105 research outputs found
Cycle groups for Artin stacks
We construct an algebraic homology functor for Artin stacks of finite type
over a field, and we develop intersection-theoretic properties.Comment: LaTeX2e, 44 page
Gromov-Witten invariants of a class of toric varieties
We describe the quantum cohomology rings of a class of toric varieties. The
description includes, in addition to the (already known) ring presentations,
the (new) analogues for toric varieties of the sorts of quantum Giambelli
formulas which exist already for Grassmannian varieties, flag varieties, etc.Comment: 22 pages, LaTeX, to appear in the Michigan Mathematical Journal,
special volume in honor of William Fulto
Quantum cohomology of orthogonal Grassmannians
Let V be a vector space with a nondegenerate symmetric form and OG be the
orthogonal Grassmannian which parametrizes maximal isotropic subspaces in V. We
give a presentation for the (small) quantum cohomology ring QH^*(OG) and show
that its product structure is determined by the ring of (P~)-polynomials. A
"quantum Schubert calculus" is formulated, which includes quantum Pieri and
Giambelli formulas, as well as algorithms for computing Gromov-Witten
invariants. As an application, we show that the table of 3-point, genus zero
Gromov-Witten invariants for OG coincides with that for a corresponding
Lagrangian Grassmannian LG, up to an involution.Comment: 20 pages, LaTeX, to appear in Compositio Mathematic
Double Schubert polynomials and degeneracy loci for the classical groups
We propose a theory of double Schubert polynomials P_w(X,Y) for the Lie types
B, C, D which naturally extends the family of Lascoux of Schutzenberger in type
A. These polynomials satisfy positivity, orthogonality, and stability
properties, and represent the classes of Schubert varieties and degeneracy loci
of vector bundles. When w is a maximal Grassmannian element of the Weyl group,
P_w(X,Y) can be expressed in terms of Schur-type determinants and Pfaffians, in
analogy with the type A formula of Kempf and Laksov. An example, motivated by
quantum cohomology, shows that there are no Chern class formulas for degeneracy
loci of ``isotropic morphisms'' of bundles.Comment: 34 pages, LaTeX; final versio
Quantum cohomology of the Lagrangian Grassmannian
Let V be a symplectic vector space and LG be the Lagrangian Grassmannian
which parametrizes maximal isotropic subspaces in V. We give a presentation for
the (small) quantum cohomology ring QH^*(LG) and show that its multiplicative
structure is determined by the ring of (Q^~)-polynomials. We formulate a
"quantum Schubert calculus" which includes quantum Pieri and Giambelli
formulas, as well as algorithms for computing the structure constants appearing
in the quantum product of Schubert classes.Comment: 27 pages, LaTeX, to appear in Journal of Algebraic Geometr
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