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Register a new userChern class formulas for $G_2$ Schubert loci
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Authors
Dave Anderson
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We define degeneracy loci for vector bundles with structure group $G_2$, and give formulas for their cohomology (or Chow) classes in terms of the Chern classes of the bundles involved. When the base is a point, such formulas are part of the theory for rational homogeneous spaces developed by Bernstein-Gelfand-Gelfand and Demazure. This has been extended to the setting of general algebraic geometry by Giambelli-Thom-Porteous, Kempf-Laksov, and Fulton in classical types; the present work carries out the analogous program in type $G_2$. We include explicit descriptions of the $G_2$ flag variety and its Schubert varieties, and several computations, including one that answers a question of W. Graham. In appendices, we collect some facts from representation theory and compute the Chow rings of quadric bundles, clarifying a previous computation of Edidin and Graham.
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In previous work, we employed a geometric method of Kazarian to prove Pfaffian formulas for a certain class of degeneracy loci in types B, C, and D. Here we refine that approach to obtain formulas for more general loci, including those coming from all isotropic Grassmannians. In these cases, the formulas recover the remarkable theta- and eta-polynomials of Buch, Kresch, Tamvakis, and Wilson. The streamlined geometric approch yields simple and direct proofs, which proceed in parallel for all four classical types. In an appendix, we develop some foundational algebra and prove several Pfaffian identities. Another appendix establishes a basic formula for classes in quadric bundles.
Using raising operators and geometric arguments, we establish formulas for the K-theory classes of degeneracy loci in classical types. We also find new determinantal and Pfaffian expressions for classical cases considered by Giambelli: the loci where a generic matrix drops rank, and where a generic symmetric or skew-symmetric matrix drops rank. In an appendix, we construct a K-theoretic Euler class for even-rank vector bundles with quadratic form, refining the Chow-theoretic class introduced by Edidin and Graham. We also establish a relation between top Chern classes of maximal isotropic subbundles, which is used in proving the type D degeneracy locus formulas.
We discuss a relationship between Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds, Fomin-Kirillov algebra, and the generalized nil-Hecke algebra. We show that nonnegativity conjecture in Fomin-Kirillov algebra implies the nonnegativity of the Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of the Bott-Samelson varieties. We also discuss the refined positivity conjectures of the Chern-Schwartz-MacPherson classes for Schubert cells motivated by the nonnegativity conjecture in Fomin-Kirillov algebra.
Let G be a split, simple, simply connected, algebraic group over Q. The degree 4, weight 2 motivic cohomology group of the classifying space BG of G is identified with Z. We construct cocycles representing the generator of this group, known as the second universal motivic Chern class. If G = SL(m), there is a canonical cocycle, defined by the first author (1993). For any group G, we define a collection of cocycles parametrised by cluster coordinate systems on the space of G-orbits on the cube of the principal affine space G/U. Cocycles for different clusters are related by explicit coboundaries, constructed using cluster transformations relating the clusters. The cocycle has three components. The construction of the last one is canonical and elementary; it does not use clusters, and provides a canonical cocycle for the motivic generator of the degree 3 cohomology class of the complex manifold G(C). However to lift this component to the whole cocycle we need cluster coordinates: the construction of the first two components uses crucially the cluster structure of the moduli spaces A(G,S) related to the moduli space of G-local systems on S. In retrospect, it partially explains why the cluster coordinates on the space A(G,S) should exist. This construction has numerous applications, including an explicit construction of the universal extension of the group G by K_2, the line bundle on Bun(G) generating its Picard group, Kac-Moody groups, etc. Another application is an explicit combinatorial construction of the second motivic Chern class of a G-bundle. It is a motivic analog of the work of Gabrielov-Gelfand-Losik (1974), for any G.
We give a short and self-contained proof of the Decomposition Theorem for the non-small resolution of a Special Schubert variety. We also provide an explicit description of the perverse cohomology sheaves. As a by-product of our approach, we obtain a simple proof of the Relative Hard Lefschetz Theorem.
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