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# Geometry of horospherical varieties of Picard rank one

R. Gonzales Department of Sciences, Pontificia Universidad Católica del Perú, San Miguel, Lima 32, Lima, Perú C. Pech School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, United Kingdom N. Perrin Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France  and  A. Samokhin Institute for Information Transmission Problems B. Karetnyi per., 19, 127994, GSP-4, Moscow, Russia
05.11.2017
###### Abstract.

We study the geometry of non-homogeneous horospherical varieties. These varieties have been classified by Pasquier and include the well known odd symplectic Grassmannians. We focus our study on quantum cohomology, with a view towards Dubrovin’s conjecture. In particular, we describe the cohomology groups of these varieties as well as a Chevalley formula, and prove that many Gromov-Witten invariants are enumerative. This enables us to prove that in many cases the quantum cohomology is semisimple. We give a presentation of the quantum cohomology ring for odd symplectic Grassmannians and construct an exceptional collection with the expected number of elements in a special case of the classification which is associated with the group .

###### 2010 Mathematics Subject Classification:
14M27,14N35,14F05,14M17

## Introduction

Schubert calculus and enumerative geometry are very classical. One of their many aspects is the study of intersection theory of projective rational homogeneous spaces with a reductive group and a parabolic subgroup. We now have a very good understanding of the cohomology of as well as a good understanding of their quantum cohomology and even their quantum -theory.

In this paper we consider intersection theory on quasi-homogeneous varieties. One of the simplest classes of non homogeneous quasi-homogeneous spaces are toric varieties, for which many results are known. A common generalisation of rational projective homogeneous spaces and toric varieties are spherical varieties. Recall that a normal variety is called -spherical for a reductive group if has a dense -orbit, where is a Borel subgroup. Spherical varieties form such a large class of varieties that we do not even have a description of their cohomology ring in general. In this paper we concentrate on a special class of spherical varieties: horospherical varieties, see Definition 1.1.2. We consider smooth, projective horospherical varieties of Picard rank one. These varieties were classified by Pasquier [35], who proved that they are either homogeneous or can be constructed in a uniform way via a triple of representation-theoretic data, where is the semisimple Lie type of the reductive group and are fundamental weights, see Section 1.3. Pasquier classified the possible triples in five classes:

1. with ;

2. ;

3. with and ;

4. ;

5. .

The third class, case (3), already attracted some attention since in this case the variety corresponds to the so called odd symplectic Grassmannian (see Subsection 1.9 and Section 5). They were studied for example in [30, 37, 38] and very recently in [31, 28].

In this paper we start a systematic study of the geometry of non homogeneous projective smooth horospherical varieties with , with a view towards Dubrovin’s conjecture. Our first results are concerned with the cohomology ring . We describe two Schubert-like cohomology basis and and prove that these bases are Poincaré dual to each other, see Proposition 1.7.3. We also give a Chevalley formula, i.e., a formula for multiplying any of the above classes with the hyperplane class generating the Picard group, see Proposition 1.8.1.

We then turn to the study of rational curves, in particular to the study of , the Kontsevich moduli space of degree , genus zero, stable maps with marked points. In particular we prove the following result (see Theorem 2.2.1).

###### Theorem 0.0.1.

Let be a non homogeneous projective smooth horospherical variety with . Then

1. is irreducible of dimension .

2. is irreducible of dimension for all in cases (2) and (3).

3. is irreducible of dimension in cases (1) for and (4), and has two irreducible components of dimension of dimension in cases (1) for and case (5).

In all cases above, the dimension of the moduli space is equal to the expected dimension . This implies that the Gromov-Witten invariants are enumerative and proves, in particular, a conjecture stated in the second author’s PhD thesis (see Corollary 3.2.3).

###### Corollary 0.0.2.

If belongs to the classes (2) or (3) of Pasquier’s list, then the Gromov-Witten invariants of are enumerative.

Using these enumerativity results, we are able to compute many Gromov-Witten invariants. In particular, we prove the following result, see Subsection 3.3 and Section 4.

###### Theorem 0.0.3.

If belongs to the classes (1) for , (2), (3) for and (5) of Pasquier’s list, then

1. there is an explicit quantum Chevalley formula;

2. the small quantum cohomology ring is semisimple.

Note that in case (3) of the classification, i.e. for symplectic Grassmannians, such a quantum (and even equivariant quantum) Chevalley formula has been obtained recently by Mihalcea and Shifler [31]. In Section 5 we use our proof of a conjecture by the second author to deduce several results on the specific case of odd symplectic Grassmannians (case (3) of Pasquier’s classification). In particular we prove quantum-to-classical results that enable to compute Gromov-Witten invariants on thanks to cohomological (thus classical) invariants on (see Theorem 5.4.2). In particular this result contains the quantum Pieri rule even if we have no closed formula for the moment. Using these results we deduce a presentation for the quantum cohomology ring . Let the Chern classes of the tautological quotient bundle on and for , define

 dr:=det(τ′1+j−i)1≤i,j≤randbr:=(τ′r)2+2∑i≥1(−1)iτ′r+iτ′r−i,

with the convention that and if or .

###### Theorem 0.0.4.

The quantum cohomology ring is generated by the classes and the quantum parameter , and the relations are

 dr=0 for m+1≤r≤2n+1−m,d2n+2−m={0if m is even,−qif m is odd,bs=(−1)2n+1−m−sqτ′2s−2n−2+m for n+2−m≤s≤n,

Finally, denote by the big quantum cohomology ring of and recall Dubrovin’s conjecture

###### Conjecture 0.0.5.

For smooth projective, is semisimple if and only if the bounded derived category has a full exceptional collection.

The above results therefore suggest to look for full exceptional collections in many horospherical varieties of the classification. Some results were already obtained. For instance, the second author proved in [38] that in case (3) for , Dubrovin’s conjecture holds. Dubrovin’s conjecture in case (2) follows from results by the third author [42] for the semisimplicity of quantum cohomology and results by Kuznetsov [26] for the derived category. Indeed, the variety in case (2) is a hyperplane section of the maximal orthogonal Grassmannian (see Subsection 1.9). In the last two sections, we consider the derived category side of Dubrovin’s conjecture. In Section 6, we construct in all cases except case (2) an exceptional vector bundle on . In Section 7 we extend this construction and construct an exceptional collection on the horospherical variety in case (5):

 (1) ⟨ˆS(−3),U(−3),OX(−3),ˆS(−2),U(−2),OX(−2),ˆS(−1),U(−1),OX(−1),ˆS,U,OX⟩

with the right number of elements. The object is a vector bundle but is a two term complex.

The exceptional collection (1) has an additional property. Namely, it is of the form , where . Such collections are called Lefschetz exceptional collections, see [26].

###### Theorem 0.0.6.

The collection (1) is a Lefschetz exceptional collection on whose length is equal to the rank of .

Acknowledgements. R.G., N.P., and A.S. would like to thank the Mathematical Institute of Düsseldorf where this work first started. C.P. thanks the University of Kent for support. R.G. and N.P. were supported by a DFG grant Project “Gromov-Witten Theorie, Geometrie und Darstellungen” (PE 2165/1-2). N.P. was supported by a public grant as part of the “Investissement d’Avenir” project, reference ANR-11-LABX-0056-LMH, LabEx LMH. A.S. was supported by the strategic research fund of the University of Düsseldorf (grant SFF F-2015/946-8) and by the SFB/TR 45 of the University of Duisburg–Essen. A.S. is also indebted to A.Kuznetsov who suggested using Hecke modifications to construct exceptional vector bundles in Sections 6-7.

Notation. Let be a complex reductive algebraic group, be a maximal torus of , and be a Borel subgroup containing . The flag variety of Borel subgroups in is denoted by . Let be the set of simple roots relative to the choice of . A parabolic subgroup of is usually denoted by ; in particular, for a simple root , denote by the minimal parabolic subgroup of associated to . For a dominant weight , the induced module is denoted by .

## 1. Horospherical varieties of Picard rank one - Geometry

In this section we recall some basic properties of smooth projective horospherical varieties of Picard rank one. Our reference is [35], see also [34, 41].

### 1.1. Horospherical varieties

Let be a complex reductive group. A -variety is a reduced scheme of finite type over the field of complex numbers , equipped with an algebraic action of . Some of the simplest -varieties are spherical varieties.

###### Definition 1.1.1.

Let be a reductive group and be a Borel subgroup. A -variety is called spherical if has a dense -orbit.

Spherical varieties admit many equivalent characterisations. For example, a normal -variety is spherical if and only if it has finitely many -orbits. We refer to [22, 41, 18, 39] and references therein for more on the geometry of spherical varieties or their classification.

We focus on a special case of spherical varieties: horospherical varieties. We give two equivalent definitions of horospherical varieties, one from a representation-theoretic point of view, the other more geometric. We refer to Pasquier’s PhD thesis [34] for details.

###### Definition 1.1.2.

Let be a -spherical variety and let be the stabiliser of a point in the dense -orbit in . The variety is called horospherical if contains a conjugate of , the maximal unipotent subgroup of contained in the Borel subgroup .

###### Remark 1.1.3.

A more geometric equivalent definition can be given as follows. Let be a -variety. Then is horospherical if there exists a parabolic subgroup , a torus which is a quotient of , and a toric -variety such that there exists a diagram of -equivariant morphisms

where is the contracted product, is birational, and is a fibration with fibers isomorphic to . The rank of the horospherical variety is the dimension of the toric variety , namely, .

### 1.2. Classification

Horospherical varieties can be classified using colored fans, see [34]. We focus here on smooth projective horospherical varieties of Picard rank one. Pasquier [35] constructed all these varieties as well as their minimal embedding.

###### Theorem 1.2.1 (Pasquier).

Let be a smooth projective -horospherical variety of Picard rank one. Then either

• is homogeneous,

or can be constructed in a uniform way from a triple belonging to the following list:

• with ;

• ;

• with and ;

• ;

• ,

where is the semisimple Lie type of and are fundamental weights.

As stated in the above theorem some smooth projective horospherical varieties of Picard rank one are homogeneous (for instance projective spaces). We will not consider them since their moduli space of curves, quantum cohomology and quantum -theory have been studied before (see [45, 21, 40, 6, 24, 25, 7, 8, 9, 12, 10, 13, 4, 5] and references therein). We will therefore focus on the second part of the list, i.e., cases (1) to (5).

### 1.3. Construction

From now on, let be a smooth projective non homogeneous horospherical variety of Picard rank one with associated triple . We introduce the following notations, which will be used throughout the paper.

###### Notation 1.3.1.

Let be the reductive group for which is horospherical. Then we denote by and the irreducible -representations with highest weights and and we let and be the corresponding highest weight vectors.

We write for the stabiliser of in and for the stabiliser of in . Both stabilizers are parabolic subgroups of . We denote by the -orbit of in and by that of in , so that and .

Pasquier [35] proves that is horospherical of rank one, and is therefore obtained as follows.

###### Proposition 1.3.2 (Pasquier).

Let be one of the varieties in cases (1) to (5) of Theorem 1.2.1. Then is the closure of the -orbit in .

### 1.4. Orbits

Let be the automorphism group of . The group acts on with three orbits: , , and the complement of in , which is open and dense in .

The group is a semi-direct product of with its unipotent radical. It acts on with two orbits, namely, and . We define . As is the unique closed orbit for we shall call the closed orbit of .

The -orbits and are projective and respectively isomorphic to and . The open -orbit is a -bundle over the incidence variety . We refer to [35] for proofs.

### 1.5. Blow-ups and projections

We refer to [35] for the results in this subsection. Let be the blow-up of in . It is obtained via base change from the blow-up of in . In particular there is a natural projection , obtained as the restriction of the projection of the former blow-up to . The projection restricts to a projection , which realises as a vector bundle over . This bundle is -equivariant and thus it is obtained from a -representation. We write for the corresponding locally free sheaf; it is the normal bundle to in . The projection realises as the projective bundle . Write for the exceptional divisor.

Exchanging the roles of and we denote by the blow-up of in . In particular there is a natural projection , which restricts to a projection . This projection realises as a -equivariant vector bundle over , which is thus obtained from a -representation. We write for the corresponding locally free sheaf; it is the normal bundle to in . The projection realises as the projective bundle . Write for the exceptional divisor.

In both cases, the exceptional divisors and are isomorphic to the incidence variety . The maps and from to and and the maps and from to and are the projections and from to and .

Summarising, we get the commutative diagrams

Let be the blow-up of in . There is a morphism which is a -bundle, and both connected components of the exceptional divisor are mapped isomorphically onto . We will denote this last map by .

### 1.6. Numerical invariants

Using the above description, it is easy to compute some numerical invariants for , and . For example, their dimensions are given in Table 1.

If is a Fano variety of Picard rank one and is an ample generator of , we denote by the unique positive integer such that . We describe , and in Table 2.

We have .

### 1.7. Cohomology classes

The description in Subsection 1.5 implies that has a cellular decomposition. It is therefore easy to describe geometric bases of the cohomology groups of .

Since and are projective and homogeneous under , we can consider their Schubert varieties and for a given Borel subgroup . These varieties define cohomology classes as follows:

 ¯¯¯σu=[Yu]∈H2ℓ(u)(Y,Z) and ¯¯¯τv=[Zv]∈H2ℓ(v)(Z,Z).

We also introduce and . Denoting by and the closed embeddings, we may define cohomology classes on as follows:

 σ′u=[Y′u]∈H2ℓ(u)(X,Z) and σu=i∗[Yu]∈H2ℓ(u)+2codimX(Y)(X,Z) τ′v=[Z′v]∈H2ℓ(v)(X,Z) and τv=j∗[Zv]∈H2ℓ(v)+2codimX(Z)(X,Z).

Using the cellular decompositions by Schubert cells in and and the cellular decomposition induced on and we obtain the following result.

###### Fact 1.7.1.

We have the following bases of .

1. The classes form a -basis of .

2. The classes form a -basis of .

###### Notation 1.7.2.

Recall that and are the projections from the incidence variety to and respectively.

1. Let , then is a Schubert variety in of the form . We define such that . For we define in a similar way: .

2. Let , then is the Schubert variety in associated to an element . For we define in a similar way.

3. Let and the hyperplane classes in and . We define coefficients and as follows:

 hY∪¯¯¯σu=∑u′∈WPYcY(u,u′)¯¯¯σu′ and hZ∪¯¯¯τv=∑v′∈WPZcZ(v,v′)¯¯¯τv′.

These coefficients are well known since they are given by the Chevalley formula for homogeneous spaces, see [14].

###### Proposition 1.7.3.

In , the singular cohomology of , the two bases and , are dual for the Poincaré duality. We have

 σ∨u=σ′u∨ and τ∨v=τ′v∨

where denotes Poincaré duality.

###### Proof.

Using the projection formula we get . But, again by the projection formula, we have where is the element in such that is Poincaré dual to in . We thus have . The same method works for the classes. We are left with proving and . To compute these intersections, we use general translates of , , and under the action of . First note that since and do not meet, we get . Next we want to compute the intersection of general -translates of and . We look for intersection points in , and . Since is a Schubert variety of dimension at most , its intersection with a general translate of is of dimension at most and therefore empty. The same argument proves that no point of the intersection is contained in . Finally, if there was an intersection point in , then using the restriction of to would give a point in the intersection of general translates of and in . The sum of their codimensions is and therefore these general translates do not meet. ∎

We state the following cup product formulas for later reference.

###### Proposition 1.7.4.

Let and with and .

1. If , then .

2. If , then .

###### Remark 1.7.5.

The condition is satisfied in case (3), and case (5). The condition is satisfied in case (1), case (3), and case (5).

###### Proof.

We prove item 1, the proof of item 2 being similar. We again look at general -translates of and . If they were meeting in , then via there would be an intersection point for general translates of and in . This is not possible for dimension reasons. These varieties can therefore only meet in and we have which is of codimension at least in . We look for the intersection of general translates of and . The sum of their codimension in is at least . This proves the result. ∎

###### Remark 1.7.6.

The same proof gives the following more precise result. Let with . Assume that , then .

### 1.8. Hasse diagram

In this subsection we describe the multiplication by the hyperplane class in .

###### Proposition 1.8.1.

We have

 h∪σ′u=τˆu+∑u′∈WPYcY(u,u′)σ′u′ and h∪τv=∑v′∈WPZcZ(v,v′)τv′.
###### Proof.

The first formula follows from the equalities and . The second formula can be directly computed in . ∎

###### Remark 1.8.2.

Similar formulas hold for the products and .

###### Example 1.8.3.

Recall Pasquier’s classification. We give in the appendix the (quantum) Hasse diagrams for Case (1), , Case (2), Case (3), and and Case (5) of Pasquier’s classification.

### 1.9. Special geometry

In two cases of Pasquier’s classification, there is a special geometric feature that we want to use: the variety is the zero locus of a section of a vector bundle. Let us fix first some notation. Denote by a connected component of the Grassmannian of maximal isotropic subspaces in endowed with a non-degenerate quadratic form. Denote by the Grassmannian of -dimensional subspaces in and let be the tautological subbundle. Finally, denote by the Grassmannian of isotropic subspaces of dimension in endowed with an antisymmetric form of rank .

###### Proposition 1.9.1.

In cases (2) and (3) of Pasquier’s classification, the variety is obtained as the zero locus of a vector bundle on a bigger homogeneous space . More precisely

1. In case (2), is a general hyperplane section of .

2. In case (3), is the zero locus of a section of in .

###### Proof.
1. We use Mukai’s classification of Fano varieties of large index (see [20]). Since is a Fano variety of dimension and index , it has to be a hyperplane section of .

2. The variety is isomorphic to , which is by definition the zero locus of a section of in .

## 2. Moduli space of curves

Let be as above a projective horospherical variety with . Let and denote by the moduli space of stable maps of genus , degree and marked points. Here the degree of a map with a nodal curve is by definition the push-forward . Identifying with this moduli space has the expected dimension:

 dimX+⟨c1(TX),d⟩+k−3=dimX+dc1(X)+k−3.

Any irreducible component of has dimension at least equal to the expected dimension. We refer to [23, 16, 27] for further properties of .

### 2.1. Curves with irreducible source

We first consider the open subset of consisting of stable maps with irreducible source. In this subsection, we describe the irreducible components of . We start with an useful lemma on maps to projective bundles. This result was proved for projective bundles of rank one in [40, Proposition 4] and the proof of the general result is very similar. We include this proof for the reader’s convenience.

###### Lemma 2.1.1.

Let be a projective bundle, i.e., , where a globally generated vector bundle over a projective variety. Let be such that (here is the tautological rank one subbundle associated to the projective bundle ).

Then the map induced by composition with realises as a non empty open subset of a projective bundle of rank over .

###### Proof.

The fiber of the map over is given by sections of . Since is rational and irreducible, we may assume that and that these sections are injective maps of vector bundles modulo scalars, where . The fiber is thus the subset of of injective maps of vector bundles. Since is globally generated, so is . This implies that our sections form a non-empty open subset of this projective space.

This construction can be realised in families, which gives us the assertion. Indeed, for let be the universal map. We will need the projection . Consider the sheaf . Since is globally generated and , we have , hence is a vector bundle of rank and is an open subset of . ∎

We use this lemma to describe the irreducible components of .

###### Proposition 2.1.2.

Let be as above. We have

1. .

2. is irreducible of dimension .

3. is irreducible of dimension .

###### Proof.

Recall from [45, 21, 40] that for a homogeneous space, the moduli space of stable maps is irreducible of the expected dimension, therefore is irreducible of dimension and is irreducible of dimension for any non negative integer . This in turn imply that and are irreducible of expected dimension. Indeed, in [40, Proposition 3] it is proved that given a globally generated vector bundle such that is irreducible of expected dimension, then so is . Since is a globally generated vector bundle, the result follows.

Let be a stable map, where is irreducible and nodal. Up to normalising, we may assume that is smooth, hence . If factors through then we are in . So we may assume that meets and need to prove that is in the closure of . For this it is enough to prove that stable maps with irreducible source having a non-trivial intersection with and form a family of dimension smaller than the expected dimension .

For this we consider stable maps from to , the blow-up of in . The map lifts to a map . We compute its degree with respect to the basis of . We have . Define . Since is semiample, we have . Since and is the intersection multiplicity of the curve with , we have .

Now we consider the open subset of stable maps with irreducible source, target and degree with respect to the basis of . We only need to assume that both and are non-negative (this is always true for non-empty moduli spaces) and . We have a natural map given by the composition with . Note that is the projective bundle and that is globally generated. We want to apply the previous lemma so we need to compute with . But a map induces an injective map of vector bundles , and the intersection of with is the vanishing locus of the first factor. In particular . Note that , where is the ample generator of . Therefore, applying the previous lemma we get that is a non-empty open subset of a projective bundle of rank over . Therefore is irreducible of dimension

 dimM∘(d,dY),k(˜XZ)=exp-dim Md,k(X)−(d−dY)(c1(X)−codimXY).

In particular, since , this dimension is smaller that the expected dimension
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