ESP in the Context of Maritime English -Challenges and Solutions -

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ESP in the Context of Maritime English - Challenges and Solutions - Naoyuki Takagi Tokyo University of Marine Science and Technology (TUMSAT)

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EQUIVARIANT COHOMOLOGY AND CHERN CLASSES OF
SYMMETRIC VARIETIES OF MINIMAL RANK
M. BRION AND R. JOSHUA
Abstract. We describe the equivariant Chow ring of the wonderful compactifi-
cation X of a symmetric space of minimal rank, via restriction to the associated
toric varietyY. Also, we show that the restrictions toY of the tangent bundleTX
and its logarithmic analogueS decompose into a direct sum of line bundles. ThisX
yields closed formulae for the equivariant Chern classes ofT andS , and, in turn,X X
for the Chern classes of reductive groups considered by Kiritchenko.
0. Introduction
The purpose of this article is to describe the equivariant intersection ring and
equivariantChernclassesofaclassofalmosthomogeneousvarieties,namely,complete
symmetric varieties of minimal rank. These include those (complete, nonsingular)
equivariant compactifications of a connected reductive group, that are regular in the
sense of [BDP90].
The main motivation for this work comes from questions of enumerative geometry
on a complex algebraic varietyM. IfM is a spherical homogeneous spherical under a
reductive group G, these questions find their proper setting in the ring of conditions
∗C (M), isomorphic to the direct limit of cohomology rings of G-equivariant com-
pactifications X of M (see [DP83, DP85]). In particular, the Euler characteristic of
any complete intersection of hypersurfaces in M has been expressed by Kiritchenko
[Ki06], in terms of the Chern classes of the logarithmic tangent bundle S of anyX
∗regular compactification X. As shown in [Ki06], these elements of C (M) are in-
dependent of the choice of X, and their determination may be reduced to the case
where X is a “wonderful variety”.
In fact, it is more convenient to work with the rational equivariant cohomology
∗ ∗ring H (X), from which the ordinary rational cohomology ring H (X) is obtainedG
∗by killing the action of generators of the polynomial ring H (pt); the Chern classes
G
∗of S have natural representatives in H (X), the equivariant Chern classes. WhenX G
∗X is a complete symmetric variety, the ring H (X) admits algebraic descriptions byG
work of Bifet, De Concini, Littelman, and Procesi (see [BDP90, LP90]).
Here we consider the case where X is a wonderful symmetric variety of minimal
rank,thatis,thewonderfulcompactificationofasymmetricspaceG/K ofrankequal
to rk(G)−rk(K). Moreover, we adopt a purely algebraic approach: we work over
an arbitrary algebraically closed field, and replace the equivariant cohomology ring
The second author was supported by a grant from the NSA and by MPI(Bonn).
16
2 M. BRION AND R. JOSHUA
∗with the equivariant intersection ring A (X) of [EG98] (for wonderful varieties overG
the complex numbers, both rings are isomorphic over the rationals).
We show in Theorem 2.2.1 that a natural pull-back map
∗ ∗ WKr :A (X)→A (Y)G T
is an isomorphism over the rationals, where T ⊂ G denotes a maximal torus con-
taining a maximal torus T ⊂ K with Weyl group W , and Y denotes the closureK K
in X of T/T ⊂G/K. Furthermore, Y is the toric variety associated with the WeylK
chambers of the restricted root system of G/K.
We also determine the images under r of the equivariant Chern classes of the
tangent bundle T and its logarithmic analogue S . For this, we show in TheoremX X
3.1.1 that the normal bundle N decomposes (as a T-linearized bundle) into aY/X
direct sum of line bundles indexed by certain roots of K; moreover, any such line
1bundle is the pull-back of O 1(1) under a certain T-equivariant morphism Y →P .P
By Proposition 1.1.1, the product of these morphisms yields a closed immersion of
the toric variety Y into a product of projective lines, indexed by the restricted roots.
In the case of regular compactifications of reductive groups, Theorem 2.2.1 is due
to Littelmann and Procesi for equivariant cohomology rings (see [LP90]); it has been
adapted to equivariant Chow ring in [Br98], and to equivariant Grothendieck rings
by Uma in [Um05]. Here we adapt the approach of [Br98], based on a precise version
of the localization theorem in equivariant intersection theory (inspired, in turn, by
a similar result in equivariant cohomology, see [GKM99]). The main ingredient is
that X contains only finitely many T-stable points and curves. This fact also plays
an essential role in Tchoudjem’s description of the cohomology of line bundles on
wonderful varieties of minimal rank, see [Tc05].
Theorem3.1.1seemstobenew,alreadyinthegroupcase; ityieldsaclosedformula
fortheimageunderr oftheequivariantToddclassofX,analogoustothewell-known
formula expressing the Todd class of a toric variety in terms of boundary divisors.
ThetoricvarietyY associatedtoWeylchambersisconsideredin[Pr90,DL94], where
its cohomology is described as a graded representation of the Weyl group; however,
its simple realization as a general orbit closure in a product of projective lines seems
to have been unnoticed.
This article is organized as follows. Section 1 gathers preliminary notions and
results on symmetric spaces, their wonderful compactifications, and the associated
toric varieties. In particular, for a symmetric space G/K of minimal rank, we study
therelationsbetweentherootsystemsandWeylgroupsofG,K, andG/K; theseare
our main combinatorial tools. In Section 2, we first describe the T-invariant points
and curves in a wonderful symmetric variety X of minimal rank; then we obtain our
∗main structure result for A (X), and some useful complements as well. Section 3G
contains the decompositions of N and of the restrictions T | , S | , togetherY/X X Y X Y
with their applications to equivariant Chern and Todd classes.
Throughoutthis article, we consider algebraic varieties over an algebraically closed
field k of characteristic = 2; by a point of such a variety, we mean a closed point.SYMMETRIC VARIETIES OF MINIMAL RANK 3
As general references, we use [Ha77] for algebraic geometry, and [Sp98] for algebraic
groups.
1. Preliminaries
1.1. ThetoricvarietyassociatedwithWeylchambers. LetΦbearootsystem
in a real vector space V (we follow the conventions of [Bo81] for root systems; in
particular, Φ is finite but not necessarily reduced). Let W be the Weyl group, Q the
∨root lattice in V, and Q the dual lattice (the co-weight lattice) in the dual vector
∗ ∗space V . The Weyl chambers form a subdivision of V into rational polyhedral
∗convex cones; let Σ be the fan of V consisting of all Weyl chambers and their faces.
∨The pair (Q ,Σ) corresponds to a toric variety
Y =Y(Φ)
∨equippedwithanactionofW viaitsactiononQ whichpermutestheWeylchambers.
The group W acts compatibly on the associated torus
∨T := Hom(Q,G ) =Q ⊗ G .m Z m
Thus, Y is equipped with an action of the semi-direct product T ·W. Note that the
character group X(T) is identified with Q; in particular, we may regard each α∈ Φ
as a homomorphism α :T →G .m
The choice of a basis of Φ,
Δ ={α ,...,α },1 r
definesapositiveWeylchamber,thedualconetoΔ. LetY ⊂Y bethecorresponding0
rT-stable open affine subset. Then Y is isomorphic to the affine space A on which0
T acts linearly with weights −α ,...,−α . Moreover, the translates w·Y , where1 r 0
w∈W, form an open covering of Y.
Inparticular,thevarietyY isnonsingular. Also,Y isprojective,asΣisthenormal
fan to the convex polytope with vertices w·v (w ∈ W), where v is any prescribed
regular element of V. The following result yields an explicit projective embedding
of Y:
Proposition 1.1.1. (i) For any α ∈ Φ, the morphism α : T → G extends to am
morphism
1f :Y →P .α
1 1 −1Moreover, f and f differ by the inverse mapP →P , z7→z .α −α
(ii) The product morphism
Y Y
1f := f :Y → Pα
α∈Φ α∈Φ
is a closed immersion. It is equivariant under T·W, where T acts on the right-hand
1side via its action on each factor P through the character α, and W acts via itsα
natural action on the set Φ of indices.
Q
1(iii) Conversely, the T-orbit closure of any point of (P \{0,∞}) is isomorphic
α∈Φ
to Y.6
6
4 M. BRION AND R. JOSHUA
(iv) Any non-constant morphismF :Y →C, whereC is an irreducible curve, factors
1through f :Y →P where f is an indivisible root, unique up to sign. Thenα α
(1.1.1) (f ) O =O 1.α ∗ Y P
Proof. (i) Since α has a constant sign on each Weyl chamber, it defines a morphism
1of fans from Σ to the fan ofP , consisting of two opposite half-lines and the origin.
This implies our statement.
(ii) The equivariance property of f is readily verified. Moreover, the product map
rY
1 rf :Y → (P )αi
i=1
1 rrestricts to an isomorphism Y → (P \{∞}) , since each f restricts to the i-th0 αi
r∼coordinate function on Y = A . Since Y = W · Y , it follows that f is a closed0 0
immersion.
(iii) follows from (ii) by using the action of tuples (t ) of non-zero scalars, viaα α∈Φ
componentwise multiplication.
(iv) Taking the Stein factorization, we may assume that F O =O . Then C is∗ Y C
normal, and hence nonsingular. Moreover, the action ofT onY descends to a unique
action on C such that F is equivariant (indeed, F equals the canonical morphism
∗ ∗Y → ProjR(Y,F L), where L is any ample invertible sheaf on C, and R(Y,F L)L
∗ n ∗denotesthesectionring Γ(Y,F L ). Furthermore,F LadmitsaT-linearization,n
1∼and hence T acts on R(Y,L)). It follows that C = P where T acts through a
character χ, uniquely defined up to sign. Thus, F induces