Let us start with a few technical propositions. For a proof, the reader is invited to consult [Br5], Propositions A3 and A4.
Proposition 2.4.1. Let X be an affine variety with a C∗-action and an attractive fixed point x. Then there exists a C∗-module V and a finite equivariant surjective morphism π:X →V such that π−1(0) ={x}.
Proposition 2.4.2. Let X be a connected variety with a nontrivial action of a torus
T and a fixed point x. Then there exists a closed irreducible T-stable curve C ⊂X
The following is a result of Brion ([Br5]) on rational smoothness and torus ac- tions.
Theorem 2.4.3. Let X be an irreducible affine T-variety with an attractive fixed point x. Then X is rationally smooth at x if and only if the following conditions hold:
(i) A punctured neighborhood of x in X is rationally smooth.
(ii) XT0 is rationally smooth at x for each subtorus T0 ⊂T of codimension one. (iii) dim(X) = ∑T0dim(XT
0
), where the (finite) sum runs over all codimension- one subtori for which XT0 6=XT.
Let X be an affine T-variety with an attractive fixed pointx. Then, by Propo- sition 2.1.6, X admits a closed equivariant embedding into its tangent space TxX. Notice that there are only a finite number of codimension-one subtori T1, . . . , Tm of T for which XTj 6=XT. Certainly, each one of them is contained in the kernel of a
weight of T in TxX. On the other hand, T acts on each XTi through its quotient T /Ti 'C∗. Becausexis an attractive fixed point ofX, we can assume, without loss of generality, that xis an attractive fixed point of each XTi, for the induced action
of C∗ 'T /Ti.
We are now ready to state what we call the Equivariant Normalization Theorem for rational cells. It is due to Brion ([Br3]) and Arabia ([Ar]).
Theorem 2.4.4. Let (X, x)be a rational cell. Then there exists a T-module V and an equivariant finite surjective map π :X →V such that π(x) = 0 and VT ={0}.
It is worth pointing out that some of the arguments to appear next are well- known constructions in algebraic geometry.
Proof of Theorem 2.4.4. We follow closely Brion’s construction ([Br3], Theorem 18). Since x is an attractive fixed point, there exists an equivariant embedding ι of X
into TxX, its tangent space at x. In other words, all the weights of T in TxX lie in an open half space of t∗. As it was emphasized before, there is only a finite collection of codimension-one subtori, say T1, . . . , Tm, for which XTj 6= XT. LetTi be one of them. Under the present circumstances, given that x is attractive, we can also assume thatx is an attractive fixed point of XTi, for the induced action of
C∗ 'T /T
i. Hence, by Proposition 2.4.1, there exists aT-equivariant finite surjective map πi : XTi → Vi, where Vi is some T-module with a trivial action of Ti. Notice that T acts on both XTi and V
i through the same character.
By construction XTi is T-stable and closed in X, so we can extend π
i to an equivariant morphism
πi :X →Vi.
Synchronizing efforts via the product map, we obtain an equivariant morphism
π :X →V,
where V is the direct sum of the Vi, sum taken over all the Ti’s above. Notice that x, being an attractive fixed point, lies in the closure all the T-orbits in X. In particular, x is contained in all the irreducible components ofπ−1(0) (i.e. π−1(0) is connected).
We now claim that the morphism π is finite. Indeed, {x} =π−1(0). For other- wise,π−1(0) would contain aT-stable curve upon whichT acts through a non-trivial character (Proposition 2.4.2). Certainly this is impossible, because π restricts to a finite morphism on each XTi.
To conclude the proof, recall that, by definition, V satisfies dim (V) =∑
Ti
Since X is rationally smooth at x, Theorem 2.4.3 (iii) dictates that X and V must have the same dimension. In conclusion,π is both dominant and surjective.
Remark 2.4.5. It is clear from the proof of Theorem 2.4.4 that if X is smooth, then the map π :X →V can be chosen to be an isomorphism.
We now specialize a result of Brion ([Br5]) to rational cells.
Corollary 2.4.6. Let (X, x) be a rational cell. Suppose that the number of closed irreducible T-stable curves on X is finite. Let n(X, x) be this number. Then
n(X, x) = dim(X).
Proof. Each closed irreducible T-stable curve Ci is the fixed point set of a unique codimension-one torus, sayTi. Since there are only a finite number of codimension- one tori, say T1, . . . , Tm, for which XTi 6= XT, then it follows from the proof of Theorem 2.4.4 that the equality below holds:
dim(X) = m ∑ j=1 dim(XTj) = m ∑ j=1 dim(Cj) =n(X, x). We are done.