Flexibility of Schubert classes

In this note, we discuss the flexibility of Schubert classes in homogeneous varieties. We give several constructions for representing multiples of a Schubert class by irreducible subvarieties. We sharpen [22, Theorem 3.1] by proving that every positive multiple of an obstructed class in a cominuscule homogeneous variety can be represented by an irreducible subvariety.     

Restriction varieties and geometric branching rules

This paper develops a new method for studying the cohomology of orthogonal flag varieties. Restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to (not necessarily isotropic) flags. They interpolate between Schubert varieties in orthogonal flag varieties and the restrictions of general Schubert varieties in ordinary flag varieties. We give a positive, geometric rule for calculating their cohomology classes, obtaining a branching rule for Schubert calculus for the inclusion of the orthogonal flag varieties in Type A flag varieties. Our rule, in addition to being an essential step in finding a Littlewood–Richardson rule, has applications to computing the moment polytopes of the inclusion of SO(n) in SU(n), the asymptotic of the restrictions of representations of SL(n) to SO(n) and the classes of the moduli spaces of rank two vector bundles with fixed odd determinant on hyperelliptic curves. Furthermore, for odd orthogonal flag varieties, we obtain an algorithm for expressing a Schubert cycle in terms of restrictions of Schubert cycles of Type A flag varieties, thereby giving a geometric (though not positive) algorithm for multiplying any two Schubert cycles.     

On the zeta function of divisors for projective varieties with higher rank divisor class group

Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than one, this is a purely p-adic function, convergent on the open unit disk. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to all of Cp. When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case. In particular, we prove a p-adic meromorphic continuation theorem which applies to a large class of varieties. Examples of such varieties are projective nonsingular surfaces defined over a finite field (whose effective monoid is finitely generated) and all projective toric varieties (smooth or singular).     

Generic vanishing index and the birationality of the bicanonical map of irregular varieties

We prove that any smooth complex projective variety with generic vanishing index bigger or equal than 2 has birational bicanonical map. Therefore, if X is a smooth complex projective variety φ with maximal Albanese dimension and non-birational bicanonical map, then the Albanese image of X is fibred by subvarieties of codimension at most 1 of an abelian subvariety of Alb X.     

Quasi-projective reduction of toric varieties

We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties. For a given toric variety X we first construct a toric quasi–projective reduction. Then we show that X has a quasi–projective reduction if and only if its toric quasi–projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a quasi–projective toric variety admits a categorical quotient in the category of quasi–projective varieties. Received October 29, 1998; in final form December 28, 1998     

Patch ideals and Peterson varieties

Patch ideals encode neighbourhoods of a variety in GL _n /B. For Peterson varieties we determine generators for these ideals and show they are complete intersections, and thus Cohen–Macaulay and Gorenstein. Consequently, we

— combinatorially describe the singular locus of the Peterson variety;

— give an explicit equivariant K-theory localization formula; and

— extend some results of [B. Kostant ‘96] and of D. Peterson to intersections of Peterson varieties with Schubert varieties.

We conjecture that the tangent cones are Cohen–Macaulay, and that their h-polynomials are nonnegative and upper-semicontinuous. Similarly, we use patch ideals to briey analyze other examples of torus invariant subvarieties of GL _n /B, including Richardson varieties and Springer fibers.     

Schubert polynomials and classes of Hessenberg varieties

Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a “Giambelli formula” expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, and we give closed combinatorial formulas for the coefficients in many cases. We introduce a closely related family of schemes called regular nilpotent Hessenberg schemes, and use our results to determine when such schemes are reduced.     

On <Emphasis Type="Italic">G</Emphasis>-rigid surfaces

Rigid algebraic varieties form an important class of complex varieties that exhibit interesting geometric phenomena. In this paper we propose a natural extension of rigidity to complex projective varieties with a finite group action (G-varieties) and focus on the first nontrivial case, namely, on G-rigid surfaces that can be represented as desingularizations of Galois coverings of the projective plane with Galois group G. We obtain local and global G-rigidity criteria for these G-surfaces and present several series of such surfaces that are rigid with respect to the action of the deck transformation group.     

Equivariant Pieri Rule for the homology of the affine Grassmannian

An explicit rule is given for the product of the degree two class with an arbitrary Schubert class in the torus-equivariant homology of the affine Grassmannian. In addition a Pieri rule (the Schubert expansion of the product of a special Schubert class with an arbitrary one) is established for the equivariant homology of the affine Grassmannians of SL _n and a similar formula is conjectured for Sp _2n and SO _2n+1. For SL _n the formula is explicit and positive. By a theorem of Peterson these compute certain products of Schubert classes in the torus-equivariant quantum cohomology of flag varieties. The SL _n Pieri rule is used in our recent definition of k-double Schur functions and affine double Schur functions.     

Quantum Giambelli formulas for isotropic Grassmannians

Let X be a symplectic or odd orthogonal Grassmannian which parametrizes isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove quantum Giambelli formulas which express an arbitrary Schubert class in the small quantum cohomology ring of X as a polynomial in certain special Schubert classes, extending the authors?? cohomological Giambelli formulas.     

Rigid Binary Relations on a 4-Element Domain

An n-ary relation ρ on a set U is strongly rigid if it is preserved only by trivial operations. It is projective if the only idempotent operations in P o l ρ are projections. Rosenberg, (Rocky Mt. J. Math. 3, 631–639, 1973) characterized all strongly rigid relations on a set with two elements and found a strongly rigid binary relation on every domain U of at least 3 elements. Larose and Tardif (Mult.-Valued Log. 7(5-6), 339–362, 2001) studied the projective and strongly rigid graphs and constructed large families of strongly rigid graphs. ?uczak and Ne?et?il (J. Graph Theory. 47, 81–86, 2004) settled in the affirmative a conjecture of Larose and Tardif that most graphs on a large set are projective, and characterized all homogenous graphs that are projective. ?uczak and Ne?et?il (SIAM J. Comput. 36(3), 835–843, 2006) confirmed a conjecture of Rosenberg that most relations on a big set are strongly rigid. In this paper, we characterize all strongly rigid relations on a set with at least three elements to answer an open question by Rosenberg, (Rocky Mt. J. Math. 3, 631–639, 1973) and we classify the binary relations on the 4-element domain by rigidity and demonstrate that there are merely 40 pairwise nonisomorphic rigid binary relations on the same domain (among them 25 are pairwise nonisomorphic strongly rigid).     

Parametrizing Shimura subvarieties of {\mathrm{A}_1} Shimura varieties and related geometric problems

This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of \({X_{a, b} = (\mathbf{H}^2)^a\times (\mathbf{H}^3)^b}\). A special case describes all Shimura subvarieties of type \({\mathrm{A}_1}\) Shimura varieties. We produce, for any \({n\geq 1}\), examples of manifolds/Shimura varieties with precisely n commensurability classes of totally geodesic submanifolds/Shimura subvarieties. This is in stark contrast with the previously studied cases of arithmetic hyperbolic 3-manifolds and quaternionic Shimura surfaces, where the presence of one commensurability class of geodesic submanifolds implies the existence of infinitely many classes.     

Desingularizations of Schubert varieties in double Grassmannians

Let X = Gr(k, V) × Gr(l, V) be the direct product of two Grassmann varieties of k-and l-planes in a finite-dimensional vector space V, and let B ? GL(V) be the isotropy group of a complete flag in V. We consider B-orbits in X, which are an analog of Schubert cells in Grassmannians. We describe this set of orbits combinatorially and construct desingularizations for the closures of these orbits, similar to the Bott-Samelson desingularizations for Schubert varieties.     

On some smooth projective two-orbit varieties with Picard number 1

We classify all smooth projective horospherical varieties with Picard number 1. We prove that the automorphism group of any such variety X acts with at most two orbits and that this group still acts with only two orbits on X blown up at the closed orbit. We characterize all smooth projective two-orbit varieties with Picard number 1 that satisfy this latter property.     

Schubert Patches Degenerate to Subword Complexes

We study the intersections of general Schubert varieties X _w with permuted big cells, and give an inductive degeneration of each such “Schubert patch” to a Stanley–Reisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. We maintain reducedness using the results of Knutson [Kn07] on automatically reduced degenerations, or through more standard cohomology-vanishing arguments. The underlying simplicial complex of the Stanley–Reisner scheme is a subword complex, as introduced for slightly di_erent purposes in Knutson–Miller [KnM05], and is homeomorphic to a ball. This gives a new proof of the Andersen–Jantzen–Soergel/Billey and Graham/Willems formulae for restrictions of equivariant Schubert classes to fixed points.     

Picard Groups of compact toric Varieties and combinatorial Classes of Fans

We consider the question what can be said about the rank of the Picard group Pic X_σ of a compact toric variety X_σ if we know only the combinatorial type of the associated fan σ. We establish upper and lower bounds for the rank of Pic X_σ and give conditions for Pic X_σ to be determined by the combinatorial type of σ. Furthermore, we show that for simple fans Pic X_σ is necessary isomorphic to {0} or Z and give an example for a compact toric variety having a trivial Picard group. Moreover in the projective case we study the relation between addition of T-invariant Cartier divisors on X_σ, taking tensor product of elements of Pic X_σ and piecewise linear functions on σ with Minkowski-addition of polytopes, where the latter operation is extended to a group operation. Finally, we explain the relation to strong cohomology in the projective case.     

Arc Spaces and Equivariant Cohomology

We present a new geometric interpretation of equivariant cohomology in which one replaces a smooth, complex G-variety X by its associated arc space J _∞ X, with its induced G-action. This not only allows us to obtain geometric classes in equivariant cohomology of arbitrarily high degree, but also provides more flexibility for equivariantly deforming classes and geometrically interpreting multiplication in the equivariant cohomology ring. Under appropriate hypotheses, we obtain explicit bijections between $ \mathbb{Z} $ -bases for the equivariant cohomology rings of smooth varieties related by an equivariant, proper birational map. We also show that self-intersection classes can be represented as classes of contact loci, under certain restrictions on singularities of subvarieties. We give several applications. Motivated by the relation between self-intersection and contact loci, we define higher-order equivariant multiplicities, generalizing the equivariant multiplicities of Brion and Rossmann; these are shown to be local singularity invariants, and computed in some cases. We also present geometric $ \mathbb{Z} $ -bases for the equivariant cohomology rings of a smooth toric variety (with respect to the dense torus) and a partial flag variety (with respect to the general linear group).     

A RATIO OF INTEGRATION BETWEEN QUOTIENTS IN GEOMETRIC INVARIANT THEORY

Let T be a maximal torus of a connected reductive group G that acts linearly on a projective variety X so that all semi-stable points are stable. This paper compares the integration on the geometric invariant theory quotient X//G of Chow classes σ to the integration on the geometric invariant theory quotient X//T of certain lifts of σ twisted by c _top(g/t), the top Chern class of the T-equivariant vector bundle induced by the quotient of the adjoint representation on the Lie algebra of G by that of T. We provide a purely algebraic proof that the ratio between any two such integrals is an invariant of the group G and that it equals the order of the Weyl group whenever the root system of G decomposes into irreducible root systems of type A _n , for various $ n\in \mathbb{N} $ . As a corollary, we are able to remove this restriction on root systems by applying a related result of Martin from symplectic geometry.