Finiteness of cominuscule quantum -theory
The product of two Schubert classes in the quantum -theory ring of a homogeneous space is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on . We show that if is cominuscule, then this power series has only finitely many non-zero terms. The proof is based on a geometric study of boundary Gromov-Witten varieties in the Kontsevich moduli space, consisting of stable maps to that take the marked points to general Schubert varieties and whose domains are reducible curves of genus zero. We show that all such varieties have rational singularities, and that boundary Gromov-Witten varieties defined by two Schubert varieties are either empty or unirational. We also prove a relative Kleiman-Bertini theorem for rational singularities, which is of independent interest. A key result is that when is cominuscule, all boundary Gromov-Witten varieties defined by three single points in are rationally connected.
2000 Mathematics Subject Classification:Primary 14N35; Secondary 19E08, 14N15, 14M15, 14M20, 14M22
The goal of this paper is to prove that any product of Schubert classes in the quantum -theory ring of a cominuscule homogeneous space contains only finitely many non-zero terms.
Let be a homogeneous space defined by a semisimple complex Lie group and a parabolic subgroup , and let denote the Kontsevich moduli space of -pointed stable maps to of degree , with total evaluation map . Given Schubert varieties in general position, there is a Gromov-Witten variety , consisting of all stable maps that send the -th marked point into for each . The Kontsevich space and its Gromov-Witten varieties are the foundation of the quantum cohomology ring of , whose structure constants are the (cohomological) Gromov-Witten invariants, defined as the number of points in finite Gromov-Witten varieties. More generally, the -theoretic Gromov-Witten invariant is defined as the sheaf Euler characteristic of , which makes sense when this variety has positive dimension. The -theoretic invariants are more challenging to compute, both because they are not enumerative, and also because they do not vanish for large degrees.
Assume for simplicity that is a maximal parabolic subgroup of , so that . The (small) quantum -theory ring is a formal deformation of the Grothendieck ring of algebraic vector bundles on , which as a group is defined by . The product of two Schubert structure sheaves is defined in terms of structure constants such that
In contrast to the quantum cohomology ring , the constants are not single Gromov-Witten invariants, but are defined as polynomial expressions of the -theoretic Gromov-Witten invariants. A result of Givental asserts that is an associative ring [givental:on]. Since the -theoretic Gromov-Witten invariants do not vanish for large degrees, the same might be true for the structure constants , in which case the product would be a power series in with infinitely many non-zero terms. When is a Grassmannian of type A, a combinatorial argument in [buch.mihalcea:quantum] shows that this does not happen; all products in are finite. In this paper we give a different geometric proof that shows more generally that all products in are finite whenever is a cominuscule homogeneous space. As a consequence, the quantum -theory ring provides an honest deformation of . The class of cominuscule varieties consists of Grassmannians of type A, Lagrangian Grassmannians, maximal orthogonal Grassmannians, and quadric hypersurfaces. In addition there are two exceptional varieties of type E called the Cayley plane and the Freudenthal variety.
Let be the minimal degree of a rational curve passing through general points of . The numbers for have been computed explicitly in [chaput.manivel.ea:quantum*1, chaput.perrin:rationality], see the table in §LABEL:sec:comin below. Our main result is the following.
Let be a cominuscule variety. Then for .
Theorem 1 holds also for the structure constants of the equivariant quantum -theory ring , see Remark LABEL:rmk:eqkfinite. The bound on is sharp in the sense that occurs in the square of a point in . In addition, this bound is also the best possible for the quantum cohomology ring that does not depend on , , and (cf. [fulton.woodward:on]).
Our proof uses that the structure constants can be rephrased as alternating sums of certain boundary Gromov-Witten invariants. Given a sequence of effective degrees such that for and , let be the closure of the locus of stable maps for which the domain is a chain of projective lines that map to in the degrees given by , the first and second marked points belong to the first projective line, and the third marked point is on the last projective line. Then any constant can be expressed as an alternating sum of sheaf Euler characteristics of varieties of the form . We use geometric arguments to show that the terms of this sum cancel pairwise whenever is cominuscule and .
Set . A key technical fact in our proof is that the general fibers of the map are rationally connected. Notice that these fibers are boundary Gromov-Witten varieties defined by three single points in , and the result generalizes the well known fact that there is a unique rational curve of degree through three general points in the Grassmannian [buch.kresch.ea:gromov-witten]. In the special case when and , it was shown in [buch.mihalcea:quantum, chaput.perrin:rationality] that the general fibers of are rational; our proof uses this case as well as Graber, Harris, and Starr’s criterion for rational connectivity [graber.harris.ea:families].
We also need to know that has rational singularities. For this we prove a relative version of the Kleiman-Bertini theorem [kleiman:transversality] for rational singularities. This theorem implies that any boundary Gromov-Witten variety in has rational singularities, for any homogeneous space . The Kleiman-Bertini theorem generalizes a result of Brion asserting that rational singularities are preserved when a subvariety of a homogeneous space is intersected with a general Schubert variety [brion:positivity].
Finally, if and are Schubert varieties in general position in a homogeneous space , we prove that is unirational and is either empty or unirational. In particular, we have and . This is done by showing that any Borel-equivariant map to a Schubert variety is locally trivial over the open cell. In particular, any single evaluation map is locally trivial.
Our paper is organized as follows. In section 2 we prove the Kleiman-Bertini theorem for rational singularities and give a simple criterion for an equivariant map to be locally trivial. These results are applied to (boundary) Gromov-Witten varieties of general homogeneous spaces in section LABEL:sec:gwvar. Section LABEL:sec:comin proves some useful facts about images of Gromov-Witten varieties of cominuscule spaces, among them that the general fibers of are rationally connected. Finally, section LABEL:sec:qkcomin applies these results to show that -theoretic quantum products on cominuscule varieties are finite.
Parts of this work was carried out during visits to the Mathematical Sciences Research Institute (Berkeley), the Centre International de Rencontres Mathématiques (Luminy), and the Max-Planck-Institut für Mathematik (Bonn). We thank all of these institutions for their hospitality and stimulating environments. We also benefited from helpful discussions with P. Belkale, S. Kumar, and F. Sottile.
2. A Kleiman-Bertini theorem for rational singularities
Let be a connected algebraic group and a -variety. A splitting of the action of on is a morphism defined on a dense open subset , together with a point , such that for all . If a splitting exists, then we say that the action is split and that is -split.
Notice that any -split variety contains a dense open orbit. Schubert varieties are our main examples of varieties with a split action.
Let be a semisimple complex Lie group, a parabolic subgroup, and the corresponding homogeneous space with its natural -action. Then is -split. Furthermore, if is a Borel subgroup and is a -stable Schubert variety, then is -split.
Let be a -stable Schubert variety, the -stable open cell, and any point. According to e.g. [springer:linear, Lemma 8.3.6] we can choose a unipotent subgroup such that the map defined by is an isomorphism. The inverse of this map is a splitting of the -action on . Since is a Schubert variety, it follows that is -split and consequently -split. ∎
Recall that a morphism is a locally trivial fibration if each point has an open neighborhood such that and is the projection to the first factor.
Let be an equivariant map of irreducible -varieties. Assume that is -split. Then is a locally trivial fibration over the dense open -orbit in , and the fibers over this orbit are irreducible.
Let and be a splitting of the -action on . Then the map defined by is an isomorphism, with inverse given by . Since is irreducible, so is . ∎
In the rest of this section, a variety means a reduced scheme of finite type over an algebraically closed field of characteristic zero. An irreducible variety has rational singularities if there exists a desingularization such that and for all . An arbitrary variety has rational singularities if its irreducible components have rational singularities, are disjoint, and have the same dimension. Zariski’s main theorem implies that any variety with rational singularities is normal. Notice also that if and have rational singularities, then so does . The converse is a special case of the following lemma of Brion [brion:positivity, Lemma 3].
Lemma 2.4 (Brion).
Let and be varieties and let be a morphism. If has rational singularities, then the same holds for the general fibers of .
The following generalization of the Kleiman-Bertini theorem [kleiman:transversality] was proved by Brion in [brion:positivity, Lemma 2] when and are inclusions and is a Schubert variety. We adapt his proof to our case.
Let be a connected algebraic group and let be a split and transitive -variety. Let and be morphisms of varieties, and assume that and have rational singularities. Then has rational singularities for all points in a dense open subset of .
It follows from Proposition 2.3 that the map defined by is a locally trivial fibration. Set and consider the diagram: