Cohomology of Vector Bundles and Syzygies
The main aim of this monograph is to calculate syzygies of some algebraic varieties. Chapters 1-4 have a preparatory character. They contain a lot of material, starting with commutative and homological algebra, introducing an important concept of a Schur functor, devoting great attention to the Grassmannians and flag varieties, and culminating with the Bott theorem on cohomology of line bundles on flag varieties. The theoretical core of the book is represented in Chapter 5, where the basic idea of the geometric technique of calculating syzygies is described. The most important role played here is by the direct image of a Koszul complex. The remaining Chapters, 6-9, are then devoted to applications. The above-mentioned technique with various necessary modifications is applied to determinantal varieties (for generic, generic symmetric and generic skew symmetric matrices), higher rank varieties (the three cases as above), nilpotent orbit closures of the adjoint action of a simple algebraic group on its Lie algebra, and to resultant and discriminant varieties. The book is designed for specialists and postgraduate students in the field. It will bring the reader into the center of contemporary research. It is very nicely written but it requires a solid background from algebra, algebraic geometry and sheaf theory. It is very pleasant that it contains many examples and many exercises. As mentioned above, to read it may require certain preparation and effort but the reading is very interesting and the book can be strongly recommended.