Res + X F F + is defined below in (1.3). According to [Je-Ki2, Definition 3.3 and Proposition 3.4], the value of Res + X

Transcription

1

2 Theorem 1.2. For any η HH (N) we have1 (1.1) κ S (η)[n red ] = c η F. Here HH (F) denotes the H-equivariant Euler class of the normal bundle ν(f), c is a non-zero constant 2, and is defined below in (1.3). According to [Je-Ki2, Definition 3.3 and Proposition 3.4], the value of on a rational function of the type (1.2) h(x, Y 1,..., Y m ) = p(x, Y 1,..., Y m ) k (m kx + i β kiy i ) where p C[X, {Y i }], m i Z \ {0} and β ki C, can be obtained as follows: regard Y 1,..., Y m as constants (i.e. as complex numbers) and set (1.3) (h) = b C Res X=b p q where on the right hand side p q is interpreted as a meromorphic function in the variable X on C. It remains to note that only residues of expressions of the type (1.2) are involved in (1.1). For we may assume that each normal bundle ν(f), F F +, is a direct sum of H-invariant line bundles and in this way the Euler class is the product of H-equivariant first Chern classes of those line bundles. Remark 1.3. For future reference, we mention that 1 X + i β iy i = 1. Remark 1.4. Guillemin and Kalkman gave another definition of the residue of h given by (1.2) (see [Gu-Ka, section 3]). Denote β k (Y ) = i β kiy i and write 1 k (m kx + β k (Y )) = 1 k m kx(1 + β k(y ) m k X ) = (m k X) 1 k k (1 β k(y ) m k X +(β k(y ) m k X )2...). Multiply the right hand side by the polynomial p, and add together all coefficients of X 1 : the result is the Guillemin-Kalkman residue. It is a simple exercise to show that the latter coincides with the residue of [Je-Ki2] (one uses the fact that the sum of the residues of the 1-form h(x)dx on the Riemann sphere equals zero). Since κ S is a ring homomorphism, from Theorem 1.2 we deduce: 1 Note that both sides of the equation are in C[Y 1,..., Y m ]. 2 The exact value of c is not needed in our paper, but the interested reader can find it in [Je-Ki2, 3]. 2

3 Corollary 1.5. κ S (η)κ S (ζ)[n red ] = (ηζ) F. Since 0 s is a regular value of µ S, S acts on µ 1 S (0) with finite stabilizers. Hence µ 1 S (0)/S has at worst orbifold singularities and in particular it satisfies Poincaré duality. From Corollary 1.5 it follows that η HH (N) is in kerκ S iff (1.4) for all ζ H H (N). (ηζ) F = 0, The main goal of our paper is to compare the latter description of ker κ S to the one given by Goldin in [Go]. We will assume that the fixed point sets N S and N H are equal, which holds for a generic choice of the subgroup S of H (see Remark 1.9 below). Theorem 1.6. (see [Go, Theorem 1.5]) We have ker κ S = K K + where K is the set of all elements of H H (N) vanishing on all components F F, and similarly for K +. The goal of our paper is to give a direct proof of the following result: Theorem 1.7. Assume that the fixed point set N H = N S is finite. A class η H H (N) satisfies (1.4) for all ζ H H (N) if and only if η K + K. It is obvious that any η K + satisfies (1.4). On the other hand, by the Atiyah-Bott- Berline-Vergne localization formula (see [At-Bo], [Be-Ve]) the sum F F (ηζ) F is a polynomial in X, {Y i }, so that its residue is zero. It follows that any η in K satisfies (1.4) as well. The hard part will be to prove that if η satisfies (1.4) for any ζ H H (N), then η is in K K +. Remark 1.8. Theorem 1.7 is a generalization of the main result of [Je]. Remark 1.9. Let M be a symplectic manifold acted on by a torus G in a Hamiltonian fashion such that the fixed point set M G is finite and let µ G : M g be the corresponding 3

4 moment map. Suppose that 0 g is a regular value of µ G and consider the corresponding symplectic reduction M//G = µ 1 G (0)/G as well as the Kirwan surjection κ : H G (M) H (M//G). Using reduction in stages, results like Theorem 1.6 or Theorem 1.7 can be used in order to obtain descriptions of ker κ. The following construction is needed: There exists a generic (in the sense of [Go, section 1]) circle S 1 G with Lie algebra s 1 such that M S 1 = M G 0 is a regular value of the moment map µ S1 : M s 1. Consider the action of G/S 1 on the reduced space M//S 1 = µ 1 S 1 (0)/S 1. As before, there exists a generic circle S 2 G/S 1 such that (M//S 1 ) S 2 = (M//S 1 ) G/S 1 0 is a regular value of the moment map µ S2 : M//S 1 s 2. The equivariant Kirwan map corresponding to the subtorus S 1 S 2 of G is the following composition of maps: HG (M) κ 1 HG/S 1 (M//S 1 ) κ 2 HG/(S 1 S 2 ) (M//(S 1 S 2 )), where we have used reduction in stages. In fact the procedure can be continued giving rise to a sequence of tori {1} = T 0 T 1 T 2... T m = G where T 1 = S 1, T 2 = S 1 S 2,... with the following properties: (M//T j 1 ) T j/t j 1 = (M//T j 1 ) G/T j 1 0 is a regular value of the moment map µ Tj /T j 1 on M//T j 1. By reduction in stages, the Kirwan map κ decomposes as (1.5) κ = κ m... κ 1 where κ j = κ Tj /T j 1 : H G/T j 1 (M//T j 1 ) H G/T j (M//T j ) is the T j /T j 1 -equivariant Kirwan map. Goldin [Go] used this decomposition in order to deduce from Theorem 1.6 the Tolman-Weitsman description of ker κ (see [To-We]). Theorem 1.7 of our paper with N = M//T j 1, H = G/T j 1 and S = T j /T j 1 shows that the same Tolman-Weitsman description of ker κ is equivalent to the obvious characterization in terms of residues arising from the components κ j of κ. 4

5 2 Expressing kerκ S in terms of residues We will give a proof of Theorem 1.7. Take ξ s a non-zero vector and f = µ ξ S = µ S, ξ : N R the function induced on N by the moment map µ S : N s. This is an H-equivariant Morse function, whose critical set is N S = N H. For any F F, we have the H-equivariant splitting of the tangent space T F N = ν f (F) ν+ f (F), determined by the sign of the Hessian on the two summands. Let e(ν f (F)), e(ν+ f (F)) HH (F) be the corresponding H-equivariant Euler classes. The following two results can be proved by using Morse theory (see for example [Go]): Proposition 2.1. Suppose η HH (N) restricts to zero on all G F for which f(g) < f(f). Then η F is some HH (pt)-multiple of e(ν f F). Proposition 2.2. For any F F there exists a class α (F) HH (N) with the following properties: 1. α (F) G = 0, for any G F which cannot be joined to F along a sequence of integral lines of the negative gradient field f (in particular, for any G F with f(g) < f(f)) 2. α (F) F = e(ν f F). In the same way there exists α + (F) HH (N) such that: 1. α + (F) G = 0, for any G F which cannot be joined to F along a sequence of integral lines of the gradient field f (in particular, for any G F with f(g) > f(f)) 2. α + (F) F = e(ν + f F). From the injectivity theorem of Kirwan which says that α HH (N) is uniquely determined by its restrictions to N H we deduce: Corollary 2.3. The set {α (F) F F} is a basis of H H (N) as a H H (pt)-module. Consider the space Ĥ H (N) consisting of all expressions of the type r F α (F), F F where r F is in the ring C(X, {Y i }) of rational expressions in X, {Y i } (i.e. quotients p/q, with p, q C[X, {Y i }], q 0). The space Ĥ H (N) is obviously a C(X, {Y i})-algebra. 5

6 Lemma 2.4. Take η HH (N) of degree d. Then we can decompose where η +, η Ĥ H (N), such that (i) η + F = 0, η F+ = 0; η = η + + η (ii) η + and η are linear combinations of α (F), F F, where the coefficients r F are rational functions whose denominators can be decomposed as products of linear factors of the type X + β i Y i, β i C. i Proof. By Corollary 2.3, we have η = F F p F α (F), where p F R[X, {Y i }] are homogeneous polynomials. Take η = F F p F α (F), η + = By Proposition 2.2, η + restricts to 0 on all G F. p F α (F). Consider the ordering F 1, F 2,... of the elements of F + such that 0 < f(f 1 ) < f(f 2 ) <.... There exists a rational function r 1 (X, {Y i }) C(X, {Y i }) such that which means that the form η F1 = r 1 (X, {Y i })e(ν f F 1), η 1 := η r 1 (X, {Y i })α (F 1 ) vanishes at F 1. There exists another rational function r 2 (X, {Y i }) C(X, {Y i }), with the property that η 1 F2 = r 2 (X, {Y i })e(ν f F 2), which implies that the form η 1 r 2 (X, {Y i })α (F 2 ) 6

7 vanishes at both F 1 and F 2. We continue this process and we get the decomposition claimed in the proposition as follows: η = η r 1 α (F 1 ) r 2 α (F 2 )..., η + = η + + r 1 α (F 1 ) + r 2 α (F 2 ) Property (ii) follows from the fact that for each F F +, the weights of the representation of S on ν f (F) are all non-zero. We are now ready to prove the main result of the paper. Proof of Theorem 1.7. Take η HH d (N) satisfying (2.1) Res + (ηζ) F X = 0 for all ζ HH (N). We consider the decomposition η = η + η + with η, η + Ĥ H (N) given by Lemma 2.4. We show that η and η + are actually in H H (N). More precisely, if η + is of the form with p G, q G C[X, Y i ] we show that η + = G F + p G q G α (G) for any G F +. q G divides p G From (2.1) and the fact that η F+ = 0, we deduce that which is equivalent to (2.2) Res + (η + ζ) F X = 0 F,G F + p G q G α (G) F ζ F = 0 7

8 for all ζ H H (N). Consider again the ordering F 1, F 2,... of the elements of F + such that f(f 1 ) < f(f 2 ) <.... We prove by induction on k 1 that q Fk divides p Fk. In (2.2) we put ζ = pα + (F 1 ), where p C[X, Y i ] is an arbitrary polynomial. Since we deduce that e(ν f F 1)e(ν + f F 1) = 1, pp F 1 q F1 = 0 for any p C[X, Y 1,..., Y m ] (we are using the fact that α + (F 1 ) F = 0 if µ(f) > µ(f 1 ) while α (G) F = 0 if µ(f) < µ(g): hence the only nonzero contribution comes from F = G = F 1 ). From Lemma 2.5 (see below) we deduce that q F1 divides p F1. Now we fix k 2, assume that q Fi divides p Fi for any i < k and show that q Fk divides p Fk. In (2.2) put ζ = pα + (F k ), with p C[X, Y 1,..., Y m ]. We obtain i j k p p F i q Fi α (F i ) Fj α + (F k ) Fj j = 0. The sum in the left hand side is over i and j (k being fixed). We divide into sums of the type Σ i := p p F i α (F i ) Fj α + (F k ) Fj, i k. q Fi j i j k If i < k, then Σ i = 0, by the hypothesis that q Fi divides p Fi and the Atiyah-Bott-Berline- Vergne localization formula for p p F i q Fi α (F i )α + (F k ) (note that α (F i ) F α + (F k ) F = 0 for any F which is not of the form F j with i j k). Finally we show that Σ k = 0, which is equivalent to for any p C[X, Y i ]. By Lemma 2.5, which concludes the proof. pp F k q Fk = 0 q Fk divides p Fk, We have used the following result: 8

9 Lemma 2.5. Let f, g be in C[X, {Y i }], where g = k (X + i β ik Y i ), β ik C. If (2.3) (p f g ) = 0, for any p C[X, {Y i }], then g divides f. Proof. Suppose that g does not divide f. We can assume that g and f are relatively prime. Then there exist p 1, p 2 C[X, {Y i }] such that From (2.3) it follows that for any p C[X, {Y i }]. Now fix k 0 and set We deduce that p 1 f + p 2 g = 1. p g = Res+ X (pp 1 f g + p 2) = 0 p = (X + β ik Y i ). k k 0 i 1 X + i β ik 0 Y i = 0. But the left hand side is actually 1 (see Remark 1.3), which is a contradiction. 3 Residues and the kernel of the Kirwan map In this section we shall give a direct characterization of the kernel of the Kirwan map in terms of residues of one variable. Let M be a compact symplectic manifold equipped with a Hamiltonian G action, where G is a torus and let κ : H G (M) H (M//G) be the Kirwan map. Consider the decomposition of κ described in Remark 1.9 (see equation (1.5)). For each j between 1 and m we consider the commutative diagram 9

10 HG/T j 1 (M//T j 1 ) κ j HG/T j (M//T j ) (3.1) π j π HT j /T j 1 (M//T j 1 ) κ j H (M//T j ) where κ j is the Kirwan map associated to the action of the circle T j /T j 1 on M//T j 1. For the same action we consider the moment map µ j, the fixed point set F j = (M//T j 1 ) T j/t j 1 and the partition of the latter into F j and Fj + which consist of fixed points where µ j is negative, respectively positive. Definition 3.1. An element η H T j /T j 1 (M//T j 1 ) is in ker res,j if Res Xj =0 F F j + (ηζ) F (X j ) = 0 for any η H T j /T j 1 (M//T j 1 ), where X j is a variable corresponding to a basis of the dual of the Lie algebra of T j /T j 1 and Res Xj =0 means the coefficient of X 1 j. For any α H G (M) we define αj = (κj κ1)(α) 0 j m. The goal of this section is to prove Theorem 3.2. Let α H G (M). Then κ(α) = 0 if and only if π j(α j 1 ) ker res,j for some j 1. Proof. If κ(α) = 0, by equation (1.5) there exists j such that α j = 0. Since α j = κ j (α j 1 ), we can use first the commutativity of the diagram (3.1) to deduce that κ j (π j α j 1 ) = 0 and then the residue formula of [Je-Ki1] and [Je-Ki2] to deduce that π j (α j 1 ) ker res,j. The opposite direction is less obvious: Take α H G (M) of degree greater than zero such that π j α j 1 ker res,j. If j = m then π j and π are the identity maps, κ j = κ j and we just have to apply the residue formula of [Je-Ki1] and [Je-Ki2] to deduce that κ m (α m 1 ) = κ(α) = 0. If j < m, by the same residue formula, we have κ j (π j α j 1 ) = 0, which implies π(κ j α j 1 ) = 0. Note that ker(π) = H G/T j (pt) so π(κ j α j 1 ) = 0 is equivalent to κ j α j 1 H G/T j (pt). The latter implies that (κ m κ j )α j 1 = 0 (since the map κ m κ j+1 sends all elements of H G/T j (pt) of degree larger than zero to 0, because the image of this map is H (M//G) and the image of the equivariant cohomology of a point under this map is the ordinary cohomology of a point). But this means κ(α) = 0 and the proof is now complete. 10

11 References [At-Bo] M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), [Be-Ve] [Go] [Gu-Ka] N. Berline and M. Vergne, Zéros d un champ de vecteurs et classes caractéristiques équivariantes, Duke Math. J. 50 (1983), R. F. Goldin, An effective algorithm for the cohomology ring of symplectic reductions, Geom. Anal. Funct. Anal. 12 (2002), V. Guillemin and J. Kalkman, The Jeffrey-Kirwan localization theorem and residue operations in equivariant cohomology, Jour. reine angew. Math. 470 (1996), [Je] L. C. Jeffrey, The residue formula and the Tolman-Weitsman theorem, preprint math.sg/ [Je-Ki1] L. C. Jeffrey and F. C. Kirwan, Localization for nonabelian group actions, Topology 34 (1995), [Je-Ki2] L. C. Jeffrey and F. C. Kirwan, Localization and the quantization conjecture, Topology 36 (1995), [Ka] [To-We] J. Kalkman, Residues in nonabelian localization, preprint hep-th/ S. Tolman and J. Weitsman, The cohomology ring of abelian symplectic quotients, preprint math.dg/ , to appear in Comm. Anal. Geom. Mathematics Department University of Toronto Toronto, Ontario M5S 3G3 Canada jeffrey@math.toronto.edu Mathematics Department University of Toronto Toronto, Ontario M5S 3G3 Canada amare@math.toronto.edu 11