Schubert varieties and degeneracy loci / William Fulton, Piotr Pragacz

This volume is an introduction to the study of degeneracy loci, a subject to which both authors have made noticeable contributions. A short historical introduction presents the central theme: determine the degree of a locus of matrices (with complex coefficients, say), restricted to certain rank conditions; more generally, and in a modern formulation, compute the homology class (or its class in the Chow ring) of the degeneracy locus of a morphism between vector bundles over an algebraic variety.
The "universal'' situation is that of Schubert varieties in a Grassmannian or a flag manifold: their classes are respectively given by the so-called Schur and Schubert polynomials, which are briefly introduced. Their properties lead, in the modern context, to the famous Thom-Porteous formula. Interesting variants are provided by Lagrangian and orthogonal Grassmannians, where the Schur Q-polynomials, defined in terms of suitable Pfaffians, make their appearance (Chapters 2 and 3). Schur polynomials were classically introduced in connection with the representation theory of symmetric and general linear groups.
Chapters 4 and 5 describe Pragacz's works on polynomials "universally supported by degeneracy loci'' (a vast generalization of the resultant of two polynomials), and on the Euler numbers of degeneracy loci, which the Gauss-Bonnet formula allows one to express in terms of characteristic classes.
Chapters 6 and 7 are devoted to flag manifolds for symplectic and orthogonal groups: the classes of their Schubert varieties cannot be expressed in terms of polynomials with such amazing properties as the famous Schubert polynomials of Lascoux and Schützenberger. Several choices can reasonably be made, each with its own particular weaknesses: both the authors present their own favorite families of Schubert polynomials for the classical groups. A related problem is to understand the intersection loci of two isotropic subbundles of a vector bundle endowed with a symplectic or a nondegenerate quadratic form: two different formulas are given for the classes of these loci. An application to Brill-Noether theory in Prym varieties follows (work of De Concini and Pragacz).
In this series of chapters, the authors juxtapose descriptions of their works in a way that does not completely allow them to avoid repetitions, but the multiplicity of points of views they offer is an evident sign of the wealth of their subject. A last chapter briefly describes some connected themes and a few open problems, the most striking being the quest for the multiplication rules of Schubert polynomials. Let us hope that, thanks to these pages of great clarity, the present volume will contribute to the opening of a golden new era in the study of degeneracy loci, as exciting as the last twenty years have been. (MathSciNet)