Halving spaces and lower bounds in real enumerative geometry
aa r X i v : . [ m a t h . A T ] J un HALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVEGEOMETRY
L ´ASZL ´O M. FEH´ER AND ´AKOS K. MATSZANGOSZ
Abstract.
We develop the theory of halving spaces to obtain lower bounds in real enumerativegeometry. Halving spaces are topological spaces with an action of a Lie group Γ with additionalcohomological properties. For Γ = Z we recover the conjugation spaces of Hausmann, Holm andPuppe. For Γ = U(1) we obtain the circle spaces . We show that real even and quaternionic partialflag manifolds are circle spaces leading to non-trivial lower bounds for even real and quaternionicSchubert problems. To prove that a given space is a halving space, we generalize results of Boreland Haefliger on the cohomology classes of real subvarieties and their complexifications. Thenovelty is that we are able to obtain results in rational cohomology instead of modulo 2. Theequivariant extension of the theory of circle spaces leads to generalizations of the results of Boreland Haefliger on Thom polynomials. Contents
1. Introduction 22. Halving spaces 42.1. Preliminaries 42.2. The definition of halving spaces 52.3. Main properties 63. Geometry 83.1. Fundamental classes of real varieties 93.2. Excess intersection 103.3. Halving cycles 144. The generalized Borel-Haefliger theorem 175. Examples 175.1. R -spaces 185.2. Quaternionic toric varieties 235.3. Constructions 236. Applications for enumerative problems 24 Mathematics Subject Classification.
Key words and phrases. conjugation spaces, equivariant cohomology, circle actions, real flag manifolds, realenumerative geometry.L. M. F. was partially supported by NKFI 112703 and 112735 as well as ERC Advanced Grant LTDBud andenjoyed the hospitality of the R´enyi Institute.A. K. M. is supported by the Hungarian National Research, Development and Innovation Office, NKFIH K 119934.
Introduction
The answer to an enumerative geometry problem over C is a natural number. In the case ofSchubert calculus, this number is completely determined by the rational cohomology ring of theGrassmannian.For an enumerative geometry problem over R , the space of generic configurations is no longerconnected, so the answer will be a finite list of natural numbers. The number of solutions forthe corresponding complex problem is an obvious upper bound for the number of real solutions.Moreover, modulo two the number of real solutions should be congruent to the number of complexsolutions (since the non-real solutions come in complex conjugate pairs).Lower bounds are more difficult to obtain, for example see [41], [35]. Several authors noticed(e.g. [13], [27]) that for some problems a cohomological calculation is available. For exampleconsider the following: Problem 1.
How many 4-dimensional linear subspaces of R intersect four given generic 4-dimensional linear subspaces in 2 dimensions? The number of solutions for the complex problem is the integral Z Gr ( C ) [ σ (2 , ] , where σ (2 , is the Schubert variety corresponding to the partition (2 , σ (2 , ] ∈ H (Gr ( C ))is the cohomology class represented by it. According to the Schubert calculus the answer is 6,which is an upper bound for the number of solutions for the real problem. The key to find alower bound is the observation that there is a degree halving isomorphism κ between the rationalcohomology ring of the real Grassmannian Gr ( R ) and the rational cohomology ring of thecomplex Grassmannian Gr ( C ). Indeed, the first ring is generated by the Pontryagin classesof the tautological subbundle, the second is generated by the Chern classes of the tautologicalsubbundle, and the assignment p i c i extends to an isomorphism. Moreover, the real Schubertvariety σ R (2 , is a cycle (it represents a cohomology class in Gr ( R )), and κ [ σ R (2 , ] = [ σ (1 , ]. Thisimplies that Z Gr ( R ) [ σ R (2 , ] = Z Gr ( C ) [ σ (1 , ] = 2 . ALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVE GEOMETRY 3
In the absence of a complex structure on Gr ( R ) this result means only that the signed sum ofthe solutions is 2, therefore 2 is a lower bound to the number of solutions.Our main goal here is to study how general this situation is. It turns out that even Grassman-nians Gr k ( R n )—and more generally even partial flag manifolds—possess a structure we will call circle space , which allows us to get lower bounds for a large family of real enumerative problems.Circle spaces are closely related to conjugation spaces, which were introduced by Hausmann, Holmand Puppe [22] to understand the topological background of the theory of Borel and Haefliger [6].The object of study in [6] is a nonsingular projective variety X over C , which is the complex-ification of its real part. Let Γ = Z act on X by complex conjugation and let X Γ denote thereal part (the Γ-fixed point set). They show in Proposition 5.15 of [6] that assuming that all x ∈ H ∗ ( X ; F ) and y ∈ H ∗ ( X Γ ; F ) can be represented by algebraic cycles defined over R andreal algebraic cycles respectively, we have a degree halving isomorphism κ between the mod 2cohomology of X and the mod 2 cohomology of the real part X Γ . This isomorphism has theproperty κ [ Z ] = [ Z Γ ] for any algebraic cycle Z , which is the complexification of its real part Z Γ .Conjugation spaces are spaces with a Γ = Z -action, satisfying a certain condition in mod 2cohomology, called the restriction equation . The nonsingular projective varieties of above withthe conjugation action are the main examples of conjugation spaces.In this paper we extend the theory of conjugation spaces to halving spaces (Definition 2.3),where the acting group Γ is allowed to be U(1) or Sp(1), and the conjugation equation becomes theanalogous condition in rational cohomology. The existence of a degree halving ring isomorphism κ between the rational cohomology ring of the fixed point set X Γ and of the halving space X canbe established analogously to the case of conjugation spaces (Theorem 2.5). Our main tool tofind examples of halving spaces is the generalized Borel-Haefliger theorem (Theorem 4.1), whichhas the following special case: Theorem 4.1.
Let
Γ = U(1) and let X be a compact orientable Γ -manifold, whose rationalcohomology groups are additively generated by good U(1) -invariant cycles [ Z i ] (Definition 3.13),such that codim R Z i = 2 codim R Z Γ i . Assume that X Γ is connected. Then X is a halving space,and the assignment sending [ Z i ] to [ Z Γ i ] determines a degree-halving multiplicative isomorphismbetween H ∗ ( X ; Q ) and H ∗ ( X Γ ; Q ) . The proof of Theorem 4.1 is an adaptation of Van Hamel’s work [40]. Using Theorem 4.1 weobtain
Theorem 5.2.
Let
Γ = U(1) and let Γ act on Fl D ( R n ) obtained by identifying C n with R n .With this action it is a circle space, with fixed point set Fl D ( C n ) . Furthermore, the degree-halvingring isomorphism κ associated to the circle space structure satisfies: κ [ σ R DI ] = 2 | I | [ σ C I ] , where [ σ C I ] ∈ H | I | (Fl D ( C N )) . and Theorem 5.7.
With the
Γ = U(1) -action defined by inner automorphisms (see Section 5.1.2),
L ´ASZL ´O M. FEH´ER AND ´AKOS K. MATSZANGOSZ Fl D ( H n ) is a circle space, with fixed point set Fl D ( C n ) . Furthermore, the degree-halving ringisomorphism κ associated to the circle space structure satisfies: κ [ σ H I ] = 2 | I | [ σ C I ] , where [ σ C I ] ∈ H | I | (Fl D ( C n )) . Using products and connected sums we can create further examples. Halving spaces with Γ =Sp(1) are less common, our examples are the octonionic flag manifolds (Theorem 5.13).Circle spaces allow us to study various real (and quaternionic) enumerative problems (the doubleSchubert problems , see Section 6) and finding lower bounds for them. A different example is
Proposition 7.4.
Given four generic linear maps α i : R → R the number of 4-dimensionalsubspaces V satisfying dim( V ∩ α i ( V )) = 2 for i = 1 , . . . , is at least 32. This example is connected with quivers and leads naturally to an equivariant variation of thetheory, giving a partial analogue (Theorem 7.5) of Borel and Haefliger’s result on mod 2 Thompolynomials [6, Theorem 6.2.].
Structure of the paper.
In Section 2, we generalize the definition of conjugation spaces [22]to U(1) and Sp(1)-actions and give the first example of a circle space.Section 3 contains the key technical results. We introduce the notion of a halving cycle andshow that if Z ⊂ X is a halving cycle, then κ [ Z ] = µ Z [ Z Γ ], where µ Z ∈ N is the excess multiplicity of Z . Our main tool is the excess intersection formula of Quillen.The main result of Section 4 is the generalized Borel-Haefliger theorem (Theorem 4.1): we givea sufficient condition (existence of a basis in cohomology, represented by halving cycles) for aΓ-manifold to be a halving space.We give examples of halving spaces in Section 5 using the generalized Borel-Haefliger theorem.We give a description of the Schubert calculus of the even real and the quaternionic flag manifolds.In Section 6 we give some applications of these results in real and quaternionic enumerativegeometry using Schubert calculus.In Section 7 we study a quiver type degeneracy locus and related enumerative problems. Inconclusion we mention an equivariant extension the generalized Borel-Haefliger theorem. Acknowledgment.
We are grateful to Bal´azs Csik´os, Matthias Franz, Tara Holm, Liviu Mare,Rich´ard Rim´anyi, Endre Szab´o and Andrzej Weber for several helpful discussions on the subjectof this paper. 2.
Halving spaces
In this section we extend the definition of conjugation spaces [22] to Γ = U(1) and Sp(1)-actionsand cohomology with Z or Q -coefficients. Most of the proofs of Hausmann, Holm and Puppe [22]generalize word by word to halving spaces, we give a discussion for the sake of completeness.2.1. Preliminaries.Definition 2.1.
Let Γ be a Lie group, and R be a ring. We say that (Γ , R ) is a halving pair , if H ∗ ( B Γ; R ) ∼ = R [ u ] with u ∈ H D ( B Γ; R ), for some D ∈ N + . ALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVE GEOMETRY 5
Example 2.2. [See also Steenrod’s polynomial realization problem [37]] • ( Z , F ) is a halving pair with D = 1, since H ∗ ( B Z ; F ) ∼ = F [ u ] for u = w ( S ) where S → R P ∞ is the tautological bundle. • (U(1) , Z ) is a halving pair with D = 2, since H ∗ ( B U(1); Z ) ∼ = Z [ u ] for u = c ( S ) where S → C P ∞ is the tautological bundle. • (Sp(1) , Z ) is a halving pair with D = 4, since H ∗ ( B Sp(1); Z ) ∼ = Z [ u ], for u = q ( S ) where S → H P ∞ is the tautological bundle and q i denotes the quaternionic Pontryagin class.Throughout the paper, we will always take cohomology with coefficients R corresponding tothe group action Γ for each halving pair(Γ , R ) ∈ { ( Z , F ) , (U(1) , Q ) , (Sp(1) , Q ) } , and u ∈ H D ( B Γ; R ) is always defined as above. To simplify the discussion (e.g. for localizationtheorems) we work with field coefficients, although some of the discussion generalizes to R = Z .If the coefficient ring R is clear from the context, then we drop it from the notation, i.e. H ∗ ( X ) = H ∗ ( X ; R ).Let X be a Γ-space, i.e. an action of Γ is given on X , and let us fix the coefficient ring R . Let H ∗ Γ ( X ; R ) = H ∗ ( B Γ X ; R ) denote the Γ -equivariant cohomology of X , where B Γ X = E Γ × Γ X is the Borel construction; let ρ : H ∗ Γ ( X ; R ) → H ∗ ( X ; R ) denote the forgetful map. A graded R -module homomorphism σ : H ∗ ( X ) → H ∗ Γ ( X ) is called a Leray-Hirsch section , if ρσ = id. Wecall σ a Leray-Hirsch section, since its existence implies that the condition of the Leray-Hirschtheorem is satisfied for the fiber bundle X → E Γ × Γ X → B Γ.We have a restriction map r : H ∗ Γ ( X ) → H ∗ ( X Γ )[ u ], where we use that the Γ-action on X Γ istrivial, so H ∗ Γ ( X Γ ) = H ∗ ( X Γ )[ u ].2.2. The definition of halving spaces.
By adapting Hausmann, Holm and Puppe’s definitionof conjugation spaces [22] to halving pairs, we obtain the central definition of this paper:
Definition 2.3.
Let X be a Γ-space, and denote by X Γ its fixed point set. Assume X has nonzerocohomology only in 2 Di degrees, and that there exists a Leray-Hirsch section σ : H ∗ ( X ) → H ∗ Γ ( X )satisfying the following degree condition :(DC) : r ( σ ( x )) is a polynomial in u of degree exactly i for all x ∈ H Di ( X ).A Γ-space X satisfying these conditions is called a halving space .Let us unravel this definition. The halving space structure involves the following maps, whichwill be used in the following: H ∗ Γ ( X ) r / / ρ (cid:15) (cid:15) H ∗ ( X Γ )[ u ] (cid:15) (cid:15) H ∗ ( X ) / / σ E E H ∗ ( X Γ )Let(1) κ : H ∗ ( X ) → H ∗ ( X Γ ) L ´ASZL ´O M. FEH´ER AND ´AKOS K. MATSZANGOSZ be the degree halving R -module homomorphism κ ( x ) := coeff( rσ ( x ) , u i ) for x ∈ H Di ( X ). Thepair ( κ, σ ) is called a cohomology frame .With this notation, the degree condition (DC) means that κ is injective and that for x ∈ H Di ( X )(2) rσ ( x ) = κ ( x ) u i + λ u i − + . . . + λ i − u + λ i , where λ j ∈ H D ( i + j ) ( X Γ ). We call equation (2) restriction equation , it is called conjugation equa-tion in [22]. For (Γ , R ) = ( Z , F ), the definition of halving spaces is the same as the definition ofconjugation spaces in [22], except that we don’t require κ to be surjective.It would be more precise to call a halving space a (Γ , R )-halving space, however when (Γ , R ) isfixed, we simply say X is a halving space. We consider halving spaces for the following halvingpairs (Γ , R ): • Hausmann-Holm-Puppe’s conjugation spaces [22] for the halving pair ( Z , F ). • Circle spaces for the halving pair (U(1) , Q ). This is the main case we will consider. • Quaternionic halving spaces for the halving pair (Sp(1) , Q ).2.3. Main properties.
To motivate the following discussion, we list some of the nice propertiesof halving spaces: • κ is a degree-halving ring homomorphism, • σ is a ring homomorphism, therefore the Leray-Hirsch isomorphism induced by σ is a H ∗ Γ = H ∗ ( B Γ; R )-algebra isomorphism, • the cohomology frame ( κ, σ ) is unique.The proof of these properties relies on the following lemma, which is implicitly used in [22], itsproof is the same, we repeat it for the sake of completeness. Lemma 2.4 (Degree Lemma) . Let X be a halving space with cohomology frame ( κ, σ ) . Let D denote the degree of the generator u ∈ H ∗ Γ . Then for x ∈ H Dk Γ ( X ; R ) x ∈ Im σ ⇐⇒ deg u ( rx ) = k Proof.
The direction ⇒ holds by definition. For the other direction, let x ∈ H Dk Γ ( X ; R ), andassume x Im σ . By the Leray-Hirsch theorem x = k X i =0 σ ( ξ i ) u k − i ) for some ξ i ∈ H Di ( X ). Since r is an R [ u ]-module morphism, rx = k X i =0 ( rσξ i ) u k − i ) = k X i =0 p i ( u ) u k − i ) where rσξ i = p i ( u ) ∈ H ∗ ( X Γ )[ u ] is a polynomial in u of degree ≤ i . Then p i ( u ) u k − i ) has degree ≤ k − i for each i . Take the smallest 0 ≤ i < k , such that ξ i = 0. Then rσξ i = κ ( ξ i ) u i + ... andsince κ is injective, rσξ i has degree i . It follows that rx is a polynomial in u of degree 2 k − i . ALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVE GEOMETRY 7
Since i < k , this is a contradiction, since rx is a polynomial of degree k by assumption. Therefore ξ i = 0 for i < k and x = σ ( ξ k ). (cid:3) In particular, by using the Leray-Hirsch theorem, Lemma 2.4 implies that if x ∈ H Dk Γ ( X ; R ),and deg u ( rx ) < k , then x = 0. Theorem 2.5.
Let X be a halving space. Then κ and σ are multiplicative.Proof. Let a ∈ H Dk ( X ), b ∈ H Dl ( X ). Set x := σ ( a ) σ ( b ), note that ρ ( x ) = ab . Then rx = r (cid:0) σ ( a ) (cid:1) r (cid:0) σ ( b ) (cid:1) = ( κ ( a ) u k + ... )( κ ( b ) u l + ... ) , so x = σ ( y ) for some y ∈ H D ( k + l ) ( X ) by the degree lemma. Since ab = ρ ( x ) = ρ ( σ ( y )) = y, so x = σ ( y ) = σ ( ab ) and by definition x = σ ( a ) σ ( b ) proving multiplicativity of σ . Using rσ ( ab ) = r (cid:0) σ ( a ) (cid:1) r (cid:0) σ ( b ) (cid:1) , the degree k + l part of the left hand side is κ ( ab ) and on the right hand side κ ( a ) κ ( b ). (cid:3) In particular, multiplicativity of σ implies that the Leray-Hirsch isomorphism induced by σH ∗ Γ ( X ) ∼ = H ∗ ( X ) ⊗ R H ∗ Γ is a ring isomorphism. Corollary 2.6 (Naturality) . Let
X, Y be halving spaces with some cohomology frames ( κ X , σ X ) and ( κ Y , σ Y ) for X and Y respectively. If f : X → Y is a Γ -equivariant map, then σ X ◦ H ∗ f = H ∗ Γ f ◦ σ Y : H ∗ ( Y ) → H ∗ Γ ( X ) and κ X ◦ H ∗ f = H ∗ f Γ ◦ κ Y : H ∗ ( Y ) → H ∗ ( X Γ )The proof proceeds similarly as the proof of multiplicativity using the degree lemma. Note thatnaturality implies uniqueness of the cohomology frame ( κ, σ ). Example 2.7.
Let X = Gr ( R n ). Consider the Γ = U(1)-action on R n by identifying it with C n and acting by complex multiplication. This induces an action on X . With this action, X is acircle space with fixed point set X Γ = C P n − . Proof.
Since the U(1)-invariant subspaces are exactly the complex subspaces, X Γ can be identifiedwith Gr ( C n ) = C P n − . In terms of characteristic classes, the ring structure can be written as H ∗ (Gr ( R n ); Q ) = Q [ x ] /x n , H ∗ ( C P n − ; Q ) = Q [ y ] /y n , where x = p ( S R ), y = c ( S C ), and S R → Gr ( R n ), S C → C P n − are the tautological bundles.Let σ be defined on the additive generators by σ ( x i ) := ( p Γ1 ( S R ) − u ) i , where p Γ i ( S R ) = p i ( B Γ S R → B Γ X ) denotes the i -th equivariant Pontryagin class. This σ is a Leray-Hirsch section,and it satisfies the degree condition (DC), which can be shown by the following computation.First, B Γ ( S R ) | X Γ = B Γ ( S R | X Γ ) = B Γ S C . L ´ASZL ´O M. FEH´ER AND ´AKOS K. MATSZANGOSZ
Since Γ acts on S C by complex multiplication, we can rewrite it as the tensor product of equivariantbundles S C = S C ⊗ C C tw , where S C denotes S C with the trivial action and C tw denotes the trivialbundle, with the nontrivial Γ-action given by complex multiplication. Then as bundles over B Γ X Γ = B Γ × X Γ , B Γ S C = S C ⊗ C τ, where B Γ C tw = τ → C P ∞ is the tautological bundle and we omit the notation for the pullbacksto the product. Therefore rp Γ i ( S R ) = p i ( B Γ S C ) = ( − i c i (( S C ⊗ C τ ) ⊗ R C )and c ∗ (( S C ⊗ C τ ) ⊗ R C ) = c ∗ ( S C ⊗ C τ ) c ∗ ( S C ⊗ C τ ) = (1 + y + u )(1 − y − u ) , so rσ ( x i ) = r ( p Γ1 ( S R ) − u ) i = (( y + u ) − u ) i = (2 yu + y ) i , hence σ satisfies the degree condition, and ( κ, σ ) is a cohomology frame with κ ( x i ) = 2 i y i . (cid:3) Similar explicit formulas can be given for even partial flag manifolds, however we will useTheorem 4.1. to prove that they are circle spaces.3.
Geometry
The original Borel-Haefliger theorem states that for a smooth complexified projective vari-ety X C , under certain conditions, the complexification map [ Z R ] [ Z C ] from H ∗ ( X R ; F ) → H ∗ ( X C ; F ) is a well defined multiplicative isomorphism, where Z R is a real subvariety. Here by real subvariety , we mean a subset Z R ⊆ X R defined by real algebraic equations whose Zariskiclosure in X C equals the set of complex points defined by the same equations – its complexifica-tion . Note, that Z R = ( Z C ) Γ is the fixed point set of the complex conjugation on Z C . In terms ofconjugation spaces, we can rephrase the Borel-Haefliger theorem as Theorem 3.1. [6] , [40] For Z C ⊆ X C with the properties above: X C is a conjugation space and (3) κ [ Z C ] = [ Z R ] . We would like to find a similar connection for halving spaces. We illustrate on the Schubertvarieties of Example 2.7, that some adjustments are necessary.Fix a complete flag F • in R n such that F i is U(1)-invariant for all i ; then F C • = ( F ≤ F ≤ . . . ≤ F n )is a complex flag, F C i = F i . Let Z be the real Schubert variety Z = σ R (2 i, i ) ( F • ). This Z isU(1)-invariant and naturally isomorpic to the Grassmannian Gr ( F n − i ) ). The fixed point set Z Γ is the set of complex lines contained in F C n − i , so Z Γ = σ C i ( F C • ), which is naturally isomorphic tothe projective space P F C n − i . If Q R → Gr ( R n ) and Q C → C P n − denote the tautological quotientbundles, then(4) [ σ R (2 i, i ) ] = p i ( Q R ) = ( − x ) i , [ σ C i ] = c i ( Q C ) = ( − y ) i , ALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVE GEOMETRY 9 with the appropriate coorientation of the submanifold σ R (2 i, i ) ⊂ Gr ( R n ), implying that insteadof (3) we have κ [ Z ] = 2 i [ Z Γ ].The goal of this section is to prove that for reasonable ’subvarieties’ Z of the halving space X we have κ [ Z ] = µ Z [ Z Γ ], where µ Z ∈ R is the excess multiplicity of Z . First we discuss the notionof topological varieties and their fundamental cohomology classes in Section 3.1.1, then reviewa part of excess intersection theory necessary to introduce the notion of excess multiplicities inSection 3.2.In this section we restrict our attention to halving manifolds , smooth manifolds which are halv-ing spaces with smooth Γ-action with (Γ , R ) being ( Z , F ) or (U(1) , Q ) and fix the correspondingdegree D = 1 ,
2. In the context of conjugation manifolds, many of their properties can be foundin [21, Section 2.7].Most of the results also hold for (Sp(1) , Q ) under additional assumptions, see Remarks 3.20and 3.24 iii).3.1. Fundamental classes of real varieties.
The Borel-Haefliger theorems involve cohomologyclasses of real and complex algebraic varieties, and there are slightly different approaches indefining these. We use the definitions of topological varieties of the original Borel-Haefliger paper[6] and van Hamel [40]. More precisely, we use a variation, in the sense that we work in cohomologyinstead of homology; in particular we use coorientability instead of orientability. By manifold andsubmanifold we always mean smooth manifold and submanifold. We expect that with enoughcare, most of the discussion generalizes to topological, even cohomological manifolds, however wewill not need this generality.This section is standard and well-known to experts, however because of the slight variations ofthese notions, we found it reasonable to include some details at least to fix terminology.3.1.1.
Topological varieties.
Borel and Haefliger define fundamental classes for a class of topo-logical spaces that we will call—following van Hamel [40]— topological varieties (or in Borel andHaefliger’s notation
V S n spaces). This is a class of topological spaces which includes analyticmanifolds and algebraic varieties; both real and complex.Throughout the discussion, fix a smooth ambient manifold X (connected, not necessarily ori-entable) and a principal ideal domain K (typically K = Z or a field F p , Q , R ), and all cohomologyis taken with K -coefficients. Definition 3.2. Σ ⊆ X has cohomological codimension codim K Σ ≥ k if H i ( X, X \ Σ) = 0 for all i ≤ k −
1. It has cohomological codimension codim K Σ = k , if codim K Σ ≥ k and codim K Σ k +1.We use the convention codim K ∅ = ∞ . If Z ֒ → X is a smooth, connected k -codimensionalsubmanifold which is not coorientable, then codim Z Z ≥ k + 1. Definition 3.3.
A connected, closed subset Z ⊆ X is a topological subvariety of codimension k if there exists an open subset U ⊆ Z which is a k -codimensional submanifold in X and itscomplement Σ := Z \ U has codim K Σ ≥ k + 1. Such a set U ⊆ Z is called a fat nonsingular set .More generally, a closed subset Z ⊆ X is a topological subvariety of codimension k if it hasfinitely many connected components, each of which is a topological subvariety of codimension k . Topological subvarieties behave similarly to algebraic ones: if Z ⊆ X is a topological subva-riety of codimension k , then we can define the set of regular points Z R , which are the pointshaving neighbourhoods that are locally submanifolds. Then the singular set Z S = Z \ Z R is con-tained in the complement Σ of any fat nonsingular set, and the long exact sequence of the triple( X, X \ Z S , X \ Σ) shows that Z R is also a fat nonsingular set. The main property of a topologicalsubvariety Z ⊆ X relevant to us, is that if Z has a fundamental class, then it is unique up tounits of K .3.1.2. Fundamental class.
Let Z ⊆ X be a topological subvariety of codimension k with fatnonsingular subset U , let y ∈ U . A normal disk D y ⊆ X to y ∈ U is a k -dimensional smoothlyembedded disk D y ⊆ X centered at y , intersecting U in the single point y transversally. Thefollowing construction is an extension of the fundamental cohomology class of submanifolds totopological subvarieties. Definition 3.4.
Let Z ⊆ X be a k -codimensional topological subvariety over K and D x a normaldisk of Z R at x . A local coorientation at x ∈ Z R is a generator of H k ( D x , D x \ x ). Definition 3.5.
Let Z ⊆ X be a k -codimensional topological subvariety over K . A funda-mental cohomology class (over K ) is an element [[ X ]] ∈ H k ( X, X \ Z ; K ) whose restriction to H k ( D x , D x \ x ; K ) is a generator for all regular points x ∈ Z R , where D x is a normal disk over x .If such a fundamental class exists, we say that Z is a cycle .This definition extends the notion of Thom class: if Z ֒ → X is a submanifold, then it is a cycleiff it is coorientable. Even if a topological subvariety Z ⊆ X is a cycle, the (non-refined) class[ Z ] ∈ H k ( X ) can be zero (although this cannot happen to the refined class [[ Z ]] ∈ H k ( X, X \ Z )).In this section we distinguish the notation of refined class [[ Z ]] and [ Z ], but later we will denoteboth by [ Z ]. The following proposition summarizes some existence and uniqueness properties offundamental classes, which follow from the previous discussion (see also [25, Section A.1.2]). Proposition 3.6.
Let Z ⊆ X be a k -codimensional topological subvariety with a fat nonsingularsubset U , Σ = Z \ U . • Uniqueness: Given a fundamental class [[ U ]] ∈ H k ( X \ Σ , X \ Z ) , Z has at most one funda-mental class [[ Z ]] restricting to [[ U ]] . • Uniqueness for K = F : If K = F , then Z has at most one fundamental class [[ Z ]] . • Existence: If U has a fundamental class [[ U ]] , and if codim K Σ ≥ k + 2 , then X has afundamental class restricting to [[ U ]] . Remark 3.7. If Z is a complex subvariety, then there is a canonical choice for the fundamentalclass: the complex structure induces an orientation on the normal bundle of the smooth part. If Z is a topological subvariety, and K = F , then by the previous proposition, the fundamentalclass is unique. However, for K = Z and Z connected we have two choices of the fundamentalclass [ Z ] according to the choice of the orientation of the normal bundle of U .3.2. Excess intersection.
Quillen introduced the excess intersection formula in the context ofcomplex cobordism in [30]. We recall Quillen’s results, define the excess weight, and then provethe excess weight lemma (Lemma 3.15).
ALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVE GEOMETRY 11
Clean intersection, excess bundle.
Smooth submanifolds
Y, Z ֒ → X are said to intersectcleanly , if their intersection W := Y ∩ Z is a submanifold and T Y | W ∩ T Z | W = T W . The excessbundle of a clean intersection is η ( Y, Z ) :=
T X | W / ( T Y | W + T Z | W ). Denoting the inclusion maps(EIF) W (cid:31) (cid:127) j / / (cid:127) _ g (cid:15) (cid:15) Z (cid:127) _ f (cid:15) (cid:15) Y (cid:31) (cid:127) i / / X the relations ν i | W ∼ = ν j ⊕ η and ν f | W ∼ = ν g ⊕ η hold: for clean intersections the defining short exactsequence of η induces0 / / ( T Y | W + T Z | W ) (cid:14) T Z | W | {z } ν g / / T X | W (cid:14) T Z | W | {z } ν f | W / / η / / ν g by the isomorphism theorems. If f, g are cooriented, thenthere is a unique compatible orientation on η such that ν f | W = ν g ⊕ η as oriented bundles. Remark 3.8. • The direct sum orientation depends on the order of ν g ⊕ η , so let us adopt this convention. • If f, i, j are cooriented, then ν g is orientable, with a unique compatible orientation satis-fying ν f | W ⊕ ν j = ν i | W ⊕ ν g as oriented bundles.3.2.2. Equivariant excess intersection formula.
For the following proposition, see Quillen [30,Proposition 3.6].
Proposition 3.9.
Let
Γ = U(1) act on X . Let Z ֒ → X be a Γ -invariant oriented smoothsubmanifold. Then Z ∩ X Γ is a clean intersection and all maps in Z Γ (cid:31) (cid:127) j / / (cid:127) _ g (cid:15) (cid:15) Z (cid:127) _ f (cid:15) (cid:15) X Γ (cid:31) (cid:127) i / / X can be compatibly oriented, and with these orientations i ∗ f ! z = g ! (cid:0) j ∗ z · e ( η ) (cid:1) in H ∗ Γ ( X Γ , X Γ \ Z Γ ) . We will be mainly interested in the special case z = 1 when we have[ Z ] | X Γ = i ∗ f ! g ! (cid:0) e ( η ) (cid:1) . Excess multiplicity.
Without going into the general theory of real representations we defineweights for the three groups we are interested in: • Γ = Z : There are two irreducible real representations, the one dimensional trival, and theone dimensional non-trival one. We define their weights to be 0 and 1 in F , respectively. • Γ = U(1): There is the one dimensional trivial representation, and for every positiveinteger n there is a 2-dimensional irreducible real representation. We define their weightsto be 0 and n in N , respectively. • Γ = Sp(1): We restrict an Sp(1)-representation to its maximal torus, which is a U(1), andthe weights are defined according to the weights of the maximal torus.The multiset of weights is denoted by W ( V ) for the Γ-representation V . So for example W ( V )is a set of non-negative numbers with multiplicities for Γ = U(1). The multiplicity of V is definedas the product µ ( V ) := Q w ∈ W ( V ) w . Remark 3.10.
We can always choose an orientation of V , such that e Γ ( V ) = µ ( V ) u k for the Γ-equivariant Euler class of V , where k depends on Γ and the dimension of V . Definition 3.11.
Let (Γ , R ) be a halving pair. Let X be a Γ-manifold, Z ⊆ X be a Γ-invarianttopological subvariety and z ∈ Z Γ R . Let η denote the excess bundle of the (clean) intersection Z Γ R = Z R ∩ X Γ . We call the elements of W ( η z ) the excess weights of Z at z . Notice that W ( η z )is a subset of the multiset of weights of ν z ( X Γ ⊂ X ). We will call the latter the normal weightsof the halving space X at z . The multiplicity µ ( η z ) of the Γ-representation η z is called the excessmultiplicity of Z at z . Remark 3.12.
For conjugation spaces and circle spaces, the slice theorem [19, Theorem B.24]implies that the excess multiplicity is not zero. For circle spaces it also implies that the rank ofthe excess bundle η ( Z ) is always even.3.2.4. Equivariant fundamental class.
Let Γ be a compact connected Lie group and Z be a Γ-invariant topological subvariety with a Γ-invariant fat nonsingular subset U and singular set Σ.For connected groups Γ, existence of a fundamental class is a nonequivariant phenomenon: if Z is a Γ-invariant cycle with fundamental class [ Z ] ∈ H k ( X, X \ Z ), then there exists a unique[ Z ] Γ ∈ H k Γ ( X, X \ Z ) restricting to [ Z ].On the other hand, this does not ensure that Z Γ is a topological subvariety, since it mighthappen that Σ Γ has too large dimension. This motivates the following definition, introduced forΓ = Z in [40]: Definition 3.13.
A Γ-invariant closed subset Z ⊆ X is a good Γ -invariant subvariety of codi-mension type ( k, l ) if • Z ⊆ X is a topological subvariety of codimension k with Γ-invariant fat nonsingular set U • Z Γ ⊆ X Γ is a (nonempty) topological subvariety of codimension l with fat nonsingular set U Γ . ALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVE GEOMETRY 13
We call such a set U a Γ -invariant fat nonsingular set . If in addition Z ⊆ X is a cycle and itsexcess multiplicity µ Z := µ ( η z ) is independent of z ∈ Z Γ R , then we say that Z is a good Γ -invariantcycle of codimension type ( k, l ). Example 3.14.
Let Z ⊆ X be a k -codimensional Γ-invariant stratified submanifold with Γ-invariant top stratum Z k . If Z Γ has a stratification whose unique top stratum is ( Z k ) Γ , then Z is a good Γ-invariant subvariety of codimension type ( k, l ) where l = codim( Z k ) Γ . If additionally Z is a Γ-invariant cycle and ( Z k ) Γ is connected, then Z is a good Γ-invariant cycle. This will berelevant in the case of real double Schubert varieties, see Theorem 5.2.For good Γ-invariant cycles, the classes [ Z ⊆ X ] Γ and [ Z Γ ⊆ X Γ ] are related by the followingLemma. Lemma 3.15 (Excess multiplicity lemma) . Let
Γ = U(1) and R = Q be the coefficients ofcohomology. Let X be a Γ -manifold and let Z ⊆ X be a good Γ -invariant cycle of codimensiontype ( k, l ) . Then Z Γ ⊆ X Γ is a cycle and it has a fundamental class [ Z Γ ⊆ X Γ ] satisfying H ∗ Γ ( X ) r / / H ∗ Γ ( X Γ ) ∼ = H ∗ ( X Γ )[ u ] ⊗ H ∗ Γ [ Z ⊆ X ] Γ ✤ / / w · [ Z Γ ⊆ X Γ ] + deg Adapting the proof of van Hamel [40, Cor 1.3(ii)] to the context of circlespaces shows that κ [ Z ] = µ [ Z Γ ], where µ is an undetermined constant. This constant is explicitlydescribed as the excess multiplicity as described in Section 3.2, and allows us to generalize theBorel-Haefliger theorem to U(1) and Sp(1)-actions. Definition 3.17. Let X be a Γ-manifold. A good Γ-invariant cycle Z ⊆ X (Definition 3.13) ofcodimension type (2 k, k ) is called a halving cycle .For example, in the algebraic case, complexified cycles Z C , are Z -halving cycles over R = F . Remark 3.18. For Γ = U(1), a halving cycle Z has codimension divisible by 4. Indeed, thecodimension of Z ⊆ X has the same parity as the codimension of Z Γ ⊆ X Γ , by Remark 3.12). ALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVE GEOMETRY 15 Now we are ready to generalize Theorem 3.1 of Borel and Haefliger to halving manifolds: Theorem 3.19. Let (Γ , R ) be ( Z , F ) or (U(1) , Q ) . Let X be a halving manifold, and Z ⊆ X bea halving cycle. Then σ [ Z ] = [ Z ] Γ , κ [ Z ] = µ Z [ Z Γ ] , where µ Z is the excess multiplicity of Z (Definition 3.11).Proof. Set codim Z = 2 Dk . By the excess multiplicity lemma (Lemma 3.15)(6) r [ Z ] Γ = w · [ Z Γ ] + deg The lemma also holds for the halving pair (Γ , R ) = (Sp(1) , Q ), if one assumesthat Z Γ ⊆ X Γ is a cycle and that w = 0, see Remark 3.16 iii).3.3.1. Coefficients of the restriction equation. Let us return for a moment to the case of conju-gation spaces; let X be a conjugation space. Franz and Puppe [14] determined the coefficients ofthe restriction equation as Steenrod squares:(7) r ( σ ( α )) = d X i =0 Sq i α · u d − i =: Sq u ( α ) , for α ∈ H d ( X ). Together with the theorem of Van Hamel [40], this proves the topologicalversion of a classical theorem of Chow [7], that [ Z ] | X Γ = [ Z Γ ] . Using Proposition 3.19, we obtaina simple proof of a weaker version of (7), namely in the algebraic case. For a similar theorem, see[5, Theorem 1.18]. Proposition 3.21. Let X be the complexification of a real algebraic variety which is smooth, andlet Z ⊆ X be a complexified subvariety which is a smooth cycle. Then rσ [ Z ] = Sq u ( κ [ Z ]) Proof. The proof can be summarized as[ Z ] Γ | X Γ == g ! ( e Γ ( η )) == g ! ( w u ∗ ( ν )) == Sq[ Z Γ ] , where g : Z Γ ֒ → X Γ and w u ∗ denotes the total homogenized Stiefel-Whitney class. 1) is the excessintersection formula. Since Z is a complexification, ν ( Z ⊆ X ) | Z Γ = ν ( Z Γ ⊆ X Γ ) ⊗ R C holds. Then the excess bundle equivariantly is η = ν ( Z Γ ⊆ X Γ ) ⊗ R i R where i R denotes thetrivial line bundle with nontrivial Z -action. This implies 2): e Γ ( ν ⊗ R i R ) = w u ∗ ( ν ) (the totalStiefel-Whitney class homogenized by powers of u ). Finally, 3) is the content of Thom’s theorem[38] saying that for i : Z ֒ → X : i ! ( w ∗ ( ν )) = Sq[ Z ] holds. (cid:3) It would be nice to have a similar description of the coefficients for the case of circle spaces,however there are no nontrivial stable rational cohomology operations, so this result has no directgeneralization.3.3.2. Poincar´e duality. In this section we give some sufficient conditions for a Γ-space X to be ahalving space. Definition 3.22. We say that a Γ-space X is almost a halving space , if X has nonzero cohomologyonly in degrees 2 Di and X is equivariantly formal with a Leray-Hirsch section σ : H ∗ ( X ) → H ∗ Γ ( X )satisfying a weaker form of the degree condition:(DC – ) for all x ∈ H Di ( X ), rσ ( x ) is a polynomial of degree at most i where r : H ∗ Γ ( X ) → H ∗ ( X Γ )[ u ] is the restriction map, u ∈ H D Γ .To put it simply, (DC – ) allows the u -degree of rσ ( x ) to be smaller than i . The following lemmacan be found (implicitly) in van Hamel [40] for the case of conjugation spaces and its proof is thesame. For the analogue of van Hamel’s theorem in the case of (Γ , R ) = (U(1) , Q ) we also have toassume Poincar´e duality/orientability: Lemma 3.23 (Injectivity lemma) . Let (Γ , R ) = (U(1) , Q ) , H ∗ Γ ∼ = Q [ u ] , u ∈ H D Γ , D = 2 . Let X be a smooth Γ -manifold which is almost a halving space with σ . If X is compact, orientableand dim X ≥ X Γ , then X satisfies the degree condition (DC) . In particular, X is a halvingspace with the same σ , and κ defined by (1) is an isomorphism.Proof. We only give a brief sketch of the main idea found in [40], for further details see [25].Let x ∈ H Dk ( X ). By Poincar´e duality there exists y ∈ H D ( n − k ) ( X ) satisfying xy = 0, heredim X = 2 Dn . Then σ ( x ) σ ( y ) = 0; degree considerations from (DC – ) show that r ( σ ( x ) σ ( y )) = κ ( x ) κ ( y ) u n . The localization theorem (restriction to the fixed-point set r : H ∗ Γ ( X ) → H ∗ ( X Γ )[ u ]is an isomorphism after inverting u ) implies that restriction to the fixed point set r : H ∗ Γ ( X ) → H ∗ ( X Γ )[ u ] is injective, therefore κ ( x ) = 0. (cid:3) Remark 3.24. i) More generally, the lemma holds for Q -Poincar´e duality spaces [2, Definition 5.1.1.] ifone replaces dim( X ) with formal dimension fd( X ). ( X is a Q -Poincar´e duality space if H top ( X ; Q ) ∼ = Q and the pairing H k ( X ) ⊗ H top − k ( X ) → H top ( X )is perfect; fd( X ) := top.) Q -Poincar´e duality is satisfied by a larger class of spaces, whichneed not be orientable nor compact. Note that the formal dimension of a manifold X canbe smaller than the dimension of X as a manifold. For example, R P n is a Q -Poincar´eduality space with formal dimension 0. More generally, all real partial flag manifolds ALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVE GEOMETRY 17 Fl R D are Q -Poincar´e duality spaces [26]. This allows us to extend the class of circle spaceexamples to the case of nonorientable Grassmannians, Gr k ( R n ) for n odd. For the sake ofconciseness we didn’t add the extra conditions to the lemma.ii) For (Γ , R ) = ( Z , F ), the same lemma holds, without having to assume orientability.Every manifold satisfies F -Poincar´e duality – indeed, van Hamel’s original proof [40] doesnot assume orientability.iii) The lemma can also be generalized to Γ = Sp(1), if one makes the additional assumptionthat the restriction to the fixed point set is an isomorphism after localization.4. The generalized Borel-Haefliger theorem Using the notions of good invariant cycles and excess multiplicity (Definitions 3.13 and 3.11) wecan state the generalized Borel-Haefliger theorem: Theorem 4.1 (Generalized Borel-Haefliger theorem) . Let (Γ , R ) be ( Z , F ) or (U(1) , Q ) and H ∗ ( B Γ; R ) ∼ = R [ u ] , u ∈ H D Γ . Let X be a smooth compact Γ -manifold, orientable if R = Q .Assume that H ∗ ( X ) is nonzero only in degrees divisible by D and has a basis of halving cycles:good Γ -invariant cycles Z i ⊆ X of codimension type (2 Dk i , Dk i ) . Then X is a halving space withcohomology frame (8) σ [ Z i ] = [ Z i ] Γ , κ [ Z i ] = µ i [ Z Γ i ] where µ i = 0 is the excess multiplicity of Z i ⊆ X .Proof. Since H ∗ ( X ) is generated by halving cycles, it is Γ-equivariantly formal. By the excessweight lemma (Lemma 3.15), rσ [ Z i ] = κ [ Z i ] u k i + deg Since conjugation spaces are discussed in [22], we concentrate on the case of circle spaces. ✲ Figure 1. The double of a Young diagram5.1. R -spaces. Our main class of examples of circle spaces are homogeneous spaces, in particularthey are all R -spaces , more commonly known as (generalized) real flag manifolds. We reserve theterminology real flag manifolds to Fl D ( R N ), which are also R -spaces.Out of the R -spaces, we have four main classes of examples of circle spaces: spheres S n , evenreal flag manifolds, quaternionic and octonionic flag manifolds. The simplest example consistsof spheres S n ; this already illustrates the idea of the proof. Let U(1) act on R n +1 as thelinear orthogonal representation, which splits into n weight one and 2 n + 1 trivial representations; S n ⊆ R n +1 is U(1)-invariant. Proposition 5.1. With this Γ -action S n is a circle space with fixed point set S n .Proof. The fixed point set of S n is S n ∩ R n +1 = S n . Since H ∗ ( S n ) = Z [ x ] / ( x ) is generated bya Γ-invariant halving cycle, namely the class of a fixed point, by the generalized Borel-Haefligertheorem, S n is a circle space. (cid:3) Real flag manifolds. Our first nontrivial class of examples are the even real flag manifolds ,i.e. flag manifolds Fl D ( R n ), where all dimensions are even. More details about this examplecan be found in [26], we give a brief summary. For D = ( d , . . . , d m ), we denote by Fl D ( R n ) themanifold of flags F • = ( F ⊆ . . . F m ), whose dimensions are dim F i = s i , where s i = P ij =1 d j .The identification of R n ↔ C n as real Γ-representations induces an action on Fl E ( R n ), E =( e , . . . , e r ). The flag manifold Fl E ( R N ) has a Schubert cell decomposition(9) Ω I ( A • ) = { F • ∈ Fl E ( R N ) : dim F i ∩ A k = r I ( i, k ) } , where I ∈ OSP( E ) = S N / ( S e × . . . × S e r )is an ordered set partition , and r I ( i, k ) = { l ∈ I ∪ . . . ∪ I i : l ≤ k } . If 2 D = (2 d , d , . . . , d r )and I ∈ OSP( D ), then the doubled ordered set partition DI ∈ OSP(2 D ) is obtained by replacingeach i ∈ I j by (2 i − , i ) ∈ DI j . A double Schubert variety σ R DI ⊆ Fl R D is a Schubert varietycorresponding to DI ∈ OSP(2 D ). In the case of the Grassmannian D = ( k, l ), DI ∈ (cid:0) k + l )2 k (cid:1) corresponds to the Young diagram obtained by subdividing each square into 2 × I ∈ (cid:0) k + lk (cid:1) , see Figure 1.The double Schubert varieties σ R DI are cycles and their classes form a basis of H ∗ (Fl R D ; Q ), see[26]. Using that they are circle spaces we can deduce their structure constants. Theorem 5.2. Let Γ act on Fl D ( R n ) obtained by identifying C n with R n . With this action itis a circle space, with fixed point set Fl D ( C n ) . Furthermore κ [ σ R DI ] = 2 | I | [ σ C I ] , ALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVE GEOMETRY 19 where [ σ C I ] ∈ H | I | (Fl D ( C N )) .Proof. Let F • be a complete flag in R n , such that F i are Γ-invariant, and let F C • denote thecorresponding complex flag in C n . By the generalized Borel-Haefliger theorem, it is enough toshow that the Schubert varieties σ R DI ( F • ) are halving cycles, have fixed point set σ C I ( F C • ) and that[ σ R DI ] form a basis. For further details see [26].We sketch why all normal weights of Fl D ( C n ) ֒ → Fl D ( R n ) are 2. Since all tangent spacesare sums of Hom-spaces, this claim can be reduced to linear algebra, namely computing weightsof the U(1)-representation Hom R ( C , C ). This representation splits into the sum of two U(1)-representations: a 2-dimensional weight 0 representation and a weight 2 representation. Theweight 0 representation corresponds to Hom C ( C , C ) (in the geometric picture this correponds tothe tangent space of the complex part) and the weight 2 representation Hom C ( C , C ) (the normalspace of the complex part). (cid:3) The corollaries below follow from Theorem 5.2, multiplicativity of κ and that κp j ( S R i ) = 2 j c j ( S C i ) . Corollary 5.3 (Littlewood-Richardson coefficients) . [ σ R DI ] · [ σ R DJ ] = X K c KIJ [ σ R DK ] where c KIJ are the complex Littlewood-Richardson coefficients [ σ C I ] · [ σ C J ] = X K c KIJ [ σ C K ] . Corollary 5.4 (Giambelli formula type description) . [ σ R DI ] = q ( p ∗ ( S R i )) ⇐⇒ [ σ C I ] = q ( c ∗ ( S C i )) , i. e. the same polynomial describes the double real Schubert classes and complex Schubert classesin terms of Pontryagin and Chern classes. Corollary 5.5. The cohomology ring of an even flag manifold can be described as follows: H ∗ (Fl R D ) = Q [ p ∗ ( S R i )] / R ( p ∗ ( S R i )) ⇐⇒ H ∗ (Fl C D ) = Q [ c ∗ ( S C i )] / R ( c ∗ ( S C i )) , where R ( x i ∗ ) denotes an ideal in the variables x ij , that is the same polynomial relations hold inthe two cohomology rings in terms of Pontryagin and Chern classes of the respective tautologicalbundles. Corollary 5.6 (Equivariant Giambelli formula) . For the case of Grassmannians Fl R D = Gr k ( R k + l ) ) , D = ( k, l ) , the (doubled) Giambelli formula holds, even Γ -equivariantly [ σ Dλ ] Γ = det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ σ Dλ ] Γ [ σ D ( λ +1) ] Γ . . . [ σ D ( λ + k ) ] Γ [ σ D ( λ − ] Γ [ σ Dλ ] Γ . . . [ σ D ( λ + k − ] Γ ... ... . . . ... [ σ D ( λ k − k ) ] Γ . . . . . . [ σ Dλ k ] Γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) where Dλ denotes the double of the partition λ ⊆ k × l and Da = (2 a, a ) for a ∈ Z .Proof. Nonequivariantly, this follows from the complex Giambelli formula and from X beinga circle space with κ [ σ R Dλ ] = 2 | λ | [ σ C λ ]. Equivariantly, this follows from σ being multiplicative(Corollary 2.5) and from the generalized Borel-Haefliger theorem, σ [ σ Dλ ] = [ σ Dλ ] Γ . (cid:3) The Grassmannians Gr K ( R N ) are also circle spaces, except when K is odd, N is even. If K and N is even, this is contained in Theorem 5.2. The remaining cases: K odd N even and K even N odd are both nonorientable, in these cases Remark 3.24, i) can be used. This gives examplesof nonorientable circle spaces. For further details, see [26].5.1.2. Galois type actions. For the next examples of halving spaces, we define U(1)-actions on H P n with fixed point set C P n and Sp(1)-actions on O P with fixed point set H P . As a firststep, let us recall the corresponding actions on the real normed division algebras. There are fourreal normed division algebras F i : R ⊆ C ⊆ H ⊆ O . In each case, there is a subgroup Γ ∼ = O( F i − ) of the R -algebra automorphisms Aut( F i ), whosefixed point set F Γ i is the previous division algebra F i − , i = 2 , , Z in the case of C , withfixed point set R . This action extends to C n and also to the complex flag manifolds Fl D ( C N ) withfixed point-set Fl D ( R n ). This is the action classically studied by Borel and Haefliger.Next, let H act on itself by inner automorphisms. Then Γ = U(1) ⊆ C = h , i i ≤ H acts on H , with fixed-point set C ≤ H . This action extends to H n therefore to H P n , and even to anyquaternionic flag manifold Fl D ( H n ), with fixed point set Fl D ( C n ).The automorphisms of the normed algebra O fixing H is isomorphic to Γ := Sp(1), and in fact O Γ = H . This induces an action on the octonionic flag manifolds, which can be seen on theirdifferent models – we describe these actions in Section 5.1.4. For additional details, see e.g. [25,Propositions B.4.1, B.4.2].5.1.3. Quaternionic flag manifolds. The flag manifold Fl D ( H n ) has a Schubert cell decomposition σ H I where I ∈ OSP( D ). Theorem 5.7. With the Γ = U(1) -action defined by inner automorphisms (see Section 5.1.2), Fl D ( H n ) is a circle space, with fixed point set Fl D ( C n ) . Furthermore κ [ σ H I ] = 2 | I | [ σ C I ] , where [ σ C I ] ∈ H | I | (Fl D ( C n )) .Proof. If F C • is a complex flag, then F • := F C • ⊗ C H is a quaternionic flag, which is Γ-invariant.Similarly to the case of real even flag manifolds, by the generalized Borel-Haefliger theorem it canbe shown that the Schubert varieties σ H I ( F • ) are halving cycles with respect to an appropriatecomplete flag F • ∈ Fl( H n ), with fixed points σ C I ( F C • ) and [ σ H I ] form a basis of rational cohomology.The normal weights are all 2. (cid:3) ALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVE GEOMETRY 21 Corollary 5.8 (Littlewood-Richardson coefficients) . [ σ H I ] · [ σ H J ] = X K c KIJ [ σ H K ] where c KIJ are the same Littlewood-Richardson coefficients as in [ σ C I ] · [ σ C J ] = X K c KIJ [ σ C K ] . Proof. Exactly the same as Corollary 5.3. (cid:3) Corollary 5.9 (Giambelli formula type description) . [ σ H I ] = q ( p ∗ ( S H i )) ⇐⇒ [ σ C I ] = q ( c ∗ ( S C i )) , where p ∗ denotes quaternionic Pontryagin classes. In words, the same polynomial describes thequaternionic and complex Schubert varieties in terms of characteristic classes. In the case of Grassmannians, this was already noticed by Pragacz and Ratajski [29]; as theyremark, the proof of the Pieri formula in [18] can be replicated in the quaternionic case implyingthe same description of the cohomology rings (complex and quaternionic) with degrees doubled.5.1.4. Octonionic flag manifolds. In this section we give some examples for quaternionic halvingspaces, i.e. halving spaces for Sp(1)-actions: the octonionic flag manifolds. We will also show thatthey are circle spaces by restricting to U(1) ≤ Sp(1).By octonionic flag manifolds we mean the following three examples: O P , O P , Fl( O )(= Fl( O )).Nonassociativity of octonions leads to the fact that there are no octonionic analogues of higherdimensional flag manifolds. We refer to [4], [16], [10], [24] for further details about octonionic flagmanifolds.Since O P ∼ = S which is easily seen to be both a circle space and a quaternionic halvingspace, we start with O P . In the case of O P a purely topological proof can be given using Hopffibrations. Proposition 5.10. The Hopf fibrations are Γ -equivariant principal G -bundles where the Γ -actionis induced by the inner automorphisms defined in Section 5.1.2. Furthermore, the Γ -fixed pointset of each Hopf fibration is the previous one: • π : S → S → S is Γ = Z -equivariant with fixed point set π : S → S → S • π : S → S → S is Γ = U(1) -equivariant with fixed point set π : S → S → S • π : S → S → S is Γ = Sp(1) -equivariant with fixed point set π : S → S → S Proof. The definition of these bundles involves the division algebra structure of F , so they arenaturally Aut( F )-equivariant. (cid:3) Corollary 5.11. O P is a halving space for both the Sp(1) -action and the U(1) -action, with fixedpoint set H P . Proof sketch. The projective planes FP can be obtained by gluing along the Hopf fibrations: R P = D a π S , C P = D a π S , H P = D a π S , O P = D a π S and the Γ-action descends to the projective planes, with fixed point set the the previous one.From the naturally occurring cell decompositions we get that each space is a halving space withthe fixed point set the previous one, in particular we get that O P is a quaternionic halving/circlespace. Alternatively, one can adapt the proof of the next section. (cid:3) O P has a description by (restricted) homogeneous coordinates as follows. The points of O P are triples ( a, b, c ) ∈ O , such that at least one of them is real, modulo the relation that two suchelements are equal if they differ by left multiplication by an element of O . The lines of O P aredefined similarly (denoted ( O P ) ∗ ), but now the equivalence relation is right multiplication. Apoint x = ( x , x , x ) ∈ O P is incident to the line l = ( l , l , l ) ∈ ( O P ) ∗ denoted x ∈ l , if x l + x l + x l = 0 for representatives chosen such that at least two of the sets { x i , l i } containa real number. The flag manifold Fl( O ) can be defined as the set of incident point-lines:Fl( O ) := { ( x, l ) : x ∈ l } ⊆ O P × ( O P ) ∗ . Remark 5.12. The description in terms of coordinates is in fact isomorphic to the usual modelof O P by the exceptional Jordan algebra h ( O ), see [15], [17], [24]. These identifications are dueto [3], [1], see also [32, Theorem 7.2], [8].The automorphisms of the normed algebra O fixing H (for further details, see [25, PropositionB.4.1.]) induces a coordinate-wise action on Y = O P with fixed point set Y Γ = H P . Sincethe action is compatible with the incidence relation, it also induces an action on X = Fl( O ) withfixed point set X Γ = Fl( H ). Theorem 5.13. With the Γ = Sp(1) -action defined above, Fl( O ) is a quaternionic halving space,with fixed point set Fl( H ) . Furthermore κ [ σ O w ] = [ σ H w ] where w ∈ S and [ σ H w ] ∈ H | w | (Fl( H )) .Proof. For d • ∈ Fl( O ) Γ = Fl( H ), the flag manifold Fl( O ) has a decomposition into Γ-invariantSchubert 8 i -cells Ω O w ( d • ), defined by incidence relations, whose fixed point sets are Ω H w ( d • ). Inparticular, the closures of the Schubert cells are Γ-invariant halving cycles σ O w ( d • ) by a dimensioncount. To see that the Sp(1)-multiplicity of the normal space ν ( H ⊆ O ) equals 1, note that thenormal Sp(1)-representation is its defining representation, since it acts freely and transitively on S (see [25, Proposition B.4.1.]). The conditions of the generalized Borel-Haefliger theorem forΓ = Sp(1) have to be checked according to Remark 4.2 ii). First, the Schubert cycles are Sp(1)-invariant halving cycles and their fixed point sets are cycles (this is straightforward). Second,Fl( O ) satisfies the localization theorem for Sp(1) by [39, Theorem III.3.8.], see also [24, Theorem1.3]. (cid:3) Remark 5.14. ALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVE GEOMETRY 23 i) By the general theory, Fl( O ) has a Bruhat cell decomposition as N -orbits, see [9], [24].This agrees with the Schubert cell decomposition—this can be verified through the Jordanalgebra model of O P .ii) These examples are also examples of circle spaces. Indeed, one can restrict the action ofΓ = Sp(1) to a ∆ = U(1)-action, such that O ∆ = H , and then the rest of the proofs arethe same.5.2. Quaternionic toric varieties. In the case of Γ = Z , smooth toric manifolds are conjuga-tion spaces [22, Example 8.7]. This example has a generalization in the context of circle spaces;quaternionic toric varieties introduced by Scott [33], which come naturally equipped with anSO(3)-action. The cohomology of a nonsingular quaternionic toric space is generated by geomet-ric cycles of degrees 4 i [34, p. 43] and is degree-doubling isomorphic to its complex counterpart[34, Theorems 3.3.2. and 5.5.1], and therefore are circle spaces.5.3. Constructions. We can construct new halving spaces out of old ones. Most ideas of [22]about conjugation spaces can be adapted, however there are some new features.Given a U(1)-action ρ : U(1) → Homeo( X ) on a space X we can rescale the action by composingit with z z k . If the action is already in this form, then we can also downscale it: For examplethe U(1)-action on the even real flag manifolds and the quaternionic flag manifolds have theproperty that − ∈ U(1) acts trivially, by downscaling we can get a new action with all normalweights equal to 1. It is elementary to check that rescaling a circle space provides a circle space.With the same proof as in [22, Proposition 4.5] one can show that Proposition 5.15. Suppose that X and Y are halving spaces and H q ( X ; R ) has finite rank forall q . Then X × Y is also a halving space. Combining product with rescaling we can construct circle spaces with prescribed normal weights.For example rescale the circle space S of Proposition 5.1 with given integers, and take the productof these. With the same proof as in [22, Proposition 4.6] one can show that Proposition 5.16. Let ( X i , f ij ) be a direct system of halving spaces which are T and f ij are Γ -equivariant inclusions. Then X = lim −→ i X i is a halving space with cohomology frame (lim ←− κ i , lim ←− σ i ) . Conjugation manifolds of the same dimension are locally isomorphic at a fixed point. This isno longer true for circle spaces, they are locally isomorphic at a fixed point only if they have thesame normal weights. So for connected sums, [22, Proposition 4.7] has to be modified: Proposition 5.17. Suppose that X and Y are circle manifolds having the same normal weights.Then the connected sum X Y is also a circle manifold. This construction provides examples of circle manifolds which are not homogeneous manifolds. Remark 5.18. No complex projective variety X can be a circle space; H ( X ) contains the non-zero hyperplane section, which violates the condition of having nonzero cohomology groups onlyin degrees 4 i . Remark 5.19. If the homogeneous space X = G/H is a circle space, then rk( G ) = rk( H ).Indeed, it is classical (e.g. [20]), that the Euler-characteristic of a homogeneous space is zeroif rk( G ) > rk( H ), which means that X has nonzero cohomology in some odd degree, againviolating the condition on even degrees. Indeed, all of the homogeneous examples above satisfythis condition. 6. Applications for enumerative problems One of the main applications of the cohomology ring structure of real flag manifolds concernsenumerative geometry, namely Schubert calculus. Whereas in the complex case enumerativeproblems are completely solved by the cohomological product of the corresponding cycles, in thereal case the product only gives a lower bound—the number of solutions depends on the givenconfiguration of the enumerative problem.6.1. Real Schubert problems. There is no general theorem (as of yet), which gives all possiblesolutions to a real Schubert problem, although certain special cases have been described, see e.g.[36], [23], [12].The interpretation of the cohomology ring given in Corollary 5.3 is a general result providinglower bounds to an infinite family of real Schubert problems, what we can call double Schubertproblems : which involves only the double Schubert varieties σ R DI defined in Section 5.1.1. Thedetails and several examples, can be found in [26, Section 7]. Let us demonstrate this techniqueon a simple example: Problem 2. How many W ∈ Gr ( R ) intersect four generic U i ∈ Gr ( R ) in 4 dimensions? This problem can be rewritten as computing the intersection of four Schubert varieties T i =1 σ λ ( U i ),where λ = (4 ). Each point of intersection inherits a sign from the orientations, therefore thecohomological product [ σ R λ ] ∈ H ∗ (Gr ( R )) gives a lower bound to the number of solutions.By Corollary 5.3, this cohomological product can be computed as [ σ C µ ] ∈ H ∗ (Gr ( C )) where µ = (2 ), and a simple verification shows that this product equals 6. In [12], we showed viaelementary techniques that this lower bound is sharp, furthermore that the number of possiblesolutions to this problem is 6 , , , 70. However, the cohomological lower bound works moregenerally for any double Schubert problem, which is a significantly larger family than the onedescribed in [12].6.2. Quaternionic Schubert problems. We can also solve quaternionic enumerative problems: Proposition 6.1. The number of solutions of a generic quaternionic Schubert problem is thesame as the corresponding complex Schubert problem.Proof. If the Schubert varieties σ j ( F H • ) are transverse (generic), then since the tangent spaces arecanonically oriented (the tangent bundle has a complex structure), the cohomology computationgives the exact number of solutions. We can conclude by the Littlewood-Richardson coefficientsof Corollary 5.8. (cid:3) For instance: ALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVE GEOMETRY 25 Problem 3. How many lines intersect four given lines in H P ? By Proposition 6.1, the answer is the same as in the complex case, which is 2. Notice thatby forgetting the Sp(1)-structure and retaining only the real linear structure, we obtain the realproblem discussed above, which has at least 6 solutions. These solutions are Sp(1)-invariant, sothey should be quaternionic. The resolution of this seeming contradiction lies in the subtle notionof genericity: the real problem obtained by forgetting the quaternionic structure is usually notreal generic , so these cases are not covered by Proposition 6.1.On this example, nongenericity can be seen explicitly as follows. To a given configuration U , . . . , U ∈ Gr ( R ), one can associate a real linear map ϕ : U → U with the property thatthe problem is real generic iff all eigenvalues of this map are distinct, and different from 0 and 1, see[12, Remark 4.14]. If the U i are complex (U(1)-invariant), the corresponding map ϕ : U → U iscomplex linear. In case the eigenvalues of ϕ as a complex map are distinct and contain no complexconjugate pairs, then the eigenvalues of ϕ as a real map are also distinct, and the problem is alsoreal generic. However, if the U i are quaternionic (Sp(1)-invariant) then ϕ is quaternionic linear.Quaternionic linear maps can be written in the formGL n ( H ) = (cid:26)(cid:18) A B − B A (cid:19) ∈ GL n ( C ) (cid:27) and their eigenvalues as a real linear map come in fours ( λ, λ, λ, λ ). Since these eigenvalues arenot distinct, the problem is not real generic, and it is also not hard to see that it has infinitelymany solutions (since these are in bijection with invariant subspaces of ϕ , [12, Corollary 2.4]).7. Further examples of halving cycles As we explained in the previous chapter, every halving cycle in an even real flag manifold canlead to a lower bound for a corresponding real enumerative question. In the case of conjugationspaces, there is a large class of examples of halving cycles, namely the complexified subvarieties.However, there is no trivial analogue of the complexification operation for U(1)-actions, so it isnot easy to find non-trivial examples of halving cycles in a circle space. For even flag manifoldswe have the even Schubert varieties and more generally Richardson varieties.Below we discuss a less obvious class of examples, obtained by using quivers. We also indicatethe enumerative consequences.7.1. Universal degeneracy loci. Let X be an even real partial flag manifold with the Γ-actionof Section 5.1.1 with fixed point set the complex partial flag manifold X Γ . Let γ ∈ N n be adimension vector, and E = ( E , E , . . . , E n ) be an n -tuple of real Γ-equivariant vector bundles ofrank 2 γ i over X such that E i | X Γ has the structure of a complex vector bundle on which Γ acts byscalar multiplication. We can construct such bundles using the various tautological subbundlesover the flag manifold. Notice that Γ acts on the bundles Hom R ( E i , E i +1 ) via conjugation withfixed point set Hom C ( E i | X Γ , E i +1 | X Γ ). Therefore the total space of the type A n real quiver bundle with dimension vector 2 γ = (2 γ , . . . , γ n ) Q R ( E ) := n − M i =1 Hom R ( E i , E i +1 )is also a circle manifold and its fixed point set is the total space of the type A n complex quiverbundle with dimension vector γQ C ( E ) := n − M i =1 Hom C ( E i | X Γ , E i +1 | X Γ ) . Let Z m be the orbit closure in the quiver representation space Rep R γ corresponding to themodule 2 m . Then we can associate to Z R m a subset Z R m ( E ) of Q R ( E ), the union of ’ Z R m -points’in all fibers. The proof of [25, Theorem 5.3.6.] implies that Proposition 7.1. Z R m ( E ) is a halving cycle with fixed point set Z C m ( E | X Γ ) . The proof is based on the orientability of the Reineke resolution of Z R m ( E ). These quiverloci are nontrivial examples of halving cycles in Q R ( E ), however they are not directly related toenumerative problems.7.2. A degeneracy locus in the Grassmannian. If we want to find new halving cycles in aneven real partial flag manifold X (and not in a bundle over X ), then we need a section of Z R m ( E )with nice properties: Observation 7.2. Let f : X → Y be a Γ -equivariant map of smooth circle manifolds, and let Z ⊂ Y be a halving cycle. If f is transversal to Z and f | X Γ : X Γ → Y Γ is transversal to Z Γ , then f − ( Z ) is also a halving cycle. We say that f is transversal to a topological subvariety Z , if it is transversal to a fat nonsingularsubset U of Z , and that f − ( U ) is a fat nonsingular subset of f − ( Z ).Suppose now that σ : X → Q R ( E ) is a Γ-equivariant section such that σ | X Γ : X Γ → Q C ( E ) isholomorphic and transversal to Z C m ( E ). If σ is transversal to Z R m ( E ), then by Observation 7.2,the degeneracy locus σ − ( Z R m ( E )) ⊂ X is a halving cycle.Such Γ-equivariant maps are not easy to find. The task of finding holomorphic sections of Q C ( E )is already quite involved, so we restrict ourselves to the example of X = Gr ( R ) with fixed pointset X Γ = Gr ( C ). Before defining these sections, let us introduce some natural subvarieties of theGrassmannian. Fix a linear map α : C → C and consider the following subvariety of Gr ( C ):Σ α = { V : dim V ∩ α ( V ) ≥ } . By forgetting the complex structure, we obtain a real linear map α R : R → R , and can similarlydefine a subvariety of Gr ( R ): Σ R α = { V : dim V ∩ α R ( V ) ≥ } . For appropriate maps α , these subvarieties are halving cycles: ALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVE GEOMETRY 27 Proposition 7.3. If α : C → C is diagonalizable with 4 different non-real eigenvalues, contain-ing no complex conjugate pairs, then Σ R α is a halving cycle, with fixed point set Σ α . These subvarieties arise as degeneracy loci as follows. Let E be the tautological subbundleand E be the tautological quotient bundle over X . Then E | X Γ is the tautological subbundleand E | X Γ is the tautological quotient bundle over Gr ( C ). The map α induces a holomorphicsection σ C α : X Γ → Hom( E | X Γ , E | X Γ ) , where σ C α ( V, v ) = [ α ( v )] ∈ C /V for V ∈ Gr ( C ) and v ∈ V . Similarly, α R also induces aΓ-equivariant section σ α : X → Hom( E , E ) , such that σ α | X Γ = σ C α . Then Σ R α = ¯Σ ( σ α ) ⊆ Gr ( R ) , where ¯Σ ( σ α ) denotes the locus where the corank of σ α is at least 2. Notice that ¯Σ is of the form Z m for an A quiver representation. The section σ α is indeed transversal to ¯Σ ( E , E ), howeverthe proof is quite technical, so instead we sketch a direct proof of the fact that Σ R α is a halvingcycle. Proof (sketch). The key is to show that Σ R α is a cycle. For this we give a stratification. Foran arbitrary subspace W ≤ R , let us introduce the notation W ′ := W ∩ α ( W ). We partition Y = ¯Σ ( σ α ) according to the dimension of V ′ . We have Y = Y ` Y ` Y , where Y i = { V ∈ Gr ( R ) : dim( V ′ ) = i } . In order to obtain a stratification, we further partition Y ; by denoting V ′′ = V ′ ∩ α ( V ′ ), let Z i := { V ∈ Y : dim( V ′′ ) = i } for i = 0 , , Y . The stratification we consider is given byΣ R α = Z a Z a Z a Y a Y . For a generic 2-dimensional subspace W we have W ′ = 0, so the assignment W 7→ h W, α ( W ) i identifies Z with an open submanifold of Gr ( R ): this is the open—and orientable—stratum of Y . If the eigenvalues of α satisfy the conditions of Proposition 7.3, then the codimensions of Z , Y and Y are greater than 2, so we don’t need to study them, and it is enough to concentrate on Z .In the remaining part, we show that Z is a one-codimensional non-orientable stratum, whichimplies that Σ R α is a cycle. For V ∈ Z we have V ′′ = h v i for some v ∈ V ; by definition α − ( v ) ∈ V ′ .Similarly, α − ( v ) ∈ V . Since V is in Z , these 3 vectors have to be independent: V has a basis ofthe form { v, α − ( v ) , α − ( v ) , z } for some z ∈ V . The assignment ( v, z ) V identifies Z with an RP bundle over an open submanifold of RP : Let U ⊂ RP be the open subset over which thebundles γ, α − ( γ ) , α − ( γ ) are independent, then we have the quotient vector bundle ξ = R / (cid:0) γ ⊕ α − ( γ ) ⊕ α − ( γ ) (cid:1) over U and Z ∼ = RP ( ξ ) . The dimension of Z is 11, one less than the dimension of Z (= dim Gr ( R )). Notice that thecomplement of U in RP has codimension higher than one, so the orientability of Z is equivalentof the vanishing of the first Stiefel-Whitney class of the virtual vector bundle˜ ξ := R ⊖ γ. An elementary calculation shows that w ( ˜ ξ ) is not zero, so we established that ¯Σ ( σ α ) is a cycle. (cid:3) An enumerative problem. Consider the following question: Given four generic linear maps α i : C → C what is the number of 2-dimensional subspaces V such that dim( V ∩ α i ( V )) = 1 for i = 1 , . . . , C α ⊂ Gr ( C ). Since this is a Thom-Porteous locus [28], a short calculation using the Giambelli-Thom-Porteous formula gives that [Σ C α ⊂ Gr ( C )] = 2 c , where c is the first Chern class ofthe tautological bundle. Then we need to intersect four general translates, and the number ofintersection points is Z Gr ( C ) (2 c ) = 32 . Now Corollary 5.3 gives the following: Proposition 7.4. Given four generic linear map α i : R → R the number of 4-dimensionalsubspaces V such that dim( V ∩ α i ( V )) = 2 for i = 1 , . . . , is at least 32. Equivariant fundamental classes. Proposition 7.1 is a result for any choice of bundles E i , so it can be translated to an equivariant statement. Note that the equivariant approach is inthe spirit of [6] for which one of the main motivation was to establish the relationship betweencomplex and real Thom polynomials mod 2. We will follow [25].The key technical point is the concept of a halving group : a group G , on which Γ acts byautomorphisms, such that the classifying space BG is a halving space in a well-defined sense. Itis showed in [25] that if G is a halving group and X is a halving space, then B G X is a halvingspace with fixed point set B G Γ X Γ . Furthermore, if Z ⊆ X is a halving cycle which is G -invariant,then κ : H ∗ G ( X ) → H ∗ G Γ ( X Γ ) maps [ Z ] G to λ i [ Z Γ ] G Γ . For instance, this result has the followingapplication [25, Theorem 5.3.6.]: Theorem 7.5. Let Q be the equioriented A n quiver. Let γ ∈ N n be a dimension vector and Z m ⊆ Rep C γ be the closure of the GL C γ -orbit corresponding to the module m = P µ ij l ij , where l ij , ≤ i ≤ j ≤ n are the indecomposable modules corresponding to the positive roots. Then [ Z C m ⊆ Rep C γ ] GL C γ = q ( c ∗ ) ⇐⇒ [ Z R m ⊆ Rep R γ ] GL R γ = q ( p ∗ ) in H ∗ ( BGL C γ ; Q ) and H ∗ ( BGL R γ ; Q ) respectively where γ = (2 γ , . . . , γ r ) and m = P µ ij l ij . ALVING SPACES AND LOWER BOUNDS IN REAL ENUMERATIVE GEOMETRY 29 The real orbit closures Z R m are defined analogously. It is also possible to define the subvarieties Z C m and Z R m using rank conditions, see e.g. [11, Lemma 4.1].The proof is essentially the same as of Proposition 7.1. A similar description can be given formatrix Schubert varieties, [25, Theorem 5.4.3.]. Remark 7.6. 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