Constructible sheaf complexes in complex geometry and Applications

TL;DR

This paper provides an accessible introduction to constructible sheaf complexes in complex geometry, illustrating their properties, applications, and connections to perverse sheaves, Morse theory, and intersection cohomology with detailed examples and proofs.

Contribution

It offers a comprehensive, example-driven overview of constructible sheaf complexes, including stability, functorial properties, and applications in complex algebraic geometry and topology.

Findings

Stability of constructible sheaves under standard functors

Relation of functors to perverse sheaves and t-structures

Applications to index theorems and intersection cohomology

Abstract

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 K\"ahler package for intersection cohomology of complex algebraic varieties, and the recent study of perverse sheaves on semi-abelian varieties.