Schubert varieties and degeneracy loci.

*(English)*Zbl 0913.14016
Lecture Notes in Mathematics. 1689. Berlin: Springer. xi, 148 p. (1998).

These notes are based upon a series of lectures that both authors had given in a summer school at Thurnau, Germany, held from June 19 to June 23, 1995. The lectures were designed to provide an introduction to the theory of Schubert varieties, at its contemporary state of knowledge, and to the related theory of degeneracy loci of vector bundle morphisms in algebraic geometry. The text under review follows closely the lectures delivered at Thurnau, the notes of which had been circulating, since then, among the community of algebraic geometers, but it has been enhanced, in its present published from, by ten additional appendices and a few up-dating remarks or footnotes. As the authors emphasize in the preface to the book, this text is neither intended to be a textbook, nor a research monograph, nor a survey on the subject. Instead, they have tried to describe what they, in their capacity of being two of the most active and competent researches in this area of algebraic geometry, regard as essential features of the whole complex of topics, each from his own point of view. The outcome is a great, huge panorama of a fascinating subject in both classical and contemporary algebraic geometry. The present text consists of nine chapters, ten appendices to them, and an utmost rich bibliography.

Chapter I starts with the classical origin of the whole subject, that is, with the description of loci of matrices of various ranks. This is followed by discussing classical and modern solutions of these old problems, including the combinatorial framework of Schur functions and Schubert polynomials. Chapter II turns to the modern generalization of the classical background, namely to morphisms of vector bundles over algebraic varieties, their degeneracy loci, and the cohomological invariants of these degeneracy loci. The fundamental case of Grassmannians and flag manifolds, together with the Schubert subvarieties associated with them, is the central topic of this chapter. Chapter III is devoted to the crucial combinatorial tools: the various kinds of symmetric functions such as Schur \(S\)-polynomials, Schur \(Q\)-polynomials, supersymmetric polynomials, and others, together with their fundamental properties and identities. Chapter IV discusses symmetric polynomials supported on degeneracy loci of vector bundle maps. The powerful general technique of Gysin maps is also explained in this chapter, and that for the important special case of Grassmannians and flag manifolds. In addition, chapters III and IV touch upon the problem of determining those polynomials that are universally supported on degeneracy loci with an explicit description of their defining ideals. Chapter V gives an application of the technique described in chapter IV to the problem of computing topological Euler characteristics of degeneracy loci and Brill-Noether loci in Jacobians of curves. The geometry of flag manifolds and determinantal formulas for Schubert varieties in the case of general homogeneous spaces associated with various classical groups are treated in chapters VI-VII. Following the correspondence method described in chapter III, degeneracy loci for generalized vector bundles (over homogeneous spaces) are investigated, too. Chapter VIII provides a particularly important application of the general theory developed in chapter VII, namely the computation of cohomology classes of some Brill-Noether loci in Prym varieties.

Although several further applications and open problems are pointed out in the course of chapters I-VIII, the concluding chapter IX is exclusively devoted to the discussion of a huge variety of other applications, related questions, and more open problems.

The following ten appendices A-J serve the purpose of making the text as self-contained as possible, on the one hand, and of indicating some closely related work that has been done since 1995, on the other hand.

Appendix A provides some background material from general intersection theory and the representation theory of degeneracy loci by symmetric polynomials. Appendix B gives a recent improvement of Fulton’s theorem on the characterization of vexillary permutations in terms of degeneracy loci. Appendix C points to the relation between degeneracy loci, Demazure’s resolution scheme for singularities, and the so-called Bott-Samelson schemes, just so for the sake of completeness. Appendix D compiles the definition and basic properties of Pfaffians, while appendix \(E\) sketches the relevant background material from the group-theoretic approach to Schubert varieties. Appendix F explains a useful Gysin formula for Grassmannian bundles, and appendix G discusses a general criterion for computing the classes of relative diagonals. A special construction for vector bundles, which is well-known and due to D. Mumford, is explained in appendix H (and used in chapter VIII). Appendix I provides a little bit of the relevant representation theory of groups and the combinatorics of Young tableaux, though this is not needed anywhere in the text. Finally, appendix J points to the very recent developments in quantum cohomology, in particular to the significance of the so-called “quantum double Schubert polynomials” introduced by I. Ciocan-Fontanine and W. Fulton (cf.: “Quantum double Schubert polynomials”, Inst. Mittag-Leffler Report No. 6 (1996-97). Throughout the entire, highly enlightening and inspiring text, the authors have focused on careful explanations of the treated material, with lots of included examples and hints to the original papers. Proofs are mostly just indicated, but always come with precise references to the original papers. The omittance of technical details is to the benefit of the non-expert reader, because this makes the beauty of the entire panorama drawn here more transparent and enjoyable. It should be mentioned that another beautiful, recent introduction to the topic of Schubert varieties and symmetric polynomials is given by the lecture notes “Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence” by L. Manivel [Cours Spécialisés, No. 3, Paris (1998; Zbl 0911.14023)].

Chapter I starts with the classical origin of the whole subject, that is, with the description of loci of matrices of various ranks. This is followed by discussing classical and modern solutions of these old problems, including the combinatorial framework of Schur functions and Schubert polynomials. Chapter II turns to the modern generalization of the classical background, namely to morphisms of vector bundles over algebraic varieties, their degeneracy loci, and the cohomological invariants of these degeneracy loci. The fundamental case of Grassmannians and flag manifolds, together with the Schubert subvarieties associated with them, is the central topic of this chapter. Chapter III is devoted to the crucial combinatorial tools: the various kinds of symmetric functions such as Schur \(S\)-polynomials, Schur \(Q\)-polynomials, supersymmetric polynomials, and others, together with their fundamental properties and identities. Chapter IV discusses symmetric polynomials supported on degeneracy loci of vector bundle maps. The powerful general technique of Gysin maps is also explained in this chapter, and that for the important special case of Grassmannians and flag manifolds. In addition, chapters III and IV touch upon the problem of determining those polynomials that are universally supported on degeneracy loci with an explicit description of their defining ideals. Chapter V gives an application of the technique described in chapter IV to the problem of computing topological Euler characteristics of degeneracy loci and Brill-Noether loci in Jacobians of curves. The geometry of flag manifolds and determinantal formulas for Schubert varieties in the case of general homogeneous spaces associated with various classical groups are treated in chapters VI-VII. Following the correspondence method described in chapter III, degeneracy loci for generalized vector bundles (over homogeneous spaces) are investigated, too. Chapter VIII provides a particularly important application of the general theory developed in chapter VII, namely the computation of cohomology classes of some Brill-Noether loci in Prym varieties.

Although several further applications and open problems are pointed out in the course of chapters I-VIII, the concluding chapter IX is exclusively devoted to the discussion of a huge variety of other applications, related questions, and more open problems.

The following ten appendices A-J serve the purpose of making the text as self-contained as possible, on the one hand, and of indicating some closely related work that has been done since 1995, on the other hand.

Appendix A provides some background material from general intersection theory and the representation theory of degeneracy loci by symmetric polynomials. Appendix B gives a recent improvement of Fulton’s theorem on the characterization of vexillary permutations in terms of degeneracy loci. Appendix C points to the relation between degeneracy loci, Demazure’s resolution scheme for singularities, and the so-called Bott-Samelson schemes, just so for the sake of completeness. Appendix D compiles the definition and basic properties of Pfaffians, while appendix \(E\) sketches the relevant background material from the group-theoretic approach to Schubert varieties. Appendix F explains a useful Gysin formula for Grassmannian bundles, and appendix G discusses a general criterion for computing the classes of relative diagonals. A special construction for vector bundles, which is well-known and due to D. Mumford, is explained in appendix H (and used in chapter VIII). Appendix I provides a little bit of the relevant representation theory of groups and the combinatorics of Young tableaux, though this is not needed anywhere in the text. Finally, appendix J points to the very recent developments in quantum cohomology, in particular to the significance of the so-called “quantum double Schubert polynomials” introduced by I. Ciocan-Fontanine and W. Fulton (cf.: “Quantum double Schubert polynomials”, Inst. Mittag-Leffler Report No. 6 (1996-97). Throughout the entire, highly enlightening and inspiring text, the authors have focused on careful explanations of the treated material, with lots of included examples and hints to the original papers. Proofs are mostly just indicated, but always come with precise references to the original papers. The omittance of technical details is to the benefit of the non-expert reader, because this makes the beauty of the entire panorama drawn here more transparent and enjoyable. It should be mentioned that another beautiful, recent introduction to the topic of Schubert varieties and symmetric polynomials is given by the lecture notes “Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence” by L. Manivel [Cours Spécialisés, No. 3, Paris (1998; Zbl 0911.14023)].

Reviewer: W.Kleinert (Berlin)

##### MSC:

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14M12 | Determinantal varieties |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14C15 | (Equivariant) Chow groups and rings; motives |