Chern class formulas for classicaltype degeneracy loci
Abstract.
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 etapolynomials of Buch, Kresch, Tamvakis, and Wilson. The streamlined geometric approch yields simple and direct proofs, which proceed in parallel for all four classical types.
Introduction
A fundamental problem asks for a formula for the cohomology (or Chow) class of a degeneracy locus, as a polynomial in the Chern classes of the vector bundles involved. In its simplest form, the answer is given by the GiambelliThomPorteous formula: the locus is where two subbundles of a given vector bundle meet in at least a given dimension, and the formula is a determinant.
The aim of this article is to prove formulas for certain degeneracy loci in classical types. One has maps of vector bundles, or flags of subbundles of a given bundle; degeneracy loci come from imposing conditions on the ranks of maps, or dimensions of intersections. The particular loci we consider are indexed by triples of tuples of integers, (in type A) or (in types B, C, and D). In type A, each specifies a rank condition on maps of vector bundles ; in other types, and are flags of isotropic or coisotropic bundles inside some vector bundle with bilinear form, and each specifies .
In each case, we will write for the degeneracy locus. Its expected codimension depends on the type. In fact, to each triple we associate a partition (again depending on type), whose size is equal to the codimension of .
The resulting degeneracy loci of type A have a determinantal formula which generalizes that of GiambelliThomPorteous. The loci corresponding to triples are exactly those defined by vexillary permutations according to the recipe of [F2]; building on work of KempfLaksov, LascouxSchützenberger, and others, it was shown in [op. cit.] that
Here each is a (total) Chern class , and is a Schur determinant; more details will be given in §1.
In other classical types, work of Pragacz and his collaborators showed that Pfaffians should play a role analogous to determinants in type A, at least for cases where there is a single bundle and all are isotropic [P1, P2, PR, LP]. More recent work of Buch, Kresch, and Tamvakis exploits a crucial insight: both determinants and Pfaffians can be defined via raising operators, and by adopting the raising operator point of view, one can define theta and etapolynomials, which interpolate between determinants and Pfaffians. These provide representatives for Schubert classes in nonmaximal isotropic Grassmannians; here one has a single isotropic , and a flag of trivial bundles , some of which may be coisotropic [BKT1, BKT2, T1]. Wilson extended this idea to define double thetapolynomials, and conjectured that they represent equivariant Schubert classes in nonmaximal isotropic Grassmannians [W]. This was proved in [IM], and, via a different method, in [TW].
In previous work, we introduced triples and studied loci defined by , with all and isotropic [AF]. The triples we introduce here are more general, in that they allow to include coisotropic bundles. Our main theorem is a corresponding generalization of the multiPfaffian formulas from [op. cit.], which includes all Grassmannian loci. The formulas are multithetapolynomials and multietapolynomials .
Theorem.
Let be a triple, and let be the corresponding degeneracy locus (of type C, B, or D).

In type C, we have

In type B, we have

In type D, we have
The entries vary by type, and along with the definitions of and , these are specified in Theorems 2, 3, and 4. For now, let us mention some special cases. When the triple has all , the loci are defined by conditions on isotropic bundles, and each formula is a Pfaffian; these are precisely the formulas found in [AF]. If the triple has all , the loci come from Grassmannians; in the type C case, we recover the formulas of [IM] and [TW], and in case the ’s are trivial, we recover the formulas of [BKT1, BKT2]. (The general type D formula includes a definition of double etapolynomial, which is new even in the Grassmannian case.^{1}^{1}1Tamvakis recently announced that he has also found such polynomials [T3].)
The structure of the argument in each type is essentially the same. First, one has a basic formula for the case where the only condition is , and furthermore is a line bundle. Next, there is the case where has rank , and the conditions are ; the formula here is easily seen to be a product, and one uses some elementary algebra to convert the product into a raising operator formula. (In type A, these loci correspond to dominant permutations.) The “main case” is where the conditions are ; such loci are resolved by a birational map from a dominant locus, and the pushforward can be decomposed into a series of projective bundles. Finally, a little more elementary algebra reduces the general case to the main case.
Proving the theorem requires only a few general facts. Some of these are treated in appendices, but we collect four basic formulas here for reference. Let be a vector bundle of rank on a variety .

If is a line bundle on , then

If is a line bundle on , for any and we have

If is a subbundle of a vector bundle , then

Let be the projective bundle, and the universal quotient bundle. For any , we have
We conclude this introduction with some remarks on the development and context of our results. The double Schubert polynomials of Lascoux and Schützenberger, which represent type A degeneracy loci, have many wonderful combinatorial properties. A problem that received a great deal of attention in the 1990’s was to find similar polynomials representing loci of other classical types. First steps in this direction were taken by Billey and Haiman, who defined (single) Schubert polynomials for types B, C, and D [BH]; double versions were obtained by Ikeda, Mihalcea, and Naruse [IMN] and studied further by Tamvakis [T1]. We should point out that in types B, C, and D, any theory of Schubert polynomials involves working in a ring with relations, but modulo these, stable formulas are essentially unique. See [AF], or the survey [T2], for more perspective on this history.
The Schubert varieties and degeneracy loci in types B, C, and D are indexed by signed permutations, and it is natural to ask whether certain signed permutations correspond to Pfaffians, by analogy with the determinantal formulas for vexillary permutations. In [AF], we identified such a class of vexillary signed permutations,^{2}^{2}2Although not obvious from the definitions, the vexillary signed permutations of [AF] correspond to certain vexillary elements of the hyperoctahedral group, studied by Billey and Lam [BL]. It should be interesting to study the permutations arising from the more general triples to be considered here; see the remark at the end of §2. defined via triples such that all . Following ideas of Kazarian [K], we proved Pfaffian formulas for vexillary loci, and also studied the relationship between these Pfaffians and the double Schubert polynomials of [IMN].
The initial impetus of the present work was to simplify and streamline the arguments of [AF]. In doing so, we found the situation was clarified by making explicit the role of raising operators. For this, we owe a great debt to the work of Buch, Kresch, and Tamvakis, whose remarkable theta and etapolynomial formulas have unequivocally demonstrated the utility of raising operators in geometry. This inspired us to apply our geometric method to more general loci, simultaneously yielding shorter and more uniform proofs of their formulas, as well as of our earlier Pfaffian formulas.
Combinatorially, the main new feature in the present article (as compared to [AF]) is that we allow triples to include negative ’s. Geometrically, this means that conditions imposed on include ones where is coisotropic; this presents some subtleties, but leads directly to the definition of theta and etapolynomials. Here we focus on these Chern class formulas, leaving the relation with Schubert polynomials to a revision of [AF].
Finally, although the recent prominence of raising operators is due to Buch, Kresch, and Tamvakis, it was Pragacz who first brought them to geometry. We take both combinatorial and geometric inspiration from his work, and dedicate this article to him on the occasion of his sixtieth birthday.
Acknowledgements. We thank Harry Tamvakis for comments on an earlier version of the manuscript.
1. Type A revisited
The determinantal formula describing degeneracy loci in Grassmann bundles — or more generally, vexillary loci in flag bundles — is, by now, quite well known; see [KL, F2] for recent versions. However, our reformulation of its setup and proof will provide a model for the (new) formulas in other types, so we will go through it in detail.
A triple of type A is the data , where each of , , and is an tuple of nonnegative integers, with
Setting and , we further require that
Associated to a triple there is a partition , defined by setting , and filling in the remaining parts minimally so that . (An essential triple specifies only the “corners” of the Young diagram for , so in a sense it is a minimal way of packaging this information.)
Given a partition and symbols , the associated Schur determinant is
For a positive integer , let be the raising operator defined by
(1) 
A simple application of the Vandermonde identity shows that
(2) 
and we will use this observation crucially in proving the degeneracy locus formula.
Here is the geometric setup. On a variety we have a sequence of vector bundles
where subscripts indicate ranks; for each , there is an induced map . The degeneracy locus corresponding to the triple is
This comes equipped with a natural subscheme structure defined locally by the vanishing of certain determinants.
Let , and set whenever . We also set , and by convention, we always take .) With this notation, the degeneracy locus formula can be stated as follows.
The case where there is only one , so , was proved by Kempf and Laksov, starting the modern search for such formulas. The general case was proved in [F2].
These formulas are to be interpreted as usual: when the bundles and the map are sufficiently generic, then has codimension equal to and the formula is an identity relating the fundamental class of with the Chern classes of and . In general, the lefthand side should be regarded as a refined class of codimension , supported on ; see [F1].
The proof proceeds in four steps.
1.1. Basic case
Assume , , , so . The locus is where vanishes, so it is the zeroes of a section of , a vector bundle of rank . Therefore
using Identity (a) to obtain the second equality.
1.2. Dominant case
Assume and for , so and . By imposing one condition at a time, we obtain a sequence
is the locus where is zero; is where also is zero; and generally is defined on by the condition that be zero. This is an instance of the basic case, so it follows that
(3) 
Writing and , an application of Identity (c) transforms (3) into
Using Identity (b), this becomes
In other words, the product (3) is equal to the determinant , where .
1.3. Main case
Assume for . There is a sequence of projective bundles
where, suppressing notation for pullbacks of bundles, is the tautological line bundle on . Let us write for the projection, and for the composition of all the ’s.
On , there is the locus where for . This is an instance of the dominant case, so in we have
(4) 
where and . Furthermore, maps birationally onto ; it is an isomorphism over the dense open set where . (Take to be the kernel.) So .
1.4. General case
Any triple can be “inflated” to with , without essentially altering the locus or the polynomial representing it. Suppose and . Inserting between the st and th positions produces a new triple with . If there is a bundle fitting into , then one easily checks . In general, it can be arranged for such an to exist by passing to an appropriate projective bundle; then the locus maps birationally to the original . The fact that is a special case of Lemma 1 of Appendix A.3.
This concludes the proof. ∎
2. Type C: symplectic bundles
A triple of type C is , with
The are allowed to be negative, but not zero, and if then . Let be the index such that , allowing and for the cases where all ’s are negative or all ’s are positive, respectively. If occurs as some , then cannot appear. As in type A, there are further conditions on , which are best described by requiring certain sequences of integers to be partitions. We will define by setting , where and are defined as follows.
First, let . If , then in what follows, we replace by the result of inserting between the st and th entries. (In this case, we must have to obtain a triple.) From now on we assume .
To define , set for . For , so , set , where is the index such that . Finally, fill in the other parts minimally subject to ; that is, for all , set when . (When indices fall out of the range , we use conventions so that the inequalities are trivial: , , and .)
To define , set
and fill in the other parts by declaring when .
With , , and thus defined, the final conditions on are the following inequalities, where :
(5) 
(6) 
and
(7) 
When , so all , condition (6) is equivalent to
and this is the only condition needed in this case.
One could define more directly, by setting
and then filling in minimally subject to the condition (7). However, it will be useful to think of as a “skew shape” . Generally, given a sequence of nonnegative integers , we will say that a partition is strict if the sequence is nonincreasing.^{3}^{3}3If all and , then a strict partition constructed from a type C triple is one so that all parts of size greater than are distinct; that is, it is a strict partition as defined in [BKT1, BKT2]. Geometrically, this means all have rank , so the locus comes from an isotropic Grassmannian. The conditions (6) and (7) can be phrased as requiring to be a strict partition, with .
For example, suppose
The meaning of these conditions is explained by the geometric setup. We have a vector bundle of rank equipped with a symplectic form, and two flags
when , the subbundles are isotropic, of rank ; when , is coisotropic, of corank ; and all the bundles are isotropic, of rank . Note that . The degeneracy locus is
The inequalities (5), (6), and (7) allow to be inflated to a triple having , so that . The role of is to bring into play the following basic fact about (co)isotropic subspaces of a dimensional symplectic vector space . Let and ; let be a coisotropic subspace, of codimension ; let be isotropic, of dimension ; and assume . Suppose the intersection has dimension . Then , so
() 
(The same holds if the bilinear form is symmetric, and has dimension or .)
The definition of is designed so that in the “dominant” case, when and , and when the subspaces are general with respect to the given conditions, we have . A more combinatorial explanation of the conditions is given in the Remark at the end of this section.
Given an integer , a sequence of nonnegative integers with , define the raising operator
Inspired by [BKT1], given symbols , and a strict partition , we define the thetapolynomial to be
When , is a Schur determinant, and when , is a Schur Pfaffian. Note that is part of the data: for example, could be written as in several ways, but the thetapolynomials depend on which is specified.
For a triple and the corresponding geometry described above, let , and for general , take where is minimal such that . Set .
Theorem 2.
We have .
The proof follows the same pattern as the one we saw in type A. As before, there are four cases. The appearance of Pfaffians in the formulas can be traced to a basic fact about isotropic subbundles: if is isotropic, then the symplectic form identifies with . This is used in the second case.
2.1. Basic case
Take , , and , so is a line bundle and we are looking at . Equivalently, is zero, so Identity (a) lets us write
where
2.2. Dominant case
Now take and , for . Write , so this is a vector bundle of rank , and we have . Letting be the locus where , we have
On , we have . Since is isotropic, we automatically have , so is defined by the condition
When , the bundle is isotropic, and this implies . In this case, is defined by , and the basic case says
with . (That is, .)
2.3. Main case
Here we only assume , for . For , we set , and for , we set , where . We have the same sequence of projective bundles as in type A,
again, is the tautological subbundle on . Write for the projection, and for the composition.
On , we have the locus . By the previous case, as a class in we have
where and
Furthermore, maps birationally onto .
This case follows, since
2.4. General case
Just as in type A, any triple can be inflated to a triple having , for , such that . The “main case” provides a thetapolynomial formula using the triple ; and Appendix A.3, Lemma 1, shows that . ∎
As mentioned above, in extreme cases the thetapolynomial is a determinant or Pfaffian. To include the case where is odd, we recall that Pfaffians can be defined for odd matrices by introducing a zeroth row, (see Appendix A.1).
Corollary.
If all , then
where the righthand side is defined to be the Pfaffian of the matrix , with