Abstract
We study the equivariant cobordism rings for the action of a torus T on smooth varieties over an algebraically closed field of characteristic zero. We prove a theorem describing the rational T-equivariant cobordism rings of smooth projective G-spherical varieties with the action of a maximal torus T of G. As an application, we obtain explicit presentations for the rational equivariant cobordism rings of smooth projective horospherical varieties of Picard number one.
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1 Introduction
Let k be a field of characteristic zero and G a linear algebraic group over k. Algebraic equivariant cobordism groups were originally introduced for smooth schemes by Deshpande [6]. This theory was developed independently for all k-schemes by Krishna as well as Heller and Malagón-López in [12, 20] and is based on the similar construction of equivariant Chow groups presented by Totaro [28] and Edidin-Graham [7]. Many properties of equivariant cobordism were proved in [12, 20] building on the theory of non-equivariant cobordism developed by Levine and Morel [22]. In [19], the theory of equivariant cobordism was presented with a special focus in the case where the underlying group is a torus.
At present, there are already some computations known for this cohomology theory. The equivariant cobordism was computed for toric varieties and flag bundles in [18, 21]. Furthermore, the localisation formula for rational equivariant Chow groups was first proved by Edidin and Graham in [7]. We will use the localisation result by Brion (cf. [4]) which was adpated from work of Goresky, Kottwitz and MacPherson (cf. [10]) in equivariant cohomology. The latter was then extended by Krishna [19] to rational equivariant cobordism which was applied in order to describe the rational equivariant cobordism rings of flag varieties and symmetric varieties in [17]. The aim of this paper is to study further classes of examples for which the rational equivariant cobordism rings can be computed. As a consequence, one obtains a presentation of the rational ordinary cobordism rings using [19, Theorem 3.4]. Now, we describe some of our main results.
In this paper, G denotes a connected reductive group over an algebraically closed field k. Additionally, all schemes are assumed to be quasi-projective k-schemes and all group actions to be linear where the latter assumption is recalled in Notations 2.1. We remark here that our main theorems do not require the assumption that the group action is linear because this is always satisfied for normal schemes (cf. [27]). Further, let T be a maximal torus of G and for a positive root α of G relative to T we denote by Lα the one-dimensional representation of T on which T acts via the weight − α. Brion obtained the first presentation of the rational T-equivariant Chow rings of smooth projective spherical G-varieties in [4, Theorem 7.3] using the equivariant intersection theory of Edidin and Graham [7] which was more recently generalised to equivariant K-theory by Banerjee and Can in [1, Theorem 1.1]. After a short recollection of some of the main known results and notions including the definitions of good pairs (cf. Section 2.3), equivariant cobordism (cf. Definition 2.6, Remark 2.7), the T-equivariant line bundle Lα over Spec k which was mentioned above and the operator ρm/n on the rational T-equivariant cobordism ring of the point \({\Omega }_{T}^{\ast }(k)_{\mathbb {Q}}\) (cf. Definition 2.4), we describe the rational T-equivariant cobordism rings of smooth projective spherical varieties (cf. Theorem 3.4) building mainly on Brion’s methods. Localisation was already used in [19, Theorem 7.8] in order to describe the rational T-equivariant cobordism rings for smooth filtrable (e.g. smooth projective spherical) schemes with finitely many T-fixed points and only finitely many T-stable curves. Our main theorem generalises this result to smooth projective spherical G-varieties with possibly infinitely many T-stable curves. In a subsequent work, these methods will lead to a generalisation of Brion’s result [4, Theorem 7.3] in equivariant Chow groups for a refined coefficient ring \(\mathbb {Z}\subseteq R\subseteq \mathbb {Q}\) which produces new computations even for (T-equivariant) Chow groups. Among others, the proof of our main result requires an explicit computation of the equivariant cobordism rings of the projective plane and the Hirzebruch surfaces coming from the pullback maps \(i^{\ast }:{\Omega }_{T}^{\ast }(X^{T^{\prime }})_{\mathbb {Q}}\to {\Omega }_{T}^{\ast }(X^{T})_{\mathbb {Q}}\) of the inclusions \(i:X^{T}\to X^{T^{\prime }}\) for all singular codimension one subtori \(T^{\prime }\) in T. Furthermore, the proof is based on the following result (cf. Theorem 2.12) describing a relation in equivariant cobordism.
Theorem 1.1
Let X be a smooth T-variety, [h : Y → X] the equivariant fundamental class of a T-stable cobordism cycle and f ∈ k(Y ) a rational T-eigenfunction with weight χ. Denote by Z0 and \(Z_{\infty }\) the zeros and poles of f, respectively, and assume that they are smooth. Then the relation
holds in \({\Omega }_{\ast }^{T}(X)\) where \(F_{{\mathbb {L}}}\) denotes the universal formal group law and \([-1]_{F_{{\mathbb {L}}}}\) is the inverse in the universal formal group law.
Using the computations of equivariant cobordism for the projective plane and the Hirzebruch surfaces \(\mathbb {F}_{n}\), we can formulate the main result (cf. Theorem 3.4) by applying the technique of localisation where the ordering of the T-fixed points in a connected component of \(X^{T^{\prime }}\) is given in the description of the T-fixed points of \(X^{T^{\prime }}\).
Theorem 1.2
For any smooth projective and spherical G-variety X, the pullback map
is injective. Moreover, the image of i∗ consists of all families \((f_{x})_{x\in X^{T}}\) such that
-
(i)
\(f_{x}\equiv f_{y}\mod {c_{1}^{T}}(L_{\chi })\) whenever x and y are connected by a T-stable curve with weight χ.
-
(ii)
\((f_{x}-f_{y})+\rho _{1/2}{c_{1}^{T}}(L_{\alpha })(f_{z}-f_{x})\equiv 0\mod {c_{1}^{T}}(L_{\alpha })^{2}\) whenever α is a positive root of G relative to T, x,y and z lie in a connected component of \(X^{\text {Ker}(\alpha )^{0}}\) isomorphic to a projective plane \(\mathbb {P}^{2}\) and x ≥ y ≥ z are ordered by their corresponding weights.
-
(iii)
\(f_{w}-f_{x}-f_{y}+f_{z}\equiv 0\mod {c_{1}^{T}}(L_{\alpha })^{2}\) whenever α is a positive root of G relative to T, w,x,y and z lie in a connected component of \(X^{\text {Ker}(\alpha )^{0}}\) isomorphic to \(\mathbb {P}^{1}\times \mathbb {P}^{1}\) and w ≥ x,y ≥ z are ordered by their corresponding weights.
-
(iv)
\(\rho _{-n/2}{c_{1}^{T}}(L_{\alpha })(f_{y}-f_{z})+\rho _{n/2}{c_{1}^{T}}(L_{\alpha })(f_{w}-f_{x})\equiv 0\mod {c_{1}^{T}}(L_{\alpha })^{2}\) whenever α is a positive root of G relative to T, w,x,y and z lie in a connected component of \(X^{\text {Ker}(\alpha )^{0}}\) isomorphic to a rational ruled surface \(\mathbb {F}_{n}\), n ≥ 1, and w ≥ x ≥ y ≥ z are ordered by their corresponding weights.
As an application of Theorem 3.4, we compute the equivariant cobordism rings of all horospherical varieties of Picard number one which were classified by Pasquier [25] and very recently studied in [9]. One particular example is the class of odd symplectic Grassmannians IG(m,2n + 1) for integers n ≥ 2 and m ∈ [2,n] which were widely studied in the past for example in [9, 26]. These computations are done by describing very precisely the T-geometry of the relevant varieties where the latter is obtained by computing the degrees of the curves in the occurring surfaces in the connected components of \(X^{T^{\prime }}\). In a subsequent work, we will also use these results in order to obtain new computations for (T-equivariant) Chow groups with refined coefficient rings.
Lastly, we recall the notion of T-equivariant multiplicities (cf. Definition 5.5) at nondegenerate fixed points x ∈ X (cf. Definition 5.1) from [4, Section 4], i.e. the tangent space TxX contains no non-zero fixed point. This will be used in order to generalise the known results for equivariant Chow rings to equivariant cobordism for smooth projective T-varieties X (cf. Proposition 5.7). To be more precise, we determine the classes [f : Y → X] for smooth varieties \(Y\subseteq X\) in which all fixed points are nondegenerate using equivariant multiplicities (cf. Lemma 5.6). In addition, we compute the classes [f : Y → X] for arbitrary projective morphisms f and smooth Y assuming that all fixed points in X and all fixed points in the fibers f− 1(x) are nondegenerate for each x ∈ XT (cf. Proposition 5.7). Furthermore, using the previous results we give the explicit example of the odd symplectic Grassmannian IG(2,5) in which the classes are computed. These methods can be extended in order to obtain a description of the T-equivariant Chow ring \(\text {CH}^{\ast }_{T}(\text {IG}(2,5))_{\mathbb {Z}[\frac {1}{2}]}\) with the refined coefficient ring \(\mathbb { Z}[\frac {1}{2}]\) which will be discussed in future work. Finally, we observe that different resolutions of singularities of singular varieties \(X_{m}\subseteq X\) coming from the filtration (2.1) of smooth projective T-varieties X lead to different classes in the equivariant cobordism ring of IG(2,5) as opposed to the equivariant Chow rings.
2 A Relation in Equivariant Cobordism
In this section, we start with the basic definitions and properties of algebraic cobordism before defining equivariant cobordism. For more details on the properties of algebraic cobordism and equivariant cobordism we refer the reader to the book of Levine and Morel [22] and the articles of Krishna [19, 20], respectively. Before we can define algebraic cobordism, we recall the definition of a formal group law and the construction of the Lazard ring \({\mathbb {L}}\) after introducing the main notations.
2.1 Notations
Let k be an algebraically closed field of characteristic zero and G a connected reductive linear algebraic group over k. We denote the category of quasi-projective equi-dimensional schemes over k by Schk and the full subcategory consisting of smooth quasi-projective equi-dimensional schemes over k by Smk. A scheme is meant to be an object of Schk. Similarly, we denote the category of quasi-projective equi-dimensional schemes over k with a G-action and G-equivariant maps by G −Schk. Frequently, these schemes will be called G-schemes. The corresponding category of smooth quasi-projective equi-dimensional G-schemes will be denoted by G −Smk. We assume all group actions to be linear, i.e. for any G-action on a scheme X there exists a finite-dimensional representation G →GL(V ) and a G-equivariant immersion \(X\to \mathbb {P}(V)\). This assumption is always fulfilled for normal schemes which was proved by Sumihiro in [27]. Furthermore, we assume all representations of G to be finite-dimensional. Lastly, throughout this article we will use the notion of T-stable subsets whereas our main sources (e.g. [4, 19]) use the term T-invariant subsets for the same property.
Definition 2.1
A commutative formal group law of rank one with coefficients in R is a pair (R,FR) consisting of a commutative ring R and a formal power series \(F_{R}(u,v)=\sum a_{ij}u^{i}v^{j}\in R[[u,v]]\) satisfying the following conditions.
-
(i)
F(u,0) = F(0,u) = u ∈ R[[u]].
-
(ii)
F(u,v) = F(v,u) ∈ R[[u,v]].
-
(iii)
F(u,F(v,w)) = F(F(u,v),w) ∈ R[[u,v,w]].
The Lazard ring \({\mathbb {L}}\) is a polynomial ring over \(\mathbb {Z}\) which is generated by infinitely but countably many variables. It is constructed as the quotient of the polynomial ring \(\mathbb {Z}[\{A_{ij}\vert (i,j)\in \mathbb {N}^{2}\}]\) by the relations obtained by imposing the conditions of a commutative formal group law on the Aij. This uniquely defines the universal commutative formal group law \(F_{{\mathbb {L}}}\) of rank one on \({\mathbb {L}}\) which is given by
where aij is the equivalence class of Aij in \({\mathbb {L}}\). We define a grading of the Lazard ring by assigning the degree i + j − 1 to the coefficient aij. The resulting graded ring will be denoted by \({\mathbb {L}}_{\ast }\). Alternatively, we could assign degree 1 − i − j to the coefficient aij in which case we denote the resulting commutative graded ring by \({\mathbb {L}}^{\ast }\). Furthermore, the graded formal power series ring (cf. [20, Section 6.3]) will be denoted by \({\mathbb {L}}[[u_{1},...,u_{n}]]_{\text {gr}}\) where in our case the variables u1,...,un always have degree 1. Its equivalent with rational coefficients is given by the graded topological tensor product \(({\mathbb {L}}[[u_{1},...,u_{n}]]_{\text {gr}})_{\mathbb { Q}}:={\mathbb {L}}[[u_{1},...,u_{n}]]_{\text {gr}}\widehat {\otimes }_{\mathbb {Z}}\mathbb {Q}\) which was described in more detail in [20].
Recall the existence of a unique formal graded power series χ(ui) \( \in {\mathbb {L}}[[u_{1},...,u_{n}]]_{\text {gr}}\) which satisfies \(F_{{\mathbb {L}}}(u_{i},\chi (u_{i}))=0\). For any positive integer \(b\in \mathbb {Z}_{\geq 1}\) we establish the following notations.
We remark that the final relation is an inductive definition of \([b]_{F_{{\mathbb {L}}}}u_{i}\). Further, it is clear that \([b]_{F_{{\mathbb {L}}}}u\) is divisible by u for any \(u\in {\mathbb {L}}[[u_{1},...,u_{n}]]_{\text {gr}}\) of degree 1.
Lemma 2.2
Let \(u\in {\mathbb {L}}{[[u_{1},...,u_{n}]]}_{\text {gr}}\) be a homogeneous element of degree 1. Then there exists an element \(g\in {\mathbb {L}}_{\mathbb {Q}}[[x]]\) such that \(u=g({[b]}_{F_{{\mathbb {L}}}}u)\) for any \(b\in \mathbb {Z}_{\geq 1}\).
Proof
Fix \(b\in \mathbb {Z}_{\geq 1}\) and write
for \(b_{i}\in \mathbb {Z}_{\geq 0}\) for all i ≥ 1. Now we construct an element ρ of degree 0 such that \(\rho \cdot [b]_{F_{{\mathbb {L}}}}u=u\) holds. By comparison of coefficients, we observe that ρ is given by
Successively replacing u with \(\rho \cdot [b]_{F_{{\mathbb {L}}}}u\) implies the claim. □
Definition 2.3
Let \(u\in {\mathbb {L}}{[[u_{1},...,u_{n}]]}_{\text {gr}}\) be a homogeneous element of degree 1. Then for \(n\in \mathbb {Z}_{\geq 1}\) we define
and furthermore, if there exist a homogeneous element \(u^{\prime }\in ({\mathbb {L}}[[u_{1},...,u_{n}]]_{\text {gr}})_{\mathbb {Q}}\) of degree 1 and a positive integer \(m\in \mathbb {Z}_{\geq 1}\) such that \([m]_{F_{{\mathbb {L}}}}u^{\prime }=u\) holds, then we define
Definition 2.4
In the setting of the above definition we define the operator ρn/m by
in \(({\mathbb {L}}[[u_{1},...,u_{n}]]_{\text {gr}})_{\mathbb {Q}}\) for any \(n\in \mathbb {Z}\setminus \{0\}\) and \(m\in \mathbb {Z}_{\geq 1}\).
Remark 2.5
The quotient ρn/mu is indeed in \(({\mathbb {L}}[[u_{1},...,u_{n}]]_{\text {gr}})_{\mathbb {Q}}\) for any \(n\in \mathbb {Z}\setminus \{0\}\) and \(m\in \mathbb {Z}_{\geq 1}\) because \(\left [\frac {1}{m}\right ]_{F_{{\mathbb {L}}}}u\in ({\mathbb {L}}[[u_{1},...,u_{n}]]_{\text {gr}})_{\mathbb {Q}}\) is homogeneous of degree 1 and therefore, \(\left [\frac {1}{m}\right ]_{F_{{\mathbb {L}}}}u=g(u)\) holds for some \(g\in {\mathbb {L}}_{\mathbb {Q}}[[x]]\) by Lemma 2.2. Further, g(u) is divisible by u by construction and thus, \(\left [n\right ]_{F_{{\mathbb {L}}}}\left (\left [\frac {1}{m}\right ]_{F_{{\mathbb {L}}}}u\right )\) is divisible by u.
2.2 Algebraic Cobordism
Let X be an equi-dimensional k-scheme. Then a cobordism cycle is given by a family [f : Y → X,L1,...,Lr] where Y is smooth and irreducible, the map f is projective and the Li are line bundles over Y where the number of line bundles may be empty. The degree of a cobordism cycle is given by \(\dim _{k}(Y)-r\). Let \(\mathcal {Z}_{\ast }(X)\) be the free graded abelian group generated by the isomorphism classes of the cobordism cycles (cf. [22, Definition 2.1.6]) where the grading is given by the degree of the cycles. Now, we impose three relations on \(\mathcal {Z}_{\ast }\) in order to define algebraic cobordism.
The first one is called the dimension axiom. Let \(\mathcal {R}_{\ast }^{\text {Dim}}(X)\) be the graded subgroup of \(\mathcal {Z}_{\ast }\) generated by the cobordism cycles [f : Y → X,L1,...,Lr] such that \(\dim _{k}(Y)<r\). We denote the corresponding quotient \(\mathcal {Z}_{\ast }(X)/\mathcal {R}_{\ast }^{\text {Dim}}(X)\) by \(\underline {\mathcal {Z}}_{\ast }(X)\).
Secondly, for a line bundle L on X and a cobordism cycle [f : Y → X, L1,...,Lr], we define the first Chern class operator on \(\underline {\mathcal {Z}}_{\ast }(X)\) by
This definition is used in order to impose the section axiom. Given a line bundle L over X and a section s : X → L which is transverse to the zero section, we denote by Z → X the closed zero-subscheme of s which is smooth by the assumption that s is transverse to the zero section. Then we define \(\mathcal {R}_{\ast }^{\text {Sect}}(X)\) to be the graded subgroup of \(\underline {\mathcal {Z}}_{\ast }(X)\) generated by elements of the form \(\widetilde {c}_{1}(L)[\text {Id}:X\to X]-[Z\to X]\). We denote the quotient \(\underline {\mathcal {Z}}_{\ast }(X)/\mathcal {R}_{\ast }^{\text {Sect}}(X)\) by \(\underline {\Omega }_{\ast }(X)\) which we refer to as algebraic pre-cobordism.
Lastly, we impose the formal group law axiom on algebraic pre-cobordism by considering the subset \(\mathcal {R}_{\ast }^{\text {FGL}}(X)\subseteq {\mathbb {L}}_{\ast }\otimes \underline {\Omega }_{\ast }(X)\) consisting of elements of the form
where L and M are line bundles over X. Finally, for the subset \({\mathbb {L}}_{\ast }\mathcal {R}_{\ast }^{\text {FGL}}(X)\subseteq {\mathbb {L}}_{\ast }\)\(\otimes \underline {\Omega }_{\ast }(X)\) which is given by elements of the form a ⊗ ρ for \(a\in {\mathbb {L}}_{\ast }\) and \(\rho \in \mathcal {R}_{\ast }^{\text {FGL}}(X)\), we define algebraic cobordism of X by
Let d be the dimension of the k-scheme X. In this case, we define \({\Omega }^{i}(X)={\Omega }_{d-i}(X)\) for all \(i\in \mathbb {Z}\).
If X is smooth, we know that Ω∗(X) is a graded \({\mathbb {L}}\)-algebra and furthermore, we obtain an established theory of Chern classes of line bundles (cf. [22]).
2.3 Equivariant Cobordism
Recall that G is a connected reductive linear algebraic group over k. Now we consider for any integer j ≥ 0 a corresponding pair (Vj,Uj) where Vj is an lj-dimensional representation of G and Uj is a G-stable open subset of Vj such that the codimension of the complement (Vj ∖ Uj) in Vj is at least j. Furthermore, we ask that G acts freely on Uj such that the quotient Uj/G is a quasi-projective scheme. Such a pair will be called a good pair for the G-action corresponding to j. It is well known that such a good pair always exists (cf. [7, Lemma 9]).
For a k-scheme X of dimension d with a G-action and an integer j ≥ 0, let (Vj,Uj) be an lj-dimensional good pair corresponding to j. Then we denote the mixed quotient of the product X × Uj by the free diagonal action of G by X ×GUj which exists as a scheme because the G-action on X is linear.
We now present one of the main results concerning actual computations of equivariant algebraic cobordism. Since the original definition is very hard to be computed in general, one can make use of the following result by Krishna [20] which will serve as our definition of equivariant cobordism.
Definition 2.6
[20, Theorem 6.1] Let {(Vj,Uj)}j≥ 0 be a sequence of lj-dimensional good pairs such that there exist G-representations (Wj)j≥ 0 with
-
(i)
Vj+ 1 = Vj ⊕ Wj as representations of G with \(\dim (W_{j})>0\) and
-
(ii)
\(U_{j}\oplus W_{j}\subseteq U_{j+1}\) as G-stable open subsets.
Then for any scheme X ∈ G −Schk of dimension d with \(g:=\dim (G)\) and any \(i\in \mathbb {Z}\), one defines
Moreover, such a sequence of good pairs always exists. For some details concerning the ring structure we refer to Remark 2.8.
Remark 2.7
One should note that the equivariant algebraic cobordism can be non-zero for any \(i\in \mathbb {Z}\) unlike the ordinary algebraic cobordism Ω∗. Furthermore, we set
If in addition X is an equi-dimensional k-scheme of dimension d with G-action, we let \({{\Omega }^{i}_{G}}(X)={\Omega }_{d-i}^{G}(X)\) and analogously \({\Omega }^{\ast }_{G}(X):= {\bigoplus }_{i\in \mathbb {Z}}{{\Omega }^{i}_{G}}(X)\). We denote the equivariant cobordism \({\Omega }^{\ast }_{G}(k)\) of the underlying ground field by S(G). Furthermore, if G is the trivial group, equivariant algebraic cobordism reduces to ordinary algebraic cobordism. Besides that, equivariant algebraic cobordism with rational coefficients is again defined by the graded topological tensor product \({\Omega }^{\ast }_{G}(X)_{\mathbb {Q}}\)\(:={\Omega }^{\ast }_{G}(X)\widehat {\otimes }_{\mathbb {Z}}\mathbb {Q}\) which was described in [20].
For any X ∈ G −Schk and a projective morphism f : Y → X in G −Schk where Y is smooth of dimension d we obtain for any j ≥ 0 and any lj-dimensional good pair (Vj,Uj) an ordinary cobordism cycle [Y ×GUj → X ×GUj] of dimension d + lj − g by [20, Lemma 5.1]. This defines a unique element \(\alpha \in {{\Omega }^{G}_{d}}(X)\) which we call the G-equivariant fundamental class of the cobordism cycle [f : Y → X].
Remark 2.8
[19, Sections 2.4–2.5] It is well known that \({\Omega }_{G}^{\ast }(X)\) is an S(G)-algebra if X is smooth. In this case, we will identify the commutative \({\mathbb {L}}\)-subalgebra of \(\text {End}_{{\mathbb {L}}}({\Omega }^{\ast }_{G}(X))\) generated by the Chern classes of vector bundles with the \({\mathbb {L}}-\)subalgebra of the equivariant cobordism ring \({\Omega }^{\ast }_{G}(X)\) via \({c_{i}^{G}}(E)\)\(\mapsto {c_{i}^{G}}(E)\left ([\text {Id}:X\to X]\right )\). Therefore, we will denote this image also by \({c_{i}^{G}}(E)\). Since we pass freely between vector bundles E and their corresponding locally free coherent sheaves we will also write \({c_{1}^{G}}(\mathcal {E})\) for a locally free coherent sheaf \(\mathcal {E}\).
From now on, we will only consider T-equivariant cobordism where T is a torus.
Theorem 2.9
[20, Proposition 6.7] Let {χ1,...,χn} be a basis of the character group of a torus T of rank n. Then the assignment \(t_{i}\mapsto {c_{1}^{T}}(L_{\chi _{i}})\) yields a graded \({\mathbb {L}}\)-algebra isomorphism
where \(L_{\chi _{i}}\) is the one-dimensional representation of T on which T acts via weight χi.
Remark 2.10
Let M be the character group of a torus T. Using Definition 2.4 one observes that
holds in \(S(T)_{\mathbb {Q}}\) for any character χ ∈ M, \(n\in \mathbb {Z}\setminus \{0\}\) and \(m\in \mathbb {Z}_{\geq 1}\) if \(\frac {n\chi }{m}\) is also a character in M.
The first step for the computations in this article is to describe a result in equivariant cobordism which is similar to the following one in equivariant Chow groups. For any T-scheme X, any closed T-stable subvariety \(Y\subseteq X\) and any rational function f on Y which is an eigenvector of T for weight χ, we have χ ⋅ [Y ] = divY(f) in the \(\text {CH}^{\ast }_{T}(k)\)-module \(\text {CH}^{\ast }_{T}(X)\) (cf. [4, Theorem 2.1]). We would like to have such a relation for smooth schemes X in equivariant cobordism and therefore, we need to properly understand the S(T)-action on \({\Omega }_{\ast }^{T}(X)\) for X ∈Smk.
Construction 2.11
Now we present a similar construction to the one introduced to prove the above relation in equivariant Chow groups in [4, Theorem 2.1]. By Proposition 2.9 we know that for any basis {χ1,...,χn} of the character group of T we have the isomorphism \({\mathbb {L}}[[t_{1},...,t_{n}]]_{\text {gr}}\cong S(T), t_{i}\mapsto {c_{1}^{T}}(L_{\chi _{i}})\), where in this case we set \(L_{\chi _{i}}\) to be the one-dimensional representation of T on which T acts via weight − χi. Hereby \({c_{1}^{T}}(L_{\chi _{i}})\) means \({c_{1}^{T}}(L_{\chi _{i}})[\text {Spec } k\to \text {Spec } k]\) where [Spec k →Spec k] is by abuse of notation the equivariant fundamental class of the ordinary cobordism cycle [Spec k →Spec k]. For any character χ and a lj-dimensional good pair (Vj,Uj) we have the line bundle (Lχ × Uj)/T → Uj/T which we denote by (Lχ)T. Since equivariant cobordism is defined via an inverse limit construction we consider the elements
By [19, Theorem 4.11] we know that the S(T)-module \({\Omega }^{T}_{\ast }(X)\) is generated by the equivariant fundamental classes of the T-stable cobordism cycles in Ω∗(X) for smooth k-schemes X. Therefore, we take one of these ordinary cobordism cycles [h : Y → X] and consider [(Y × Uj)/T → (X × Uj)/T] in the j-th component of the equivariant fundamental class which we denote by [Y → X]j for some good pair (Vj,Uj). For the morphism g : (X × Uj × Uj)/T → Uj/T, that is induced by the second projection p2 : Uj × Uj → Uj, we use the exterior product on equivariant cobordism which was described in the proof of [20, Theorem 5.2] and thus, we obtain
in \({\Omega }_{\ast }^{T}(X)\). This equation will be important in the proof of the following theorem.
Theorem 2.12
Let X be a smooth T-variety, [h : Y → X] the equivariant fundamental class of a T-stable cobordism cycle and f ∈ k(Y ) a rational T-eigenfunction with weight χ. Denote by Z0 and \(Z_{\infty }\) the zeros and poles of f, respectively, and assume that they are smooth. Then the relation
holds in \({\Omega }_{\ast }^{T}(X)\) where \(F_{{\mathbb {L}}}\) denotes the universal formal group law and \([-1]_{F_{{\mathbb {L}}}}\) is the inverse in the universal formal group law.
Proof
We consider the rational function f on Y. One may observe that
is a rational section of the line bundle h∗g∗(Lχ)T. For this line bundle with the given rational section, we can also write
by the known correspondence between Cartier divisors and pairs (L,s) consisting of a line bundle and a rational section. We simplify by setting \(L_{0}=\mathcal {O}_{(Y\times U_{j}\times U_{j})/T}(Z_{0})\) and \(L_{\infty }=\mathcal {O}_{(Y\times U_{j}\times U_{j})/T}(Z_{\infty })\). By the smoothness assumption we know that the corresponding sections of L0 and \(L_{\infty }\) coming from the rational section s are transverse to the zero sections of L0 and \(L_{\infty }\), respectively. Furthermore, the zero-subschemes of these sections are T-stable and hence they define cobordism cycles whose equivariant fundamental classes are in \({\Omega }_{\ast }^{T}(Y)\). In the following computation we will use [22, Definition 2.1.2] axiom (A3) and [22, Definition 2.2.1] axiom (Sect). We know further by [22, Proposition 5.2.1] that the Chern class operator \(\widetilde {c}_{1}(L)\) on a smooth scheme X is given by \(\widetilde {c}_{1}(L)(\eta )=c_{1}(L)\cdot \eta \) for η ∈Ω∗(X). Lastly, we have the embeddings of the zero-subschemes i0 : (Z0 × Uj × Uj)/T → (Y × Uj × Uj)/T and similarly \(i_{\infty }:(Z_{\infty }\times U_{j}\times U_{j})/T \to (Y\times U_{j}\times U_{j})/T\). Using all those properties, we obtain
Furthermore, the sum is finite since the ordinary first Chern classes are nilpotent. We conclude the claim because taking the limit on these elements commutes with the pushforward h∗ by the definition of the equivariant pushforward maps. □
We recall the morphism g : (X × Uj × Uj)/T → Uj/T and the given line bundle (Lχ × Uj)/T → Uj/T which we denote by (Lχ)T (cf. Construction 2.11). In the sequel, we will only consider the cases in which there exists a global section of the line bundle h∗g∗(Lχ)T which is transverse to the zero section. In this particular case, the terms containing \(Z_{\infty }\) disappear and one obtains the following statement.
Corollary 2.13
Assume there exists a global section s of the line bundle h∗g∗(Lχ)T which is transverse to the zero section. In this case, the relation
holds in \({\Omega }_{\ast }^{T}(X)\) where Z0 is the zero-subscheme of s on Y.
For equivariant Chow groups the above relations generate all relations as explained in the following result of Brion.
Theorem 2.14
[4, Theorem 2.1] Let X be a variety with an action of a torus T. The \(\text {CH}^{T}_{\ast }(k)\)-module \(\text {CH}^{T}_{\ast }(X)\) is defined by generators [Y ], where \(Y\subseteq X\) is a T-stable subvariety, and by relations [divY(f)] = χ ⋅ [Y ], where f is a non-constant rational function on Y which is an eigenvector of T of weight χ.
Remark 2.15
Similarly to Theorem 2.14 we know that \({\Omega }_{T}^{\ast }(X)\) is generated by the equivariant fundamental classes of the T-stable cobordism cycles in Ω∗(X) by [19, Theorem 4.11] for a smooth variety X with an action of a torus. At present the author does not know whether the equivariant cobordism rings \({\Omega }^{\ast }_{T}(X)\) are given by the equivariant fundamental classes of T-stable cobordism cycles in Ω∗(X) modulo the previously described relations from Proposition 2.12, but it might be enough for smooth projective varieties X with an action of a torus.
2.4 Localisation at Fixed Points
We now prove a lemma which will be useful in the sequel for comparing the equivariant algebraic cobordism with respect to a torus T and its quotient T/F by a finite subgroup F.
Lemma 2.16
Let T be a torus of rank n and F be a finite subgroup. Then we have a graded \({\mathbb {L}}\)-algebra isomorphism
Proof
Let {χ1,...,χn} be a basis of the character group of T such that the basis of the character group of T/F is then given by {a1χ1,...,anχn} for positive integers a1|a2|⋯|an. Using Proposition 2.9 we know that there is an isomorphism \({\Omega }_{T}^{\ast }(k)\cong {\mathbb {L}}[[t_{1},...,t_{n}]]_{\text {gr}}\) map** \({c_{1}^{T}}(L_{\chi _{i}})\mapsto t_{i}\) where \(L_{\chi _{i}}\) is the one-dimensional representation of weight − χi. Furthermore, we have \({\Omega }_{T/F}^{\ast }(k)\cong {\mathbb {L}}[[t^{\prime }_{1},...,t^{\prime }_{n}]]_{\text {gr}}\) for \(c_{1}^{T/F}(L_{a_{i}\chi _{i}})\mapsto t^{\prime }_{i}\). Since we consider the \(L_{\chi _{i}}\) as the one-dimensional representations of T and similarly those of T/F, we know that
holds in \({\Omega }_{T}^{\ast }(k)\) where \(F_{{\mathbb {L}}}\) denotes again the universal formal group law in cobordism. On the other hand, we know that we can take \({c_{1}^{T}}(L_{a_{i}\chi _{i}})\) as generators of \({\Omega }_{T}^{\ast }(k)_{\mathbb {Q}}\) instead of \({c_{1}^{T}}(L_{\chi _{i}})\) as soon as we consider rational coefficients by Lemma 2.2. This leads to the desired isomorphism. □
Remark 2.17
The preceding lemma implies the same statement for equivariant Chow groups and furthermore we remark that the finite subgroup F has order a1⋯an. Lastly, the statement also holds if we only take coefficients in \(\mathbb {Z}[1/p_{1},...,1/p_{\ell }]\) where p1,..,pℓ are the primes occurring in the prime factorisation of an. Therefore, we only have to invert a finite number of primes in order to obtain the isomorphism of Lemma 2.16.
To finish this introductory section, let T be a torus and X ∈ T −Schk. We recap some basic notation for T-filtrable schemes and relevant applications which were presented by Krishna [19]. We say that X is T-filtrable if the fixed point subscheme XT is smooth and projective and if there is an ordering \(X^{T}={\coprod }_{m=0}^{n} Z_{m}\) of the connected components Zm of the fixed point subscheme such that there is a filtration of X by T-stable closed subschemes
with \(Z_{m}\subseteq W_{m}:=X_{m}\setminus X_{m-1}\) and maps ϕm : Wm → Zm for all 0 ≤ m ≤ n which are all T-equivariant vector bundles such that the inclusions Zm↪Wm are the 0-section embeddings. One should note that if X is T-filtrable then so is every closed subscheme Xm. We remark that this definition coincides with Brion’s definition in [4, Section 3] for smooth projective schemes with finitely many T-fixed points which will be the objects of our main interest. The following result which is a consequence of the Bialynicki-Birula decomposition will be essential for our understanding of the equivariant cobordism of smooth projective varieties.
Theorem 2.18
[2, Theorem 4.3] Let X be a smooth projective variety with an action of a torus T and finitely many T-fixed points. Then X is T-filtrable.
The following proposition is very useful for computing equivariant cobordism. As opposed to the localisation theorem for equivariant Chow groups (cf. [4, Theorem 3.3]) one has to assume that the fixed point scheme consists only of finitely many isolated points in order to formulate the equivalent statement in equivariant cobordism.
Theorem 2.19
[19, Theorem 7.6, Theorem 7.1] Let X be a smooth T-filtrable scheme with an action of a torus T. Further, let XT consist of finitely many fixed points x1,...,xs and let i : XT↪X denote the inclusion of the fixed point subscheme. Then the pullback map \(i^{\ast }:{\Omega }^{\ast }_{T}(X)\to {\Omega }^{\ast }_{T}(X^{T})\) is injective and its image is the intersection of the images of
where \(T^{\prime }\) runs over all subtori of codimension one in T.
Remark 2.20
In a subsequent work, we will show that the proof of the previous proposition can be adapted such that the result also holds for a refined coefficient ring \(\mathbb {Z}\subseteq R \subseteq \mathbb {Q}\) which leads to new refined computations even for (T-equivariant) Chow groups.
Next, we present the analogous result of [4, Theorem 3.4] in equivariant cobordism.
Theorem 2.21
[19, Theorem 7.8] Let X be a smooth T-filtrable scheme where a torus T acts with finitely many fixed points x1,...,xs and finitely many T-stable curves. Then the image of
is the subalgebra of \((f_{1},...,f_{s})\in S(T)^{s}_{\mathbb {Q}}\) such that \(f_{i}\equiv f_{j}\mod {c_{1}^{T}}(L_{\chi })\) whenever xi and xj are connected by a stable irreducible curve where T acts through the weight χ.
Lastly, we recall an important statement which relates T-equivariant cobordism and ordinary cobordism for k-varieties with a T-action.
Theorem 2.22
[19, Theorem 3.4] Let T be a torus acting on a k-variety X. Then there is an isomorphism
If X is smooth, this is an \({\mathbb {L}}\)-algebra isomorphism.
Remark 2.23
The previous proposition was proved by Krishna in [19] using the localisation sequence in equivariant cobordism. There is work in progress (cf. [16, Theorem 7.3]) fixing a gap in the original proof of the localisation sequence (cf. [12, Theorem 20]). This gap was first mentioned in [15, Example 12.18]. Note that our computations do not rely on this proposition, but only on some weaker version of the previous result for T-filtrable varieties with finitely many T-fixed points. The latter is well known to be true using [13, Theorem 2.5], [19, Corollary 4.8] and the fact that T-filtrable varieties with finitely many T-fixed points are cellular.
3 Equivariant Cobordism of Spherical Varieties
Throughout this section let G be a connected reductive group, \(B\subseteq G\) a Borel subgroup and \(T\subseteq B\) a maximal torus. Recall that a normal G-variety X containing a dense B-orbit is called spherical. We remark further that spherical G-varieties have only finitely many T-fixed points (cf. [5, Lemma 2.2]). This section is based on [4].
Definition 3.1
A subtorus \(T^{\prime }\subseteq T\) is regular if its centraliser
is equal to the torus T. If this is not the case, we call the subtorus \(T^{\prime }\) singular.
Remark 3.2
Following [14, Corollary B, Section 26.2] a subtorus \(T^{\prime }\) of codimension one is singular if and only if it is the identity component of the kernel of some positive root α of (G,T). In this case we will write \(T^{\prime }=\text {Ker}(\alpha )^{0}\). Then α is unique and the group \(C_{G}(T^{\prime })\) is the product of \(T^{\prime }\) with a subgroup \(S(\alpha )\subseteq G\) isomorphic to SL2 or to PSL2. Then the fixed point locus \(X^{T^{\prime }}\) is equipped with an action of
since \(S(\alpha )\cap T^{\prime }\) is either of order one or two. Furthermore, we have \(T=T^{\prime }T(\alpha )\) for a maximal subtorus T(α) of S(α), the image of the coroot of α. As above, \(T^{\prime }\cap T(\alpha )\) is a finite group F(α) of order one or two. Clearly, F(α) acts trivially on \(X^{T^{\prime }}\) and hence the T-action on \(X^{T^{\prime }}\) factors through an action of the corresponding quotient \(T/F(\alpha )\cong (T^{\prime }\times T(\alpha ))/(F(\alpha )\times F(\alpha ))\).
In the following proposition, we will analyse the components of the fixed point subschemes \(X^{T^{\prime }}\) for regular and singular codimension one subtori \(T^{\prime }\subseteq T\). Recall that the surface \(\mathbb {F}_{n}=\mathbb {P}(\mathcal {O}_{\mathbb {P}^{1}}\oplus \mathcal {O}_{\mathbb {P}^{1}}(n))\) is called the n-th Hirzebruch surface.
Theorem 3.3
[4, Proposition 7.1] Let X be a smooth projective spherical G-variety and let \(T^{\prime }\subseteq T\) be a subtorus of codimension one.
-
(i)
Each irreducible component of \(X^{T^{\prime }}\) is a spherical \(C_{G}(T^{\prime })\)-variety.
-
(ii)
If \(T^{\prime }\) is regular, then the fixed point set \(X^{T^{\prime }}\) is at most one-dimensional.
-
(iii)
If \(T^{\prime }\) is singular, then \(X^{T^{\prime }}\) is at most two-dimensional. Furthermore, any two-dimensional connected component of \(X^{T^{\prime }}\) is either a Hirzebruch surface \(\mathbb {F}_{n}\) where \(C_{G}(T^{\prime })\) acts through the natural action of SL2, or the projective plane \(\mathbb {P}^{2}\) where \(C_{G}(T^{\prime })\) acts through the projectivization of a non-trivial SL2-module of dimension three.
In this section we want to generalise the presentations of the equivariant Chow rings of smooth projective spherical G-varieties (cf. [4, Theorem 7.3]) to equivariant algebraic cobordism. In order to be able to generalise those, we need to compute the equivariant algebraic cobordism of the projective plane and the Hirzebruch surfaces as required by Proposition 3.3 and Proposition 2.19. Using notation as in Proposition 2.21, we can now formulate the main result of this section which is the analogue of [4, Theorem 7.3] and which will be proved later on in this section. We remark that the ordering of the T-fixed points in \(\mathbb {P}^{2}\) and \(\mathbb {F}_{n}\), n ≥ 0, which is used in the upcoming theorem, is discussed after Remark 3.5 where the T-fixed points of \(X^{T^{\prime }}\) are described in more detail.
Theorem 3.4
For any smooth projective and spherical G-variety X, the pullback map
is injective. Moreover, the image of i∗ consists of all families \((f_{x})_{x\in X^{T}}\) such that
-
(i)
\(f_{x}\equiv f_{y}\mod {c_{1}^{T}}(L_{\chi })\) whenever x and y are connected by a T-stable curve where T acts through the weight χ.
-
(ii)
\((f_{x}-f_{y})+\rho _{1/2}{c_{1}^{T}}(L_{\alpha })(f_{z}-f_{x})\equiv 0\mod {c_{1}^{T}}(L_{\alpha })^{2}\) whenever α is a positive root of G relative to T, x,y and z lie in a connected component of \(X^{\text {Ker}(\alpha )^{0}}\) isomorphic to a projective plane \(\mathbb {P}^{2}\) and x ≥ y ≥ z are ordered by their corresponding weights.
-
(iii)
\(f_{w}-f_{x}-f_{y}+f_{z}\equiv 0\mod {c_{1}^{T}}(L_{\alpha })^{2}\) whenever α is a positive root of G relative to T, w,x,y and z lie in a connected component of \(X^{\text {Ker}(\alpha )^{0}}\) isomorphic to \(\mathbb {P}^{1}\times \mathbb {P}^{1}\) and w ≥ x,y ≥ z are ordered by their corresponding weights.
-
(iv)
\(\rho _{-n/2}{c_{1}^{T}}(L_{\alpha })(f_{y}-f_{z})+\rho _{n/2}{c_{1}^{T}}(L_{\alpha })(f_{w}-f_{x})\equiv 0\mod {c_{1}^{T}}(L_{\alpha })^{2}\) whenever α is a positive root of G relative to T, w,x,y and z lie in a connected component of \(X^{\text {Ker}(\alpha )^{0}}\) isomorphic to a rational ruled surface \(\mathbb {F}_{n}\), n ≥ 1, and w ≥ x ≥ y ≥ z are ordered by their corresponding weights.
Remark 3.5
We will see later in the proof that condition (i) in the preceding proposition comes from Proposition 2.21. Further, the formulation of the equations in the conditions (ii) and (iv) slightly differs from the one in Brion’s description. We need to introduce the terms ρn/2, \(n\in \mathbb {Z}\setminus \{0\}\), from Definition 2.4 because of the universal formal group law in cobordism. Besides that, as opposed to the formulation of Brion, we have to distinguish between the cases \(\mathbb {F}_{0}\) and \(\mathbb {F}_{n}\), n ≥ 1, again due to the universal formal group law. In the case of a smooth projective spherical G-variety X with only finitely many T-stable curves, the statement of the theorem can be also obtained from Proposition 2.21 because the cases (ii)–(iv) do not occur if the variety has only finitely many T-stable curves.
T-Fixed Points of \(X^{T^{\prime }}\)
Now we want to compute equivariant cobordism for projective planes and Hirzebruch surfaces. Therefore, we describe the irreducible components of \(X^{T^{\prime }}\) for singular codimension one subtori \(T^{\prime }\) coming from Proposition 3.3 in some more detail.
We start with the description of the T-fixed points in \(X^{T^{\prime }}\). Let D be the torus of diagonal matrices in SL2 and let α be the positive root. At first, we want to consider the two cases of \(\mathbb {P}(V)\) for a non-trivial SL2-module V of dimension three arising from case (ii) of the previous theorem. Set \(V_{n+1}:=\text {Sym}^{n+1}(k^{2})\). Let V = V0 ⊕ V1 be one of the non-trivial SL2-modules of dimension three. The weights of D in V are − α/2,0 and α/2 with the induced group action of D on V. We denote by x,y and z the corresponding fixed points of D in \(\mathbb {P}(V)\). To be more explicit, the fixed points x = [1 : 0 : 0],y = [0 : 1 : 0] and z = [0 : 0 : 1] correspond to the weights α/2,0,−α/2, respectively. Thus, we identify \({\Omega }^{\ast }_{D}(\mathbb {P}(V)^{D})_{\mathbb {Q}}\) with \(S(D)_{\mathbb {Q}}^{3}\).
Similarly, for the other non-trivial SL2-module \(V=V_{2}=\mathfrak {sl_{2}}\) of dimension three, the corresponding weights are α,0 and − α where the corresponding fixed points are again x = [1 : 0 : 0],y = [0 : 1 : 0] and z = [0 : 0 : 1], respectively.
Next, we consider the case \(\mathbb {F}_{0}=\mathbb {P}^{1}\times \mathbb {P}^{1}\) with D-action given by
We denote by w and z the D-fixed points ([1 : 0],[1 : 0]) and ([0 : 1],[0 : 1]), respectively. Further, we denote the remaining two D-fixed points ([1 : 0],[0 : 1]) and ([0 : 1],[1 : 0]) by x and y, respectively.
Lastly, we have a look at the rational ruled surface \(\mathbb {F}_{n}\), n ≥ 1, which is the closure of the SL2-orbit SL2 ⋅ [v1 + vn+ 1] in \(\mathbb {P}(V)\) for V := V1 ⊕ Vn+ 1 where v1 ∈ V1 and vn+ 1 ∈ Vn+ 1 denote the two highest weight vectors, respectively. The open SL2-orbit cannot contain any D-fixed points. Since the two closed SL2-orbits are given by SL2 ⋅ [v1] and SL2 ⋅ [vn+ 1], which are both projective lines, we observe that \(\mathbb {F}_{n}\) has four D-fixed points w,x,y and z with corresponding weights (n + 1)α/2,α/2,−α/2 and − (n + 1)α/2, respectively, by the induced D-action on \(\mathbb {F}_{n}\). Therefore, we can identify \({\Omega }^{\ast }_{D}({\mathbb {F}_{n}^{D}})_{\mathbb {Q}}\) with \(S(D)_{\mathbb {Q}}^{4}\).
Theorem 3.6
Let X be a Hirzebruch surface \(\mathbb {F}_{n}\) or a projective plane \(\mathbb {P}(V)\) as above.
-
(i)
The image of the pullback
$$ \begin{array}{@{}rcl@{}} i^{\ast}: {\Omega}^{\ast}_{D}(\mathbb{F}_{n})_{\mathbb{Q}}\to S(D)^{4}_{\mathbb{Q}} \end{array} $$consists of all \((f_{w},f_{x},f_{y},f_{z})\in S(D)^{4}_{\mathbb {Q}}\) such that
$$ \begin{array}{@{}rcl@{}} f_{w}\equiv f_{x}\equiv f_{y}\equiv f_{z}&\mod& {c_{1}^{D}}(L_{\alpha})\text{ and } \end{array} $$(3.1)$$ \begin{array}{@{}rcl@{}} f_{w}-f_{x}-f_{y}+f_{z}\equiv 0&\mod& {c_{1}^{D}}(L_{\alpha})^{2} \end{array} $$(3.2)hold for n = 0 and of all \((f_{w},f_{x},f_{y},f_{z})\in S(D)^{4}_{\mathbb {Q}}\) such that
$$ \begin{array}{@{}rcl@{}} f_{w}\equiv f_{x}\equiv f_{y}\equiv f_{z}&\mod& {c_{1}^{D}}(L_{\alpha}) \text{ and }~~~~~ \end{array} $$(3.3)$$ \begin{array}{@{}rcl@{}} \rho_{-n/2}{c_{1}^{D}}(L_{\alpha})(f_{y}-f_{z})+\rho_{n/2}{c_{1}^{D}}(L_{\alpha})(f_{w}-f_{x})\equiv 0&\mod& {c_{1}^{D}}(L_{\alpha})^{2} \end{array} $$(3.4)hold for n ≥ 1.
-
(ii)
Moreover, the image of
$$ \begin{array}{@{}rcl@{}} {\Omega}^{\ast}_{D}(\mathbb{P}(V))_{\mathbb{Q}}\to S(D)^{3}_{\mathbb{Q}} \end{array} $$consists of all (fx,fy,fz) such that
$$ \begin{array}{@{}rcl@{}} f_{x}\equiv f_{y}\equiv f_{z}&\mod& {c_{1}^{D}}(L_{\alpha})\text{ and }~~~~ \end{array} $$(3.5)$$ \begin{array}{@{}rcl@{}} (f_{x}-f_{y})+\rho_{1/2}{c_{1}^{D}}(L_{\alpha})(f_{z}-f_{x})\equiv 0&\mod& {c_{1}^{D}}(L_{\alpha})^{2} \end{array} $$(3.6)hold.
Remark 3.7
In the above statement, there is one equation more than needed to keep the symmetry in the arguments. For example, in the \(\mathbb {F}_{n}\) case we could remove the equation \(f_{w}\equiv f_{x} \mod {c_{1}^{D}}(L_{\alpha })\).
Proof
We first consider the case of \(\mathbb {P}(V)\) for V = V0 ⊕ V1. Since i∗ is a ring homomorphism, the class \([\mathbb {P}(V)\to \mathbb {P}(V)]\) maps to (1,1,1). We remark that the closures of the Bialynicki-Birula cells are smooth in the cases of the projective plane and the Hirzebruch surfaces. Now we compute the images of the closures of the Bialynicki-Birula cells, i.e. the images of the equivariant fundamental classes of the D-stable cobordism cycles \([(yz)\to \mathbb {P}(V)]\) and \([z\to \mathbb {P}(V)]\) because these images generate the equivariant cobordism ring of \(\mathbb {P}(V)\) as a subalgebra of \(S(D)^{3}_{\mathbb {Q}}\) (cf. [19, Corollary 4.8]). We have a look at the pullback
where \(i^{\ast }_{x}[Y\to \mathbb {P}(V)]\) denotes the pullback of the class \([Y\to \mathbb { P}(V)]\) under the inclusion ix of the corresponding fixed point in \(\mathbb {P}(V)\). To compute \(i_{z}^{\ast }[Y\to \mathbb {P}(V)]\) we can replace \(\mathbb {P}(V)\) by any open D-stable neighbourhood Uz of z. In this case we choose Uz to be the affine chart of \(\mathbb {P}(V)\) in which the coordinate associated to z does not vanish. We introduce the coordinates a,b and c for V such that our coordinates for Uz become a/c and b/c. Therefore, D acts linearly on Uz with weights α and α/2. We choose f(a/c,b/c) = a/c which is an eigenfunction of D of weight − α. In this situation, we can apply Corollary 2.13 because f gives rise to the section
which is transverse to the zero section with zero-subscheme Z0 = (yz) ∩ Uz. Thus, we know that
holds in \({\Omega }^{\ast }_{D}(U_{z})_{\mathbb {Q}}\). Pulling back to \({\Omega }^{\ast }_{D}(z)_{\mathbb {Q}}\) yields \(i^{\ast }_{z}[(yz)\cap U_{z}\to U_{z}]={c_{1}^{D}}(L_{-\alpha })\). We can apply the same argument for the pullback \(i^{\ast }_{y}[(yz)\to \mathbb {P}(V)]\) by choosing Uy to be the open affine neighbourhood of y such that the coordinate associated to y does not vanish. Thus, D acts linearly on Uy with weights α/2 and − α/2. We take f(a/b,c/b) = a/b which is an eigenfunction of D of weight − α/2. Therefore, we conclude \(i^{\ast }_{y}[(yz)\cap U_{y}]={c_{1}^{D}}(L_{-\alpha /2})\) by the same argument as above.
Finally, we consider the last closure of the Bialynicki-Birula cells, i.e. the point z. Clearly, z is the complete intersection of the two lines (yz) and (xz). Therefore, we want to compute the pullback of [(xz) ∩ (yz) ∩ Uz → Uz] = [z → Uz]. We want to apply the same argument again using the relation
from Corollary 2.13 where z = (xz) ∩ (yz) ∩ Uz is the zero-subscheme of the section defined by the eigenfunction g(a/c,b/c) = b/c of D of weight − α/2 on (yz) ∩ Uz. Using the equality \(i^{\ast }_{z}[(yz)\cap U_{z}\to U_{z}]={c_{1}^{D}}(L_{-\alpha })\), we obtain the pullback
in \({\Omega }^{\ast }_{D}(z)_{\mathbb {Q}}\). The images of the D-stable cobordism cycles coming from the closures of the Bialynicki-Birula decomposition generate the equivariant cobordism ring as a subring of \(S(D)^{3}_{\mathbb {Q}}\) by [19, Corollary 4.8] as already mentioned above. Therefore, the image of the pullback \(i^{\ast }: {\Omega }^{\ast }_{D}(\mathbb {P}(V))_{\mathbb {Q}}\to S(D)^{3}_{\mathbb {Q}}\) is generated by the images
These images satisfy the equations (3.5) and (3.6) which can be seen by again expressing \({c_{1}^{D}}(L_{\alpha /2})\) as a formal power series in the variable \({c_{1}^{D}}(L_{\alpha })\) with rational coefficients. For the following computation and similar ones upcoming in the sequel of this proof, we remark that any element which is divisible by \({c_{1}^{D}}(L_{\alpha })\) will be also divisible by \({c_{1}^{D}}(L_{n\alpha /m})\) for \(n\in \mathbb {Z}\setminus \{0\}\) and \(m\in \mathbb {Z}_{\geq 1}\) if \(\frac {n\alpha }{m}\) is also a character in M because we can again express the former Chern class in terms of the latter one and factor out. Therefore, for an element \((f_{x},f_{y},f_{z})\in S(T)_{\mathbb {Q}}^{3}\) satisfying the given equations, we have
which completes the proof in the case V = V0 ⊕ V1.
The computation for V = V2 can be done similarly. We obtain
which satisfy the equations (3.5) and (3.6). This can be verified using the properties of the formal group law. Again, we obtain
which completes the proof for V = V2 since the last coefficient is well defined using the properties of the formal group law and the equations (3.5) and (3.6). More precisely, the quotient \({c_{1}^{D}}(L_{-\alpha })/{c_{1}^{D}}(L_{-2\alpha })\) has the same coefficients as \(\rho _{1/2}{c_{1}^{D}}(L_{\alpha })\) and the only difference will be the variable \({c_{1}^{D}}(L_{-2\alpha })\) in the first quotient as opposed to \({c_{1}^{D}}(L_{\alpha })\) in the second one. As we consider the reduction modulo \({c_{1}^{D}}(L_{\alpha })^{2}\), we only need to take the first two summands of \({c_{1}^{D}}(L_{-\alpha })/{c_{1}^{D}}(L_{-2\alpha })\) into account. Therefore, \({c_{1}^{D}}(L_{-\alpha })/{c_{1}^{D}}(L_{-2\alpha })\) differs from \(\rho _{1/2}{c_{1}^{D}}(L_{\alpha })\) only by a factor of − 2 in the second summand. We consider the difference
which is a product containing a factor \({c_{1}^{D}}(L_{\alpha })(f_{z}-f_{x})\). This implies that the above difference vanishes modulo \({c_{1}^{D}}(L_{\alpha })^{2}\) because of the equation (3.5) which finishes the argument.
Next, we consider the case \(\mathbb {F}_{0}=\mathbb {P}^{1}\times \mathbb {P}^{1}\) for which we choose Uw to be an open D-stable neighbourhood of w = ([1 : 0];[1 : 0]). We get (t− 2b/a,t− 2v/u) for coordinates ([a : b];[u : v]) which implies that D acts linearly on Uw with weight − α. The class \([\mathbb {F}_{0}]\in {\Omega }^{\ast }_{D}(\mathbb {F}_{0})_{\mathbb {Q}}\) again maps to (1,1,1,1) and we want to compute the remaining images of the closures of the Bialynicki-Birula cells.
Therefore, we take the closure (wx) of one of the remaining Bialynicki-Birula cells. We choose f(b/a,v/u) = b/a to be an eigenfunction of D of weight α. By Corollary 2.13 we obtain
in \({\Omega }^{\ast }_{D}(U_{w})_{\mathbb {Q}}\). Pulling this relation back yields \(i^{\ast }_{w}[(wx)\cap U_{w}\to U_{w}]={c_{1}^{D}}(L_{\alpha })\). With the eigenfunction f(b/a,u/v) = b/a and an open D-stable neighbourhood Ux of the fixed point x we obtain \(i^{\ast }_{x}[(wx)\cap U_{x}\to U_{x}]={c_{1}^{D}}(L_{\alpha })\).
For the pullbacks of (wy) we take the eigenfunction f(b/a,v/u) = v/u of D of weight α on the open D-stable Uw from above, but in this case we have V (f) = (wy) ∩ Uw and therefore, \(i^{\ast }_{w}[(wy)\cap U_{w}\to U_{w}]={c_{1}^{D}}(L_{\alpha })\). Similarly, we obtain \(i^{\ast }_{y}[(wy)\cap U_{y}\to U_{y}]={c_{1}^{D}}(L_{\alpha })\).
Lastly, we consider the pullback of the point w which is again the complete intersection of (wy) and (wx). By the same argument as in the above cases, we get
whereas the other pullbacks of the class of the point w vanish. We summarise that the image of i∗ is determined by the image of the basis, displayed below
which satisfies the equations (3.1) and (3.2).
Conversely, for an element \((f_{w},f_{x},f_{y},f_{z})\in S(T)^{4}_{\mathbb {Q}}\) fulfilling the equations we have
which completes the proof for the case \(\mathbb {F}_{0}\).
In the following, we consider the case \(\mathbb {F}_{n}\) for n ≥ 1. The class \([\mathbb {F}_{n}]\in {\Omega }^{\ast }_{D}(\mathbb {F}_{n})_{\mathbb {Q}}\) is again mapped to (1,1,1,1).
Now we compute the remaining pullbacks of the closures of the Bialynicki-Birula cells. We choose again an open D-stable neighbourhood Uw of the fixed point w = [0 : 0 : 1 : 0 : ... : 0]. The induced D-action on Uw is given by (t−nx0/y0,t− 2y1/y0) for coordinates [x0 : x1 : y0 : y1 : ... : yn+ 1] and therefore, D acts linearly on Uw with weights − nα/2 and − α. We choose f(x0/y0,y1/y0) = y1/y0 to be an eigenfunction of D of weight α. By the relations on the coordinates in \(\mathbb {F}_{n}\) we obtain V (f) = (wx) ∩ Uw with the given notations of the D-fixed points. As above, we get \(i^{\ast }_{w}[(wx)\cap U_{w}\to U_{w}]={c_{1}^{D}}(L_{\alpha })\). One may observe that the pullback does not depend on the choice of coordinates for Uw. For the point x = [1 : 0 : ... : 0] and a D-stable nighbourhood Ux we choose the eigenfunction f(x1/x0,y1/x0) = x1/x0 of D of weight α which leads to \(i^{\ast }_{x}[(wx)\cap U_{x}\to U_{x}]={c_{1}^{D}}(L_{\alpha })\).
For the pullback of (xy) let Ux be given by coordinates (y0/x0,x1/x0) and take the eigenfunction f(y0/x0,x1/x0) = y0/x0 of weight − nα/2 which leads to V (f) = (xy) ∩ Ux and therefore, to \(i^{\ast }_{x}[(xy)\cap U_{x}\to U_{x}]={c_{1}^{D}}(L_{-n\alpha /2})\). For the coordinates (y0/x1,yn+ 1/x1) for Uy and the eigenfunction f(y0/x1,yn+ 1/x1) = yn+ 1/x1 we get V (f) = (xy) ∩ Uy and hence, \(i^{\ast }_{y}[(xy)\cap U_{y}\to U_{y}]={c_{1}^{D}}(L_{n\alpha /2})\).
Finally, we consider the pullback of the point w by introducing an eigenfunction on (wx) ∩ Uw. We choose g(x0/y0,y1/y0) = x0/y0 which is an eigenfunction of weight nα/2. This leads to V (g) = (wz) ∩ (wx) ∩ Uw = w and thus, we obtain
in \({\Omega }^{\ast }_{D}(U_{w})_{\mathbb {Q}}\) again by Corollary 2.13.
We conclude \(i_{w}^{\ast }[w\to U_{w}]={c_{1}^{D}}(L_{n\alpha /2}){c_{1}^{D}}(L_{\alpha })\). Thus, we obtain the image of i∗ which is determined by the image of the basis, displayed below
satisfying the equations (3.3) and (3.4).
Conversely, let \((f_{w},f_{x},f_{y},f_{z})\in S(T)^{4}_{\mathbb {Q}}\) be an element fulfilling the equations. This leads to
which completes the proof in the case of \(\mathbb {F}_{n}\) because of the equations (3.3) and (3.4) and the above mentioned fact that an element which is divisible by \({c_{1}^{D}}(L_{\alpha })\) will be also divisible by \({c_{1}^{D}}(L_{n\alpha /m})\) for \(n\in \mathbb {Z}\setminus \{0\}\) and \(m\in \mathbb {Z}_{\geq 1}\) if \(\frac {n\alpha }{m}\) is a character in M since we consider rational coefficients. □
Remark 3.8
The equations given in Proposition 3.6 reduce to Brion’s equations given in [4, Proposition 7.2] for rational equivariant Chow rings. In order to be able to compute rational equivariant cobordism rings one has to consider the universal formal group law and not the additive formal group law which simplifies the computations in the Chow group case.
Next, we want to prove Theorem 3.4 which is a refinement of [4, Theorem 7.3]. The proof in the Chow group case (cf. [4, Theorem 2.1]) proceeds using explicit generators and relations which are not available for algebraic cobordism (cf. Remark 2.15). Instead, our proof will use some known results on T-filtrable varieties and their equivariant algebraic cobordism rings.
Proof Proof of Theorem 3.4
In order to compute the ring structure of \({\Omega }_{T}^{\ast }(X)_{\mathbb {Q}}\) we need to apply Proposition 2.19 to the given variety X. Due to Proposition 3.3 we know which fixed point subschemes \(X^{T^{\prime }}\) can occur and therefore, we distinguish between codimension one subtori \(T^{\prime }\) with \(\dim X^{T^{\prime }}\leq 1\) and those with \(\dim X^{T^{\prime }}=2\).
Recall that for a subtorus \(T^{\prime }\) with \(\dim X^{T^{\prime }}\leq 1\), there are only finitely many T-stable curves in \(X^{T^{\prime }}\) and furthermore, in the setting of a smooth projective spherical G-variety X, we only have finitely many T-fixed points by [5, Lemma 2.2]. This implies that the assumptions of Proposition 2.21 are fulfilled and thus, we can apply Proposition 2.21 to \(X^{T^{\prime }}\) which leads to case (i).
Now we consider the case where \(\dim X^{T^{\prime }}=2\) for which we know that \(X^{T^{\prime }}\) is either a projective plane or a Hirzebruch surface \(\mathbb {F}_{n}\). The T-orbits in \(X^{T^{\prime }}\) are always one-dimensional and thus, the surfaces occurring in (ii)–(iv) must consist of infinitely many T-stable curves. For these cases we need some different results. We claim that \({\Omega }_{\ast }^{T}(X^{T^{\prime }})_{\mathbb {Q}}\cong {\Omega }_{\ast } ^{T/F(\alpha )}(X^{T^{\prime }})_{\mathbb {Q}}\) holds where F(α) is given as in Remark 3.2. We will use [19, Theorem 4.7] in order to prove our claim. This theorem states that we have an isomorphism of S(T)-modules \({\Omega }_{\ast }^{T}(X^{T^{\prime }})\cong {\Omega }_{\ast }(X^{T^{\prime }})[[t_{1},...,t_{r}]]_{\text {gr}}\) where r is the rank of T and ti corresponds to \({c_{1}^{T}}(L_{\chi _{i}})\) for a basis {χ1,...,χr} of the character group of T which is chosen such that the character group of T/F(α) is given by {a1χ1,...,arχr} for positive integers a1|⋯|ar. We remark that \(X^{T^{\prime }}\) is also a T/F(α)-filtrable variety as F(α) acts trivially on \(X^{T^{\prime }}\) and therefore, the T-action factors through a T/F(α)-action. As above, we obtain the isomorphism \({\Omega }_{\ast }^{T/F(\alpha )}(X^{T^{\prime }})\cong {\Omega }_{\ast }(X^{T^{\prime }})[[t_{1},...,t_{r}]]_{\text {gr}}\) of S(T/F(α))-modules where ti here corresponds to \(c_{1}^{T/F(\alpha )}(L_{a_{i}\chi _{i}})\), but as we are considering rational coefficients we have \(S(T)_{\mathbb {Q}}\cong S(T/F(\alpha ))_{\mathbb {Q}}\) by Lemma 2.16. This implies the claim and using the same argument for the torus \(T^{\prime }\times T(\alpha )\) we obtain
where \({U^{1}_{j}}\) and \({U^{2}_{j}}\) are the corresponding parts of the sequences of good pairs \(\{({V^{1}_{j}},{U^{1}_{j}})\}_{j\geq 0}\) and \(\{({V^{2}_{j}},{U^{2}_{j}})\}_{j\geq 0}\) for T(α) and \(T^{\prime }\), respectively. In this case we know that \({U_{j}^{2}}/T^{\prime }\) are products of projective spaces by the choice of good pairs in the proof of [19, Lemma 6.1]. As a product of projective spaces, the \({U_{j}^{2}}/T^{\prime }\) are cellular which means that we can use a special version of a Künneth formula (cf. [12, Proposition 7]) from line 4 to 5. The ordinary cobordism vanishes for negative degrees and therefore the inverse limit and the finite sum commute in our setting. We conclude the proof by Proposition 3.6. □
Remark 3.9
Proposition 2.22 leads to an abstract description of the rational ordinary algebraic cobordism ring of any smooth projective and spherical G-variety X. Using this result, we would be able to describe \({\Omega }^{\ast }(X)_{\mathbb {Q}}\) explicitly if we could compute all the classes in \({\Omega }_{T}^{\ast }(X)_{\mathbb {Q}}\). We will come back to this problem in Section 5.
Now we want to have a look at the specific example of IG(2,5) for which we can use Theorem 3.4 in order to compute its rational equivariant cobordism ring.
Example 3.10
Let V = k5 be given with standard basis e1,...,e5. Recall (cf. [26, Section 1]) that the odd symplectic Grassmannian X = IG(2,5) is given by
where ω is an antisymmetric form of maximal rank on V and Gr(2,5) denotes the usual Grassmannian embedded into \(\mathbb {P}(\bigwedge ^{2} k^{5})\) via the Plücker embedding. We know (cf. [26, Section 1]) that the odd symplectic Grassmannian does not depend on the given form ω and thus, we choose the form
which has kernel e3. It is well known (cf. [25, Theorem 0.1]) that all odd symplectic Grassmannians are smooth, projective and horospherical (cf. Definition 4.11). We will consider the natural torus action of \(T\subseteq \text {Sp}_{4}\) on IG(2,5) which leads to eight T-fixed points in IG(2,5). These are given by
in \(\mathbb {P}(\bigwedge ^{2} k^{5})\) whereas the points x15 = [e1 ∧ e5] and x24 = [e2 ∧ e4] are not in IG(2,5).
The positive roots of (Sp4,T) are given by ε1 − ε2,ε1 + ε2,2ε1 and 2ε2 and thus, the identity component of the kernel of a given positive root is a singular codimension one subtorus. A short computation shows that the fixed point subschemes \(X^{\text {Ker}(\varepsilon _{1}-\varepsilon _{2})^{0}}\) and \(X^{\text {Ker}(\varepsilon _{1}+\varepsilon _{2})^{0}}\) consist of three T-stable curves and the remaining two isolated fixed points, respectively. Lastly, the fixed point subschemes \(X^{\text {Ker}(2\varepsilon _{1})^{0}}\) and \(X^{\text {Ker}(2\varepsilon _{2})^{0}}\) consist of two projective planes \(\mathbb {P}^{2}\) and a T-stable curve, respectively.
The other codimension one subtori are regular. Those are given by \(T^{\prime }=\text {Ker}(\chi )^{0}\) for some primitive character χ of T which is not a multiple of a root and further, they will not contribute to the computations of cobordism since one may verify that \(X^{T^{\prime }}=X^{T}\) holds for those \(T^{\prime }\).
These precise descriptions of the fixed point subschemes lead to the equations describing the image of \(i^{\ast }: {\Omega }_{T}^{\ast }(\text {IG}(2,5))_{\mathbb {Q}}\to {\Omega }_{T}^{\ast }((\text {IG}(2,5))^{T})_{\mathbb {Q}}\). Using Theorem 3.4 the equations are given by
These equations give a complete description of the rational equivariant algebraic cobordism ring of IG(2,5).
Furthermore, we would like to identify the elements in the algebra \({\Omega }_{T}^{\ast }((\text {IG}(2,5))^{T})_{\mathbb {Q}}\) with geometric T-stable cobordism cycles in \({\Omega }_{T}^{\ast }(\text {IG}(2,5))_{\mathbb {Q}}\). As an example, we consider the T-stable projective plane \(\mathbb {P}^{2}_{14,34,45}\) which is defined to be the projective plane in the connected component of \(X^{\text {Ker}(2\varepsilon _{1})^{0}}\) containing the fixed points x14,x34 and x45 in IG(2,5). This leads to
for some \(n_{1},n_{2},n_{3}\in S(T)_{\mathbb {Q}}\) of degree zero. We do not know at this point which particular choice of the ni determines the pullback of the class \([\mathbb {P}^{2}_{14,34,45}\to \text {IG}(2,5)]\), but one of those tuples is certainly its image under the pullback map i∗. We will come back to this problem in the sequel of this article (cf. Section 5).
4 Equivariant Cobordism of Horospherical Varieties of Picard Rank One
In this section, we let G be a connected reductive linear algebraic group, B a fixed Borel subgroup with maximal torus T and W = N(T)/T the Weyl group. We want to compute the equivariant algebraic cobordism of smooth projective horospherical varieties of Picard number one. We begin this section by recalling the description of the T-stable curves in flag varieties (cf. [8]) which will be important in order to describe the geometry of horospherical varieties. After that, we will define horospherical varieties and recall some of their basic notions as well as their geometry. Excellent references for the geometry of horospherical varieties are for example [9, 25]. Using these descriptions, we will be able to describe the rational equivariant cobordism ring of horospherical varieties of Picard number one in terms of Theorem 3.4.
4.1 T-Stable Curves in Flag Varieties
In this section, we will recall the main notions and results on T-stable curves in flag varieties G/P from [8, Section 3]. We denote by R = R+ ∪ R− the positive and negative roots and by S the simple roots. Furthermore, we denote by sα the reflections in W which are indexed by positive roots α. These are simple reflections if α is in S. For a subset \(I\subseteq S\), let WI be the subgroup of W which is generated by the reflections sα for α in I. In addition, let \(P_{I}={\coprod }_{w\in W_{I}}BwB\) and \(R^{+}_{P_{I}}\) be the set of positive roots that can be written as sums of roots in I. This is the well known correspondence between parabolic subgroups PI of G containing B and subsets \(I\subseteq S\). The length ℓ(w) of an w ∈ W is the minimum number of simple reflections whose product is w.
For any u ∈ W/WI we let \(X(u)=\overline {BuP_{I}/P_{I}}\) be the corresponding Schubert variety which is of dimension ℓ(u) where ℓ(u) denotes the unique minimum length of a representative of u in W. We denote its cohomology class [X(u)] ∈ H∗(G/PI) by σ(u). Furthermore, for any u ∈ W/WI we denote by x(u) = uPI/PI the corresponding T-fixed point in G/PI. The Schubert classes of dimension one have the form σ(sβ) as β varies over S ∖ I. We define a degree d to be a non-negative integral combination \(d=\sum d_{\beta } \sigma (s_{\beta })\). The degrees are the classes of curves on G/PI. For any positive root α, we write \(\alpha =\sum n_{\alpha \beta }\beta \) as the positive integral combination of simple roots β. Then we define the degree d(α) of α by
Remark 4.1
If hα = 2α/(α,α) and ωβ is the fundamental weight corresponding to β, then hα(ωβ) = nαβ(β,β)/(α,α) which implies
Lemma 4.2
[8, Lemma 3.1] If w is in WI, then we have d(w(α)) = d(α).
For any positive root α which is not in \(R^{+}_{P_{I}}\), there is a unique T-stable curve Cα in G/PI that contains the points x(1) and x(sα). We know that Cα = S(α) ⋅ PI/PI where S(α) is the 3-dimensional subgroup of G whose Lie algebra is \(\mathfrak {g_{\alpha }}\oplus \mathfrak {g_{-\alpha }}\oplus [\mathfrak {g_{\alpha }},\mathfrak {g_{-\alpha }}]\).
Lemma 4.3
[8, Lemma 3.4] The degree [Cα] of Cα is d(α).
Definition 4.4
We say that two unequal elements u and v in W/WI are adjacent if there is a reflection sα in W for α ∈ R+ such that v = sαu. In this case we define d(u,v) to be the degree d(α).
Remark 4.5
If u and v are adjacent, then for any w ∈ W, the elements wu and wv are also adjacent and d(wu,wv) = d(u,v) holds.
Lemma 4.6
[8, Lemma 4.2] Elements u and v in W/WI are adjacent if and only if x(u)≠x(v) and there is a T-stable curve C connecting x(u) and x(v). In this case, the curve C is unique, isomorphic to \(\mathbb {P}^{1}\) and its degree is equal to d(u,v).
Remark 4.7
A general T-stable curve in G/PI has the form w ⋅ Cα for some \(\alpha \in R^{+}\setminus R^{+}_{P_{I}}\) and w ∈ W. This curve is the unique T-stable curve connecting x(w) = w ⋅ x(1) and x(wsα) = w ⋅ x(sα).
Example 4.8
We consider the flag variety G2/Pα where α and β denote the simple roots of G2, β being the long root. The flag variety G2/Pα is a 5-dimensional quadric whose geometry was also studied in [9]. The positive roots are given by
Furthermore, we know that Wα is generated by sα which has order 2. Thus, we have 6 T-fixed points in G2/Pα and they are indexed by the elements of W/Wα. From the above we know that for any \(\gamma \in R^{+}\setminus R^{+}_{P_{\alpha }}\) there exists a unique T-stable curve connecting x(1) and x(sγ). A short computation shows that any two distinct elements of W/Wα are adjacent. This implies that there is a T-stable curve connecting any two of the T-fixed points in G2/Pα which leads to a total of 15 T-stable curves in the flag variety G2/Pα. Similar computations can be done for the flag variety G2/Pβ.
Later on, we will be interested in the weight of a T-stable curve C and also its degree. To obtain those one can use the following lemma.
Lemma 4.9
[8, Lemma 2.1] Let a torus T act on a curve \(C\cong \mathbb {P}^{1}\) with two different T-fixed points p and q and let L be a T-equivariant line bundle on C. Let χp and χq be the weights of T acting on the fibers Lp and Lq, respectively, and ψp the weight of T acting on the tangent space of C at p. Then we have
where n is the degree of L on C.
Remark 4.10
In our case, and more specifically in the previous Example 4.8, the degree can also be obtained by Remark 4.1. These computations lead to 6 T-stable curves of degree 1, 6 T-stable curves of degree 3 and 3 T-stable curves of degree 2 in the flag variety G2/Pα in Example 4.8.
4.2 Geometry of Horospherical Varieties of Picard Number One
In this section, we focus on the class of horospherical varieties which is a special case of spherical varieties. We give two equivalent definitions and refer to [9, 23,24,25] for more details on the geometry of horospherical varieties.
Definition 4.11
Let X be a normal G-variety.
-
(i)
Let \(H\subseteq G\) be a closed subgroup containing the unipotent radical U of B. In this case, the homogeneous space G/H is said to be horospherical.
-
(ii)
We call X horospherical if it contains an open orbit isomorphic to a horospherical homogeneous space.
Remark 4.12
The Bruhat decomposition implies that this open orbit isomorphic to G/H contains an open orbit under the action of the Borel subgroup and therefore, a horospherical variety is spherical.
Now we give the second definition of horospherical varieties using a more geometric description.
Remark 4.13
A horospherical homogeneous space G/H can be equivalently described as a torus bundle over a flag variety G/P with fiber P/H. In this situation we have P = NG(H) by [24, Proposition 2.2]. Furthermore, one has P = TH = BH for all maximal tori T and all Borel subgroups B contained in P.
Definition 4.14
For a horospherical homogeneous space G/H, we call the dimension of the fiber P/H the rank of G/H. Furthermore, for a horospherical variety X, the rank of X is defined as the rank of its open G-orbit.
In this article, we focus on smooth projective horospherical varieties of Picard number one which have been classified by Pasquier [25] in the following theorem.
Theorem 4.15
[25, Theorem 0.1] Let G be a connected reductive algebraic group. Let X be a smooth projective horospherical G-variety with Picard number one. Then one of the following cases can occur
-
(i)
X is homogeneous.
-
(ii)
X is horospherical of rank 1. Its automorphism group is a connected non-reductive linear algebraic group, acting with exactly two orbits.
Moreover, in the second case X is uniquely determined by its two closed G-orbits Y and Z, isomorphic to G/PY and G/PZ, respectively, and (G,PY,PZ) is one of the triples of the following list.
-
(1)
(Bn,P(ωn− 1),P(ωn)) for n ≥ 3
-
(2)
(B3,P(ω1),P(ω3))
-
(3)
(Cn,P(ωm),P(ωm− 1)) for n ≥ 2 and m ∈ [2,n]
-
(4)
(F4,P(ω2),P(ω3))
-
(5)
(G2,P(ω1),P(ω2)
Here we denote by P(ωi) the maximal parabolic subgroup of G corresponding to the fundamental weight ωi where we use the notations from Bourbaki [3].
Remark 4.16
In our notation P(ωi) will always be the maximal parabolic subgroup \(P_{S\setminus \alpha _{i}}\) for the simple root αi associated to the fundamental weight ωi.
Lemma 4.17
[25, Lemma 1.2] Let G/H be a horospherical homogeneous space. Up to isomorphism of varieties, there exists at most one smooth projective G/H-embedding with Picard number one.
In the sequel, we will be only interested in the cases which are not homogeneous because the cobordism for homogeneous varieties has been studied before in [17]. Therefore we recall the construction from [9, Section 1.3].
Let X be a smooth projective horospherical but non homogeneous variety of Picard number one with associated triple (G,PY,PZ). In this case, we denote the previous triple also by (G,P(ωY),P(ωZ)) for the corresponding fundamental weights ωY and ωZ. Furthermore, the dense orbit is given by \(G/H=G\cdot [v_{Y}+v_{Z}]\subseteq \mathbb {P}(V_{Y}\oplus V_{Z})\) where VY and VZ are the irreducible G-representations with highest weights ωY and ωZ and the corresponding highest weight vectors vY and vZ. We conclude by the construction that PY and PZ are the stabilisers of [vY] and [vZ] in \(\mathbb {P}(V_{Y})\) and \(\mathbb {P}(V_{Z})\) and that Y and Z are the G-orbits of [vY] and [vZ] in \(\mathbb {P}(V_{Y})\) and \(\mathbb {P}(V_{Z})\), respectively.
The T-fixed points of X are given by the T-fixed points of the two closed G-orbits. Now, we will be analysing the T-stable curves and the fixed point subschemes \(X^{T^{\prime }}\) for some given X in order to be able to use Theorem 3.4 to obtain the rational equivariant cobordism of X. In the previous section, we have already seen how to determine the T-stable curves in the closed orbits G/PY and G/PZ which are flag varieties. Next, we will analyse the T-stable curves meeting the dense open orbit G/H for any smooth projective horospherical variety X of Picard number one. We will use the diagram
where π is the corresponding \(\mathbb {C}^{\ast }\)-bundle.
Definition 4.18
Let C be a T-stable irreducible curve in the dense open orbit G/H. Then we define S := π− 1(π(C)) to be the preimage of π(C).
Lemma 4.19
Let C be a T-stable irreducible curve in the dense open orbit G/H. Then S is given by one of the following cases.
-
(i)
S is the curve C itself.
-
(ii)
S is a surface containing C.
Proof
Let C be a given T-stable irreducible curve in the dense open orbit G/H. Then, π(C) is also T-stable. The following two cases can occur for π(C).
-
(i)
π(C) = {∗} is a point. Without loss of generality, we can choose this point to be the B-fixed point in G/P := G/(PY ∩ PZ) where PY ∩ PZ is the same as the normaliser NG(H) which was mentioned in Remark 4.13. The B-fixed point is 1 ⋅ P/P and therefore the closure \(\overline {C}\subseteq X\) of the fiber π− 1(1 ⋅ P/P) = C is the line joining the B-fixed points 1 ⋅ PY/PY = [vY] ∈ Y and 1 ⋅ PZ/PZ = [vZ] ∈ Z because π is a \(\mathbb {C}^{\ast }\)-bundle and B-fixed points are mapped to B-fixed points via the projections pY and pZ. The other lines will be obtained by the Weyl group action. Those lines are T-stable by assumption.
-
(ii)
π(C) is a T-stable irreducible curve. Without loss of generality, we can choose π(C) = S(α) ⋅ P/P for some positive root α which is not in \(R^{+}_{P}\) where S(α) is the 3-dimensional subgroup of G whose Lie algebra is \(\mathfrak {g_{\alpha }}\oplus \mathfrak {g_{-\alpha }}\oplus [\mathfrak {g_{\alpha }},\mathfrak {g_{-\alpha }}]\). This curve joins the B-fixed point x(1) and x(sα) = sα ⋅ P/P. Then we obtain a surface S := π− 1(π(C)) because π is a \(\mathbb {C}^{\ast }\)-bundle. This surface S contains C and is T-stable since π is equivariant. The other curves are obtained by the Weyl group action.
□
Lemma 4.20
Any surface in a connected component of \(X^{T^{\prime }}\) for a singular codimension one subtorus \(T^{\prime }=\text {Ker}(\alpha )^{0}\) for some positive root α is of the form \(\overline {S}\subseteq X\) for some T-stable curve C and S = π− 1(π(C)).
Proof
Let A be a surface in \(X^{T^{\prime }}\). This is itself a connected component of \(X^{T^{\prime }}\) and A is a spherical \(C_{G}(T^{\prime })\)-variety by Proposition 3.3. We know that A ∩ G/H≠∅ because the two closed orbits Y ≅G/PY and Z≅G/PZ contain only finitely many T-stable curves. Now let a ∈ A. Then we have \(ta=tt^{\prime }a=t^{\prime }ta\) for all \(t,t^{\prime }\in T\) which implies that A is T-stable. We have \(A\cap G/H\subseteq \pi ^{-1}\pi (A\cap G/H)\) and the reversed inclusion is also true because A is T-stable and T acts transitively on the fibers of π as P/H is a quotient of T. Therefore, the whole fiber must be in A ∩ G/H. Furthermore, A has only zero- and one-dimensional T-orbits since \(T/T^{\prime }\) is one-dimensional. The image under π of those orbits is either a T-fixed point or the T-stable irreducible curve π(A ∩ G/H). We conclude that there must be a T-stable curve \(C\subseteq A\cap G/H\) such that A ∩ G/H = π− 1π(C) because if the T-orbits were only the fibers then there would be infinitely many T-fixed points in G/P. □
In the following, we want to analyse which surfaces S are a connected component in some \(X^{T^{\prime }}\) for some codimension one subtorus \(T^{\prime }\). Therefore, we formulate the following lemma.
Definition 4.21
Let X be a smooth projective horospherical G-variety of Picard number one of the form (G,P(ωY),P(ωZ)). Then we denote by χ := ωY − ωZ the difference of the two fundamental weights ωY and ωZ.
Lemma 4.22
For any smooth projective horospherical variety X of Picard number one we have the following properties.
-
(1)
The only T-stable curves in X meeting the open orbit G/H occurring as a connected component of \(X^{T^{\prime }}\) for some codimension one subtorus \(T^{\prime }\) are of the form \(\overline {\pi ^{-1}(z)}\) where z ∈ G/(PY ∩ PZ) is a T-fixed point.
-
(2)
The surfaces occurring in \(X^{T^{\prime }}\) only arise from codimension one subtori of the form \(T^{\prime }=\text {Ker}(w\alpha )^{0}=\text {Ker}(w\chi )^{0}\) for some positive root α which is a non-zero multiple of χ and some w ∈ W.
Proof
As above, we have the B-fixed point 1 ⋅ P/P in G/P := G/(PY ∩ PZ). We need to consider the previously discussed case from Lemma 4.19 (ii). Therefore, we assume that there exists a T-stable curve \(C\subseteq G/H\) such that a general point in the T-stable curve π(C) has the form w ⋅ u−α(x) ⋅ P/P where u−α(x) denotes the corresponding element in the root subgroup U−α. A general point in S = π− 1π(C) has the form
for \(t\in P/H=\mathbb {C}^{\ast }\). Now we consider the T-action on those points for z ∈ T:
This implies that a point z acts trivially if and only if w− 1zw ∈ H = Ker(χ) and z ∈Ker(wα) hold. This implies by the Weyl group action on the character group that this is equivalent to z ∈Ker(wχ) ∩Ker(wα). Since we only investigate the actions of subtori of T on general points in S, we can deduce that z must be in Ker(wχ)0 ∩Ker(wα)0.
If Ker(wχ)0≠Ker(wα)0 holds, then Ker(wχ)0 ∩Ker(wα)0 has codimension two in T. Therefore, we obtain a T-stable surface S or a T-stable curve in π− 1π(C). It remains to check whether those are fixed by some codimension one subtorus. If the closure of one of those was a connected component of \(X^{T^{\prime }}\), then the stabiliser of any point in \(X^{T^{\prime }}\) would have at most codimension one in T, but the stabiliser of a general point in S and therefore also in every potential T-stable curve in π− 1π(C) is precisely Ker(wχ) ∩Ker(wα). Therefore, the stabiliser of a general point would be of codimension two in T and thus, the closure of the T-stable surface S is not a connected component of \(X^{T^{\prime }}\) and there exists no T-stable curve in π− 1π(C) whose closure is a connected component of \(X^{T^{\prime }}\).
If \(\text {Ker}(w\chi )^{0}=\text {Ker}(w\alpha )^{0}=T^{\prime }\) holds, then the closure of S is some connected component of \(X^{T^{\prime }}\) because z acts trivially on a general point of S. This implies property (2) because in this case α must be a non-trivial multiple of χ.
Now, let \(\overline {C}\) be a T-stable curve in \(X^{T^{\prime }}\) for some codimension one subtorus \(T^{\prime }\subseteq T\). If \(\overline {C}\) is in one of the two closed orbits, then we know that it is a T-stable curve in some flag variety and thus, it is a connected component of \(X^{T^{\prime }}\). Otherwise, the curve \(\overline {C}\) meets the open orbit G/H along a T-stable curve C. By Lemma 4.19, C is either a fiber of π or C is contained in the T-stable surface S = π− 1π(C). In the latter case, we distinguish whether α is non-trivial multiple of χ or not. As discussed above, there exists no T-stable curve in S whose closure is a connected component of \(X^{T^{\prime }}\) if α is not a non-trivial multiple of χ. We also showed above that the closure of S is itself a connected component of \(X^{T^{\prime }}\) containing C if α is a non-trivial multiple of χ. We conclude that \(\overline {C}\) is never a connected component of \(X^{T^{\prime }}\) if C is contained in the T-stable surface S. This implies property (1). □
Algorithm
We analyse the occurring surfaces in \(X^{T^{\prime }}\). As we have seen above, we need to consider roots α which are non-zero multiples of the difference χ of the two fundamental weights ωY and ωZ up to the Weyl group action. After that, we look at the curves in the closed orbits Y and Z. Up to Weyl group action, these are given by \(S(\alpha ) [v_{\omega _{Y}}]\) which connect \(s_{\alpha }[v_{\omega _{Y}}]\) and \([v_{\omega _{Y}}]\) in Y and similarly in Z. Thus, we need to compute sα(ωY) = ωY − (α∨,ωY)α and similarly for ωZ. Then we will know how many T-fixed points we have in this connected component of \(X^{T^{\prime }}\) and in which orbits they occur. If we obtain 3 T-fixed points then we obtain a projective plane and if we obtain 4 T-fixed points, then we will have a Hirzebruch surface \(\mathbb {F}_{n}\). We remark that sα(ωY) = ωY holds if and only if (α∨,ωY) vanishes. Now it remains to be checked which Hirzebruch surface \(\mathbb {F}_{n}\) we obtain in the case of 4 T-fixed points. For details concerning the geometry of Hirzebruch surfaces as ruled surfaces we refer the reader to [11, Chapter V]. Let \(S=\mathbb {F}_{n}\). Then the two closed SL2-orbits are T-stable curves, and we may assume that their degrees x and y satisfy x ≤ y. Now we want to determine the non-negative integer n and the embedding of S. Let \(p:S\to \mathbb {P}^{1}\) be the ruling. Then p has a section C0 such that the self-intersection is \({C_{0}^{2}}=-n\). It is furthermore well known that this is the only section of p with negative (non-positive) self-intersection for n > 0 (n = 0). Let f be a fiber of p. We know that the Picard group of S is generated by C0 and f. Furthermore, we have f.C0 = 0 and f2 = 0. Geometrically, the morphism p describes the projection to one of the closed SL2-orbits. Thus, we may choose f to be the line \([\lambda v+\mu v^{\prime }]\subseteq \mathbb {P}(V\oplus V^{\prime })\) where v ∈ V and \(v^{\prime }\in V^{\prime }\) are the highest weights vectors. So let L = aC0 + bf be the line bundle defining the embedding of S. Following [11, Chapter V, Corollary 2.18] we know that L is very ample if and only if a > 0 and b > an. In addition, we have two special sections of p which are the two closed SL2-orbits, i.e. the curves C and \(C^{\prime }\) of degrees x and y, respectively. As they are sections of p, the intersection with f is 1 and therefore, they are of the form C = C0 + cf and \(C^{\prime }=C_{0}+c^{\prime }f\). For the embedding \(\phi :S\to \mathbb {P}(V\oplus V^{\prime })\) we know that \(L=\phi ^{\ast } (\mathcal {O}(1))\). Further, for a general hyperplane H in \(\mathbb {P}(V\oplus V^{\prime })\) we have H.f = 1 because f is a line in \(\mathbb {P}(V\oplus V^{\prime })\). We also have H.C = x and \(H.C^{\prime }=y\) in \(\mathbb {P}(V\oplus V^{\prime })\) by the assumption on the degrees of the curves C and \(C^{\prime }\). Considering now the line bundle L on S, we compute (H ∩ S).f = 1 for the divisor H ∩ S whose class corresponds to the line bundle L. This can then be also written as L.f = 1. Thus, we have 1 = L.f = (aC0 + bf).f = a. Similarly, we can pull \(\mathcal {O}(1)\) back to \(\mathbb {P}(V)\) and \(\mathbb {P}(V^{\prime })\) which are the curves C and \(C^{\prime }\), respectively. We obtain (H ∩ C).C = x and \((H\cap C^{\prime }).C^{\prime }=y\) which can be written as L.C = x and \(L.C^{\prime }=y\). Further, we have \(C.C^{\prime }=0\) because the two closed SL2-orbits do not meet. On the other hand, we have
which leads to
These observations yield C2 = 2c − n = x − y ≤ 0. This leads to C = C0 because C0 is the unique irreducible curve with non-positive self-intersection (or x = y and n = 0). We conclude c = 0 and hence, n = y − x which also holds for n = 0 and y = x. Thus, the embedding is given by L = C0 + yf and the curve \(C^{\prime }\) is \(C^{\prime }=C_{0}+(y-x)f\).
We consider some examples of the classification of Pasquier which are given by triples (G,PY,PZ). We will study their geometry using the above algorithm and the classification of Bourbaki [3].
Example 4.23
In this example we will discuss three of the possible cases from Proposition 4.15.
-
(i)
Firstly, we consider type (1), i.e. (Bn,P(ωn− 1),P(ωn)) for n ≥ 3. The fundamental weights are given by
$$ \begin{array}{@{}rcl@{}} \omega_{n-1}&=&\varepsilon_{1}+...+\varepsilon_{n-1}\\&=&\alpha_{1}+2\alpha_{2}+...+(n-2)\alpha_{n-2}+(n-1)(\alpha_{n-1}+\alpha_{n})\text{ and }\\ \omega_{n}&=&1/2(\varepsilon_{1}+...+\varepsilon_{n})\\&=&1/2(\alpha_{1}+2\alpha_{2}+...+n\alpha_{n}) \end{array} $$for αi = εi − εi+ 1, 1 ≤ i ≤ n − 1, and αn = εn. Therefore, we have
$$ \begin{array}{@{}rcl@{}} \chi&=&\omega_{n-1}-\omega_{n}\\&=&1/2(\varepsilon_{1}+...+\varepsilon_{n-1}-\varepsilon_{n})\\&=&1/2(\alpha_{1}+2\alpha_{2}+...+(n-1)\alpha_{n-1}+(n-2)\alpha_{n}). \end{array} $$The positive roots are given by εi for 1 ≤ i ≤ n and εi ± εj for 1 ≤ i < j ≤ n. This implies that there will not be any surface in \(X^{T^{\prime }}\) because there is no root which is a non-zero multiple of χ.
-
(ii)
Secondly, we consider type (3), i.e. (Cn,P(ωm),P(ωm− 1)) with integers n ≥ 2 and m ∈ [2,n]. The fundamental weights are given by
$$ \begin{array}{@{}rcl@{}} \omega_{i}&=&\varepsilon_{1}+...+\varepsilon_{i}\\&=&\alpha_{1}+2\alpha_{2}+...+(i-1)\alpha_{i-1}+i(\alpha_{i}+\alpha_{i+1}+...+\alpha_{n-1}+\frac{1}{2}\alpha_{n}) \end{array} $$for 1 ≤ i ≤ n and αi = εi − εi+ 1, 1 ≤ i ≤ n − 1, and αn = 2εn. Therefore, we have
$$ \begin{array}{@{}rcl@{}} \chi&=&\omega_{m}-\omega_{m-1}\\&=&\varepsilon_{m}\\&=&\alpha_{m}+...+\alpha_{n-1}+\frac{1}{2}\alpha_{n}. \end{array} $$The positive roots are given by εi ± εj for 1 ≤ i < j ≤ n and 2εi for 1 ≤ i ≤ n. Thus, there is a positive root which is a non-zero multiple of χ namely α := 2εm. Consequently, we have
$$ \begin{array}{@{}rcl@{}} \alpha^{\vee}=\frac{2\alpha}{(\alpha,\alpha)}=\frac{2\cdot 2\varepsilon_{m}}{(2\varepsilon_{m},2\varepsilon_{m})}=\varepsilon_{m} \end{array} $$and therefore we obtain
$$ \begin{array}{@{}rcl@{}} (\alpha^{\vee},\omega_{m})=(\varepsilon_{m},\varepsilon_{1}+...+\varepsilon_{m})=1 \end{array} $$and
$$ \begin{array}{@{}rcl@{}} (\alpha^{\vee},\omega_{m-1})=(\varepsilon_{m},\varepsilon_{1}+...+\varepsilon_{m-1})=0. \end{array} $$This implies that we have 3 T-fixed points and that we obtain a projective plane in \(X^{T^{\prime }}\). We recover the odd symplectic Grassmannian IG(m,2n + 1) and thus, in particular Example 3.10 in the case m = n = 2.
-
(iii)
Lastly, we consider type (5), i.e. the triple (G2,P(ω1),P(ω2)). The fundamental weights are given by
$$ \begin{array}{@{}rcl@{}} \omega_{1}&=&-\varepsilon_{2}+\varepsilon_{3}\\&=&2\alpha_{1}+\alpha_{2}\text{ and }\\ \omega_{2}&=&-\varepsilon_{1}-\varepsilon_{2}+2\varepsilon_{3}\\ &=&3\alpha_{1}+2\alpha_{2} \end{array} $$for α1 = ε1 − ε2 and α2 = − 2ε1 + ε2 + ε3. Therefore, we have
$$ \begin{array}{@{}rcl@{}} \chi&=&\omega_{1}-\omega_{2}\\&=&\varepsilon_{1}-\varepsilon_{3}\\&=&-\alpha_{1}-\alpha_{2}. \end{array} $$The positive roots are given by α1,α2,α1 + α2,2α1 + α2,3α1 + α2 and 3α1 + 2α2. Thus, α := −χ is a positive root and consequently, we have
$$ \begin{array}{@{}rcl@{}} \alpha^{\vee}&=&\frac{2\alpha}{(\alpha,\alpha)}=\frac{2(\varepsilon_{3}-\varepsilon_{1})}{(\varepsilon_{3}-\varepsilon_{1},\varepsilon_{3}-\varepsilon_{1})}=\varepsilon_{3}-\varepsilon_{1}. \end{array} $$Therefore, we obtain
$$ \begin{array}{@{}rcl@{}} (\alpha^{\vee},\omega_{1})=(\varepsilon_{3}-\varepsilon_{1},-\varepsilon_{2}+\varepsilon_{3})=1 \end{array} $$and
$$ \begin{array}{@{}rcl@{}} (\alpha^{\vee},\omega_{2})=(\varepsilon_{3}-\varepsilon_{1},-\varepsilon_{1}-\varepsilon_{2}+2\varepsilon_{3})=3. \end{array} $$This implies that we have 4 T-fixed points and that we obtain a Hirzebruch surface \(\mathbb {F}_{2}\) by the Algorithm and Lemma 4.3 where the latter ensures that (α∨,ω1) and (α∨,ω2) give us the degrees of the curves in the two closed orbits Y and Z, respectively.
After having described the T-stable points and curves on these smooth projective horospherical varieties of Picard number one, we can describe their equivariant algebraic cobordism rings. This will be done using Theorem 3.4.
Example 4.24
Here, we will give the equivariant cobordism rings of the previous three cases. Therefore, we will in general consider as usual the injective map
-
(i)
At first, we consider the case (Bn,P(ωn− 1),P(ωn)) for n ≥ 3. For any element \(w^{\prime }\in W/W_{S\setminus \alpha _{n-1}}\) we denote by \(y(w^{\prime }):=w^{\prime }P(\omega _{n-1})/P(\omega _{n-1})\) the corresponding T-fixed point in Y and similarly by \(z(w^{\prime \prime }):=w^{\prime \prime }P(\omega _{n})/P(\omega _{n})\) the T-fixed point in the closed orbit Z for any \(w^{\prime \prime }\in W/W_{S\setminus \alpha _{n}}\). The equations for the closed orbits Y and Z are given by
$$ \begin{array}{@{}rcl@{}} f_{y(w\cdot s_{\alpha})}&\equiv& f_{y(w)}\qquad \mod {c_{1}^{T}}(L_{w\omega_{n-1}-ws_{\alpha}\omega_{n-1}}) \end{array} $$(4.2)$$ \begin{array}{@{}rcl@{}} f_{z(w\cdot s_{\beta})}&\equiv& f_{z(w)} \qquad \mod {c_{1}^{T}}(L_{w\omega_{n}-ws_{\beta}\omega_{n}}) \end{array} $$(4.3)for \(\alpha \in R^{+}\setminus R^{+}_{P(\omega _{n-1})},\beta \in R^{+}\setminus R^{+}_{P(\omega _{n})}\) and w ∈ W which is true as the difference of the weights associated to the T-fixed points is a multiple of the weight of the corresponding curve and we consider rational coefficients. We have seen above that there are no surfaces in this particular case. Therefore, the last equations are given by the T-stable curves joining the two closed orbits. These are given by
$$ \begin{array}{@{}rcl@{}} f_{y(w)}\equiv f_{z(w)} \mod {c_{1}^{T}}(L_{w\omega_{n-1}-w\omega_{n}}) \end{array} $$(4.4)for w ∈ W. This describes completely the equivariant algebraic cobordism \({\Omega }^{\ast }_{T}(X)_{\mathbb {Q}}\) in case (1).
-
(ii)
Secondly, we consider the case (Cn,P(ωm),P(ωm− 1)) for n ≥ 2 and m ∈ [2,n]. The equations from the curves in the closed orbits can be obtained as in (4.2) and (4.3). Furthermore, the equations from the T-stable curves joining the closed orbits can be obtained as in (4.4). As we have seen in Example 4.23, we need to choose α := 2εm to be the positive root which is a multiple of χ = ωm − ωm− 1 in order to obtain a surface in \(X^{T^{\prime }}\) for \(T^{\prime }=\text {Ker}(\alpha )^{0}\). The reflection sα acts trivially on the T-fixed point z(1) and therefore, we obtain the T-fixed points z(1),y(1) and y(sα) in \(X^{T^{\prime }}\). Having a look at the weights of the T-stable curves in the resulting surface \(\mathbb {P}^{2}\), we can identify the T-fixed points z(1),y(1) and y(sα) with y,x and z, respectively, where we consider the T-action on \(\mathbb {P}^{2}\) given by t ⋅ [x : y : z] = [tx : y : t− 1z]. For any w ∈ W this leads to the equation
$$ \begin{array}{@{}rcl@{}} (f_{y(w)}-f_{z(w)})+\rho_{1/2}{c_{1}^{T}}(L_{w\alpha})(f_{y(w\cdot s_{\alpha})}-f_{y(w)})\equiv 0 \mod {c_{1}^{T}}(L_{w\alpha})^{2}. \end{array} $$This completes the description of the equivariant algebraic cobordism in case (3). Furthermore, we remark that we recover precisely the description of the rational equivariant algebraic cobordism of IG(2,5) from Example 3.10 for m = n = 2.
-
(iii)
Lastly, we consider case (5) which is given by the triple (G2,P(ω1),P(ω2)) for ω1 = 2α1 + α2 and ω2 = 3α1 + 2α2. The curves can be described as above for the previous cases. In order to obtain surfaces in \(X^{T^{\prime }}\) we need to choose α := −χ by Example 4.23. Therefore, we obtain the T-fixed points y(1),y(sα),z(1) and z(sα) contained in a Hirzebruch surface \(\mathbb {F}_{2}\) which has been described in Example 4.23. By that example we know that we have a T-stable curve of degree 1 in Y and one of degree 3 in Z. By verifying the weights, we can identify y(1),y(sα),z(1) and z(sα) with x,y,w and z, respectively, using the notion from Proposition 3.6. For any \(w^{\prime }\in W\) we define \(\xi _{w^{\prime }\cdot s_{\alpha }}:=\left (f_{y(w^{\prime }\cdot s_{\alpha })}-f_{z(w^{\prime }\cdot s_{\alpha })}\right )\) and \(\xi _{w^{\prime }}:=\left (f_{z(w^{\prime })}-f_{y(w^{\prime })}\right )\) which leads to the equations
$$ \begin{array}{@{}rcl@{}} \rho_{-2/2}{c_{1}^{T}}(L_{w^{\prime}\alpha})\xi_{w^{\prime}\cdot s_{\alpha}}+\rho_{2/2}{c_{1}^{T}}(L_{w^{\prime}\alpha})\xi_{w^{\prime}}\equiv 0\mod {c_{1}^{T}}(L_{w^{\prime}\alpha})^{2}. \end{array} $$We remark that the coefficients \(\rho _{2/2}{c_{1}^{T}}(L_{w^{\prime }\alpha })\) and \(\rho _{-2/2}{c_{1}^{T}}(L_{w^{\prime }\alpha })\) are given by 1 and \(\frac {[-1]_{F_{{\mathbb {L}}}}{c_{1}^{T}}(L_{w^{\prime }\alpha })}{{c_{1}^{T}}(L_{w^{\prime }\alpha })}\), respectively. This completes the description of \({\Omega }_{T}^{\ast }(X)_{\mathbb {Q}}\) in case (5).
5 Equivariant Multiplicities at Nondegenerate Fixed Point in Cobordism
In this section, we want to generalise some results for equivariant Chow groups from [4, Section 4] to equivariant algebraic cobordism.
Definition 5.1
Let X be a scheme with a T-action. We call a T-fixed point x ∈ X nondegenerate if the tangent space TxX contains no non-zero fixed point. Equivalently, 0 is not a weight for the T-module TxX. The weights of this module counted with their equivariant multiplicities will be called the weights of x in X.
Remark 5.2
[4, Section 4.1] We have Tx(XT) = (TxX)0 where (TxX)0 denotes the zero weight space. Therefore, any T-fixed point in a nonsingular T-variety is nondegenerate if and only if it is isolated. Thus, for the class of smooth projective and spherical varieties all T-fixed points are nondegenerate.
Before we start to prove the main analogues of [4, Section 4] we recall two important statements which were proved by Krishna [19]. Let M be the character group of T. Recall that S(T)[M− 1] is the graded ring obtained by inverting all non-zero linear forms \({\sum }_{j=1}^{n}m_{j}t_{j}\) which was described in more detail in [19, Section 6]. For a smooth k-scheme X with a torus action, we denote \({\Omega }_{T}^{\ast }(X)\otimes _{S(T)}S(T)[M^{-1}]\) by \({\Omega }_{T}^{\ast }(X)[M^{-1}]\).
Theorem 5.3
[19, Proposition 3.1] Let G be a linear algebraic group and f : Y → X be a regular G-equivariant embedding in G −Schk of pure codimension d and let NY/X denote the equivariant normal bundle of Y inside X. Then one has
for every \(\eta \in {\Omega }^{G}_{\ast }(Y)\).
Corollary 5.4
[19, Corollary 7.3] Let X be a smooth T-filtrable variety with an action of a torus T and i : XT → X be the inclusion of the fixed point subscheme. Then the pushforward map \(i_{\ast }: {\Omega }_{\ast }^{T}(X^{T})_{\mathbb {Q}}\to {\Omega }_{\ast }^{T}(X)_{\mathbb {Q}}\) becomes an isomorphism after base change to \(S(T)_{\mathbb {Q}}[M^{-1}]\).
We recall that the equivariant cobordism module of disconnected varieties is the direct sum of the equivariant cobordism modules of the connected components.
Definition 5.5
Let X be a smooth T-filtrable variety with an action of a torus T. Further, let \([Y\to X]\in {\Omega }_{\ast }^{T}(X)_{\mathbb {Q}}\) and x ∈ X be an isolated T-fixed point. We distinguish between isolated fixed points and connected components \(F\subseteq X^{T}\) which are not an isolated point. For any isolated fixed point we define the T-equivariant multiplicity \(e_{x,X}[Y\to X]\in S(T)_{\mathbb {Q}}[M^{-1}]\) of X at x to be given by the equality
which holds in \({\Omega }_{\ast }^{T}(X)_{\mathbb {Q}}[M^{-1}]\) for some \(e_{F}\in S(T)_{\mathbb {Q}}[M^{-1}]\) and \([F^{\prime }\to F]\break \in {\Omega }_{\ast }^{T}(F)_{\mathbb {Q}}\).
Lemma 5.6
Let X be a smooth T-filtrable scheme with a T-action. Furthermore, let \(Y\subseteq X\) be a closed smooth T-filtrable subvariety. For the class [f : Y → X] in the \(S(T)_{\mathbb {Q}}\)-algebra \({\Omega }_{T}^{\ast }(X)_{\mathbb {Q}}\) and any T-fixed point y ∈ Y which is nondegenerate in X, we have
in \({\Omega }^{T}_{\ast }(X)_{\mathbb {Q}}[M^{-1}]\) where χ1,...,χm are the weights of y in Y.
Proof
First, we remark that the assumption of y ∈ YT being nondegenerate in X implies that y lies in XT and is nondegenerate in Y . Next, we consider the equality
coming from Definition 5.5. For j : YT → Y, we apply j∗ on both sides. Using Proposition 5.3 and the Whitney sum formula, we obtain
which leads to
for all isolated \(y^{\prime }\in Y^{T}\). Now, we apply f∗ to (5.1) and thus, we have
On the other hand, by Definition 5.5, we have the equality
Let i : y → X be the inclusion of the isolated fixed point y in X. Applying i∗ to the two equations above implies
Hence, comparing the coefficients leads to
which implies the claim. □
Next, we consider classes [Y → X] of the \(S(T)_{\mathbb {Q}}\)-algebra \({\Omega }_{T}^{\ast }(X)_{\mathbb {Q}}\) for which Y is not necessarily a closed smooth T-filtrable subvariety of X. This generalises [4, Proposition 4.3] in the setting of smooth T-filtrable varieties with a T-action.
Theorem 5.7
Let X,Y be smooth T-filtrable varieties with a T-action such that [f : Y → X] is a class in the \(S(T)_{\mathbb {Q}}\) -algebra \({\Omega }_{T}^{\ast }(X)_{\mathbb {Q}}\) . Let x ∈ X be a nondegenerate fixed point. Assume further that all fixed points in the fiber f − 1 (x) are nondegenerate. Then we have
Proof
Let j : U → X be the inclusion of some open T-stable neighbourhood of x. By potential shrinking we may assume that x is the unique T-fixed point in X. Using Definition 5.5 in \({\Omega }_{T}^{\ast }(X)_{\mathbb {Q}}\) we obtain
We have \(j^{\ast }[\widetilde {F}^{\prime }\to X]=0\) if \(\Im (\widetilde {F}^{\prime })\subseteq X\) does not contain x. Therefore, pulling back along j yields
On the other hand, we have
Applying the pushforward f∗ to the equation results in
Again, \(j^{\ast }[F^{\prime }\to X]=0\) and \(j^{\ast }[y^{\prime }\to X]=0\) for any \(y^{\prime }\in Y^{T}\) if \(f(y^{\prime })\neq x\). Thus, applying the pullback j∗ yields
Due to the fact that [x → U] = [y → U] holds in \({\Omega }_{T}^{\ast }(U)_{\mathbb {Q}}[M^{-1}]\) for any y ∈ YT with f(y) = x, we obtain
Thus, the corresponding coefficients in \(S(T)_{\mathbb {Q}}[M^{-1}]\) must coincide which implies the claim. □
Example 5.8
We want to determine the corresponding classes of IG(2,5) in \(S(T)^{8}_{\mathbb {Q}}\), see Example 3.10. Therefore, we consider the Bialynicki-Birula decomposition coming from Brion’s definition of T-filtrable varieties in [4, Section 3]. We choose the generic one-parameter subgroup t↦diag(t2,t,t− 1,t− 2). Using Definition 5.5 and Lemma 5.6, one can compute the pullbacks of the fixed points which are given by
where x45 is the most attractive fixed point, i.e. the fixed point whose Bialynicki-Birula cell is open, and ε1,ε2 are given as in Example 3.10. We recall the filtration (2.1), i.e. the Xi are T-stable closed subschemes of IG(2,5) for all 0 ≤ i ≤ 7. Further, we remark that the Wm = Xm ∖ Xm − 1, 0 ≤ m ≤ 7, are precisely the Bialynicki-Birula cells which are affine spaces in our situation. Lastly, using Lemma 5.6 and by computing the weights on stable neighbourhoods of the fixed points we deduce
where X2 and \(X_{2}^{\prime }\) are the two projective planes obtained by attaching the affine planes X2 ∖ X1 and X3 ∖ X2 to the projective line X1, respectively. Therefore, X3 is the union of the projective spaces X2 and \(X_{2}^{\prime }\) meeting in the projective line X1. Let \(\widetilde {X}_{3}\) be the normalisation of X3 which is smooth. Then the pullback \(i^{\ast }[\widetilde {X}_{3}\to \text {IG}(2,5)]\) is given by the sum of i∗[X2 →IG(2,5)] and \(i^{\ast }[X_{2}^{\prime }\to \text {IG}(2,5)]\).
In the sequel, we set Ei to be the vector space generated by the first i basis vectors of \(\mathbb {C}^{5}\). For the sake of completeness, we remark that X0,X1,X2 and \(X_{2}^{\prime }\) are given by
Now, we will consider the singular subscheme \(X_{4}\subseteq \text {IG}(2,5)\) which is obtained by attaching the \(\mathbb {A}^{3}=X_{4}\setminus X_{3}\) containing the fixed point x25 to X3. Geometrically, X4 can be identified with a cone over a surface with only one singular point x12. The pullback to smooth T-fixed points in X4 works similar as in the previous cases. Therefore, we only consider the pullback to the singular fixed point x12. One can compute the blow up of the point x12 in X4 explicitly and check that there are four T-fixed points in the exceptional divisor E. Using Proposition 5.7, we need to compute the weights of the four T-fixed points in \(E\subseteq \widetilde {X}_{4}\). These weights can be seen from the computation directly. Using Proposition 5.7 and Definition 5.5 leads to
We remark that this element reduces to the correct one in Chow rings and that the pullback \(i^{\ast }_{x_{12}}[\widetilde {X}_{4}\to \text {IG}(2,5)]\) is an element in \(S(T)_{\mathbb {Q}}\). Alternatively, one could check that the geometric descriptions of X4 and \(\widetilde {X}_{4}\) are given by
We consider now the closed subscheme \(X_{5}\subseteq \text {IG}(2,5)\) which is obtained by attaching the cell \(\mathbb {A}^{3}=X_{5}\setminus X_{4}\) containing the fixed point x34 to X4. A short computation shows that the planes containing x12,x13,x14 and x12,x13,x23 are singular in X5. Normalising yields X4 and \(X_{4}^{\prime }:=X_{3}\cup (X_{5}\setminus X_{4})\). We remark that \(X_{4}^{\prime }\) is given by the equations e4 ∧ e5 = e3 ∧ e5 = e2 ∧ e5 = 0 which implies
One may observe that any isotropic subspace V2 in E4 has to remain isotropic when considering \(\overline {V_{2}}:=(V_{2}+E_{4}^{\perp })/E_{4}^{\perp }\subseteq E_{4}/E_{4}^{\perp }\), but since \(E_{4}/E_{4}^{\perp }=\langle e_{2},e_{4}\rangle \) holds, we obtain
We claim that a resolution \(\widetilde {X}_{4}^{\prime }\)of \(X_{4}^{\prime }\) is given by
This is birational to \(X_{4}^{\prime }\) via the second projection. Now, we consider the map
which is a \(\mathbb {P}^{1}\)-fibration over \(\mathbb {P}^{1}\times \mathbb {P}^{1}\). Therefore, \(\widetilde {X}_{4}^{\prime }\) is smooth and projective. The only singular point in \(X_{4}^{\prime }\) is x13 and thus, we want to compute \(i_{x_{13}}^{\ast }[\widetilde {X}_{4}^{\prime }\to \text {IG}(2,5)]\) using Proposition 5.7. The T-fixed points in the exceptional divisor are given by
Exemplarily, we compute the weights for the first T-fixed point in the exceptional divisor, i.e. for \(\widetilde {x_{1}}:=(E_{1},\langle e_{1},e_{3}\rangle ,E_{3})\). Therefore, we consider the morphism h and the tangent space \(T_{h(\widetilde {x_{1}})}\mathbb {P}^{1}\times \mathbb {P}^{1}=T_{[1:0];[0:1]}\mathbb { P}^{1}\times \mathbb {P}^{1}\) which leads to the weights − ε1 and ε1. The last weight can be seen in the tangent space \(T_{\widetilde {x_{1}}}(h^{-1}(E_{1},E_{3}))=T_{[0:1]}\mathbb {P}(e_{2},e_{3})\). This leads to the weight ε2. We summarise that the weights of \(\widetilde {x_{1}}\) in \(\widetilde {X}_{4}^{\prime }\) are given by − ε1,ε1 and ε2. The weights of the other T-fixed points in the exceptional divisor can be computed similarly. Therefore, for any T-fixed point x ∈ X5 one can compute
Lastly, we consider the singular subscheme \(X_{6}\subseteq X\) which is given by
We claim that a resolution \(\widetilde {X}_{6}\) of X6 is given by
Again, this is birational to X6. Now, we want to show smoothness of \(\widetilde {X}_{6}\). We consider the map
whose fibers are
where \(V_{4}\subseteq V_{1}^{\perp } \Leftrightarrow V_{1}\subseteq V_{4}^{\perp }\) holds. Consider now the projection
which is a \(\mathbb {P}^{2}\)-bundle over \(\mathbb {P}^{1}\) because \(V_{4}^{\perp }\) is two-dimensional. Thus, f− 1(V4) is smooth and therefore, \(\widetilde {X}_{6}\) is smooth and projective.
Now, we want to apply Proposition 5.7 to obtain the pullback \(i_{x}^{\ast }[\widetilde {X}_{6}\to \text {IG}(2,5)]\) for the singular T-fixed points x ∈ X6. The singular T-fixed points in X6 are x12, x13 and x23. The T-fixed points in the exceptional divisor which map to x12 are given by (E1, E2,E4) and (e2, E2, 〈E3, e5〉). For the other two singular T-fixed points we obtain three T-fixed points in the exceptional divisor, e.g. (e1, 〈e1, e3〉, E4),(e3, 〈e1, e3〉, E4) and (e3, 〈e1, e3〉, 〈E3, e5〉) are the T-fixed points in the fiber of x13. Exemplarily, we compute the weights for one of the T-fixed points in the fiber of x12, i.e.\(\tilde {x}:=(E_{1},E_{2},E_{4})\). Therefore, we consider the morphism f and the tangent space \(T_{f(\tilde {x})}\mathbb {P}^{1}=T_{[1:0]}\mathbb {P}(e_{4},e_{5})\). Thus, we obtain the weight − ε1 + ε2. Next, we need to compute the weights in \(T_{\tilde {x}}(f^{-1}(E_{4}))\). Therefore, we consider the morphism g and the tangent space \(T_{g(\tilde {x})}\mathbb {P}^{1}=T_{[1:0]}\mathbb {P}(e_{1},e_{3})\) which leads to the weight − ε1. Lastly, we consider the tangent space \(T_{\tilde {x}}(g^{-1}(E_{1}))\) which are the two-dimensional spaces containing e1 and contained in E4. Thus, we obtain the last weights from \(T_{\tilde {x}}(g^{-1}(E_{1}))=T_{[1:0:0]}\mathbb {P}(e_{2},e_{3},e_{4})\). This leads to the weights − ε2,− 2ε2. We summarise that the weights of \(\tilde {x}\) in \(\widetilde {X}_{6}\) are given by − ε1 + ε2,−ε1,−ε2 and − 2ε2. Similarly, one can compute all the other weights and apply Proposition 5.7 to finish the computation. This then determines the whole ring structure of \({\Omega }_{T}^{\ast }(\text {IG}(2,5))_{\mathbb {Q}}\) and allows us to multiply classes.
Remark 5.9
Assuming we could determine the pullback at singular points using the equations given in Example 3.10 and the weights of the tangent spaces at smooth points as in Chow rings (cf. [4, Section 4]), we would be able to determine the class \([\widetilde {X}_{4}\to \text {IG}(2,5)]\) uniquely for an arbitrary T-equivariant resolution of singularities ofX4. A long computation shows that one cannot even determine a unique class in K-theory only using the weights of the tangent spaces at smooth fixed points because of the multiplicative formal group law. In fact it is not even known whether these (not uniquely determined) classes correspond to the resolutions of singularities of X4. The fact that one cannot determine the class \([\widetilde {X}_{4}\to \text {IG}(2,5)]\) uniquely is natural because two different resolutions of singularities determine two different classes in cobordism. For example, one could also consider another resolution of singularities of X4 given by
which is a \(\mathbb {P}^{2}\)-fibration over \(\mathbb {P}^{1}\). The exceptional locus of \(\widetilde {X}^{\ast }_{4}\) over X4 is a \(\mathbb {P}^{1}\) over the singular point x12. A computation shows that the classes \(i_{x_{12}}^{\ast }[\widetilde {X}_{4}\to \text {IG}(2,5)]\) and \(i_{x_{12}}^{\ast }[\widetilde {X}^{\ast }_{4}\to \text {IG}(2,5)]\) do not coincide, although they both reduce to the same one in Chow rings.
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Acknowledgements
I am deeply indebted to Nicolas Perrin for providing key ideas behind the present results which were clarified in very helpful discussions. I would like to thank Michel Brion for his detailed explanations of some of his results which were used in this article. I am also very grateful to Jens Hornbostel for his valuable comments and suggestions during various stages of this work. Further, I am thankful for the comments and remarks from an anonymous referee that considerably improved the exposition of the manuscript. Additionally, I would like to thank Herman Rohrbach, Christoph Spenke and Thomas Hudson for many helpful discussions.
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Open Access funding enabled and organised by Projekt DEAL. The author is supported by the DFG Research Training Group 2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology.
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July, H. Equivariant Cobordism of Smooth Projective Spherical Varieties. Transformation Groups (2022). https://doi.org/10.1007/s00031-022-09775-z
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DOI: https://doi.org/10.1007/s00031-022-09775-z