The book covers a majority of basic notions and results in theory of constructive sheaves on complex spaces and it provides a rich amount of geometrical examples and applications. The author takes the reader from simpler older results to the most powerful and general results currently available. Due to a modest size of the book, some proofs have to be omitted and substituted by references to other sources. In the first chapter, a brief introduction to theory of derived categories and derived functors is given, including an example of derived categories of coherent sheaves on algebraic varieties. The second chapter starts with a general discussion of sheaves and hypercohomology. It contains various versions of the de Rham theorem, a discussion of direct and inverse images of sheaves and the Leray spectral sequences, basic properties of local systems (both topological and analytical aspects). In the third chapter, the author treats Poincaré-Verdier duality and related topics. The fourth chapter describes constructible sheaves and their properties, including characteristic cycles on smooth manifold. Perverse sheaves form a main topic of the book, they are presented in the fifth chapter. Basic properties of perverse sheaves and an explicit description of germs of perverse sheaves on a smooth curve in terms of easy linear algebra are given (including a brief introduction into theory of D-modules). The last chapter contains several applications of perverse sheaves in geometry. The book is well written, I would like to recommend it to anybody interested in the topic.

Reviewer:

jbu