. Maxim | Searcharxiv

arXiv2026

Euclidean distance degree defect of singular projective varieties

The unit Euclidean distance degree and the generic Euclidean distance degree are two well-studied invariants of projective varieties. These quantities measure the algebraic complexity of nearest-point problems on a variety, and in many examples arising in optimization, engineering, statistics, and data science, there is a significant gap between them. We refer to this difference as the defect of the Euclidean distance (ED) degree. In this paper, we provide a constructible enhancement and a topological formula for the defect of the ED degree of an arbitrary complex projective variety, extending our previous results from the smooth setting. Since the generic Euclidean distance degree is typically more tractable, our approach offers a new method for computing ED degrees in broad generality.

math.AG↗

arXiv2021

Constructible sheaf complexes in complex geometry and Applications

We present a detailed introduction of the theory of constructible sheaf complexes in the complex algebraic and analytic setting. All concepts are illustrated by many interesting examples and relevant applications, while some important results are presented with complete proofs. This paper is intended as a broadly accessible user's guide to these topics, providing the readers with a taste of the subject, reflected by concrete examples and applications that motivate the general theory. We discuss the stability of constructible sheaf complexes under the standard functors, and explain the relation of these functors to perverse sheaves and the perverse $t$-structure. We introduce the main results of stratified Morse theory in the framework of constructible sheaves, for proving the basic vanishing and finiteness results. Applications are given to various index theorems, the functorial calculus of characteristic cycles of constructible functions, and to weak Lefschetz and Artin-Grothendieck type theorems. We recall the construction of Deligne's nearby and vanishing cycle functors, prove that they preserve constructible complexes, and discuss their relation with the perverse $t$-structure. We finish this paper with a description and applications of the Kähler package for intersection cohomology of complex algebraic varieties, and the recent study of perverse sheaves on semi-abelian varieties.

math.AG↗