HIRZEBRUCH χ y GENERA OF THE HILBERT SCHEMES OF SURFACES BY LOCALIZATION FORMULA

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1 HIRZEBRUCH χ y GENERA OF THE HILBERT SCHEMES OF SURFACES BY LOCALIZATION FORMULA KEFENG LIU, CATHERINE YAN, AND JIAN ZHOU Abstract. We use the Atiyah-Bott-Berine-Vergne ocaization formua to cacuate the Hirzebruch χ y genus χ y(s [n] ), where S [n] is the Hibert schemes of points of ength n of a surface S. Combinatoria interpretation of the weights of the fixed points of the natura torus action on (C ) [n] is used. 1. Introduction 1.1. Genera of compex manifods. Let Ω Ω U Q be the compex bordism ring with coefficients in Q. Given a ring R, a compex genera with vaues in R is a ring homomorphism φ : Ω R. Reca that according to Minor [11] and Novikov, two compex manifods are compex cobordant if and ony if they have the same Chern numbers. For a cosed compex manifod M, Hirzebruch [10] defined its χ y genus by χ y (M) y p χ(m, Λ p T M). p,q 0 y p ( 1) q dim H q (M, Λ p T M) p 0 Reca that for any hoomorphic vector bunde V M, χ(m, V ) q 0( 1) q dim H q (M, V ). This can be extended to Ω to obtain a compex genus with vaues in C[y]. Dijkgraaf, Moore, Verinde and Verinde [4] introduced a compex genus as foows. For any vector bunde V over M, define the forma sums Λ q V k 0 q k Λ k V, S q V k 0 q k S k V, where Λ k and S k denote the k-th exterior and symmetric product respectivey. Let dim C M d, set E q,y y ( d Λ yq n 1T M Λ y 1 q nt M S q nt M S q nt M ). n 1 One can write E q,y q m y E m,. m, It is easy to see that E m, is a hoomorphic vector bunde of finite rank. Define χ(m; q, y) q m y χ(m, E m, ). m, 1

2 KEFENG LIU, CATHERINE YAN, AND JIAN ZHOU By Hirzebruch-Riemann-Roch theorem, χ(m; q, y) M ch(e q,y )τ(m), where τ(m) is the Todd cass of M. Hence χ(m; q, y) can be expressed in term of Chern numbers of M, therefore it is a compex cobordism invariant. By mutipicative properties of Λ q and S q, it is easy to see that χ(m; q, y) is indeed a genus. It is easy to see that χ(m; 0, y) y d χ y (M). 1.. A conjecture on Hibert schemes of surfaces. Let S be a smooth agebraic surface over C, denote by S [n] the Hobert scheme of zero dimensiona subscheme of ength n. By a we-known resut of Fogarty [7], S [n] is smooth and projective of dimension n. Cosey reated to S [n] is the n-th symmetric product S (n) which is the quotient of the Cartesian product S n by the natura action of permutation group on n objects. There is a morphism π : S [n] S (n) which gives a crepant resoution of S (n). String theory on orbifods motivates a definition of χ(s (n) ; q, y). By physica arguments, Dijkgraaf et a [4] showed that χ(m (n) 1 ; q, y) (1 q m y ) c(nm,) where c(m, ) are given by,m 0, χ(m; q, y) m 0, c(m, )q m y. They conjecture that for a K3 surface or an eian surface S, χ(s (n), q, y) χ(s [n] ; q, y). Therefore, one is ed to the foowing conjectured formua for agebraic surfaces: χ(s [n] 1 (1) ; q, y) (1 q m y ) c(nm,) with c(m, ) given by,m 0, χ(m; q, y) m 0, c(m, )q m y Reduction to CP and CP 1 CP 1. Foowing a recent paper by Eingsrud, Göttsche and Lehn [5], write H(S) [S [n] ]. They have shown that the cobordism cass [H(S)] Ω depends ony on the compex cobordism cass of S Ω. They aso have shown that if [S] a 1 [S 1 ] + a [S ] for some smooth agebraic surfaces S, S 1, S and rationa numbers a a and a, then we have H(S) H(S 1 ) a1 H(S ) a. Minor [11] showed that Ω is a poynomia agebra freey generated by the cobordism casses [CP n ] for positive integers n. Hence for any surface S, [S] a[cp ] + b[cp 1 CP 1 ]

3 HIRZEBRUCH χ y GENERA OF THE HILBERT SCHEMES 3 for some rationa number a and b. As a resut, for any genus φ : Ω R, we have () φ(h(s)) φ(h(cp )) a φ(h(cp 1 CP 1 )) b. In particuar, to prove (1), it suffices to prove it for CP and CP 1 CP 1. expained in Zhou [1], when q 0, (1) reduces to χ y (H(S)) exp χ y m(s) p m (3) 1 (yp) m. m m 0 This formua has been proved by Göttsche-Soerege [8] and Cheah [3] by different methods from agebraic geometry. Einsgrud et a gave a new proof using (). They made use of a resut of Carre-Lieberman [] on the Hodge numbers of Käher manifods which admit C -action with isoated fixed points Locaization formua and combinatorics. The motivation of this paper is to find a method to prove (3) which can be generaized to prove (1). The first named author proposed to use the Atiyah-Bott-Berine-Vergne ocaization theorem, this paper contains the detais for the case of χ y genera. The genera case wi appear in a separate pubication. Haiman [9] has used ocaization theorem for the natura torus action on the Hibert schemes of CP to give a geometric interpreation of t, q-cataan numbers. Our cacuations in this paper shares some simiar feature with his. The interpretation of the weights in terms of Young diagrams greaty simpies the notations used in the origina version. A new feature is the use of its which shoud correspond to the geometric its under the torus action.. Preinaries.1. Locaization of χ y genus. For a cosed compex manifod M, d χ y (M) ch Λ y (T x M)T (M) (1 ye xj j ) 1 e, xj M M j1 where T (M) denotes the Todd cass of M, {x 1,, x d } denote the forma Chern roots of T M. Assume that M admits a torus action with isoated fixed points {F 1,, F m }, and at F i, the weights of the action are {w i,1,, w i,d }. Then by Atiyah-Bott-Berine-Vergne ocaization theorem [1], we have d m j1 χ y (M) (1 w ) i,j m d ye wi,j 1 e w i,j 1 ye wi,j (4) d j1 w i,j 1 e. wi,j i1 Denote by χ y (M) Fi the contribution from F i. i1 j1.. Partitions. For any natura number n, reca the set of partions of n is r P(n) {(b 0, b 1,, b r ) : b 0 b r 1 > b r 0, b j n}. Extend the definition to P(0) {(0)}. For a partition P (b 0, b 1,, b r ) P(n), n > 0, set r(p ) r. j0 As

4 4 KEFENG LIU, CATHERINE YAN, AND JIAN ZHOU Aso set r((0)) 0. There is a one-to-one correspondence between P(n) and the foowing set: {(N 1, N, ) Z : N m 0, m mn m n}, given by setting N m (P ) to be the number of m s in P P(n). Ceary, r(p ) n N m. There is aso a one-to-one correspondence between P(n) and the set of Young diagrams with n boxes. Since there is no danger of confusion, the Young diagram of a partion P wi aso be denoted by P. Denote by B(P ) the set of boxes in a Young diagram P. Notice that the transpose of a Young diagram is aso a Young diagram, so one obtains an invoution t : P(n) P(n). Ceary r(p ) is the number of rows of the Young diagram of P, and b 0 r(p t ). Given an box e in a Young diagram P, the number of boxes on the right of it is caed its arm, whie the number of boxes beow it is caed its eg. Denote by a(e) and (e) the arm and the eg of e respectivey. Notice that (5) card{e B(P ) : a(e) 0} r(p ). Introduce the foowing notation: for P P(n), I(P ) {(i, j, s) 1 i j, b j s b j 1 1}. The set I(P ) is in one-to-one correspondence with the set of the boxes of the transpose of Y (P ). Indeed, (i, j, s) I(P ) corresponds to the box at (s + 1, i). It is easy to see that j i is the arm of this box, whie b i 1 s 1 is the eg..3. Fixed points and weights of the Hibert schemes of C. There is a natura T -action on C : (λ, µ) (x, y) (λx, µy), for λ, µ C, (x, y) C. The ony fixed point of this action is (0, 0). There is an induced action on (C ) [n]. According to Eingsrud and Strømme [6], the fixed points are in one-to-one correspondence with partitions P (b 0, b 1,, b r ) of n. Furthermore, in the representation ring of (C ), (6) T (λ i j 1 µ bi 1 s 1 + λ j i µ s bi 1 ), (i,j,s) I(P ) Here we have used the notation: λ m µ n means the one-dimensiona representation on which (λ, µ) acts as mutipication by λ m µ n. With the notation in., we rewrite (6) as (7) T (λ a(e) 1 µ (e) + λ a(e) µ (e) 1) ). e B(P t ) Note that such combinatoria interpretation of the weights has appeared in Haiman [9]. Ceary, the ove resuts sti hod if λ and µ are two ineary independent weights on T.

5 HIRZEBRUCH χ y GENERA OF THE HILBERT SCHEMES 5 3. Hirzebruch χ y genera of Hibert schemes of surfaces 3.1. Weights of the torus action on the Hibert schemes of CP. Consider the T -action on CP given by (8) (, t ) [z 0 : z 1 : z ] [z 0 : z 1 : t z ], where, t S 1, [z 0 : z 1 : z ] CP. Ceary the fixed points are x 0 [1 : 0 : 0], x 1 [0 : 1 : 0] and x [0 : 0 : 1]. For i 0, 1,, et U i {[z 0 : z 1 : z ] CP : z i 0}. Then U i C. Indeed, on U 0, the isomorphism is given by x z1 z 0, y z z 0. The action (8) corresponds on to the foowing action on C : (9) (, t ) 0 (x, y) ( x, t y). Simiary, near x 1 and x, we get the foowing actions: (10) (11) (, t ) 1 (x, y) ( 1 x, t y), (, t ) (x, y) ( 1 t x, t y). There is an induced T -action on the Hibert scheme CP [n]. We now reca the description of its fixed point set by Eingsrud and Strømme [6]. If Z CP [n] is a fixed point of this action, then the support of Z is contained in {x 0, x 1, x }. Hence we may write Z Z 0 Z 1 Z, where Z i is supported in P i. Let n i be the ength of O Zi, then Z i can be regarded as a fixed point in U [ni i, hence it corresponds to a partition P i P(n i ) by.3. Therefore the fixed point set on CP [n] is in one-to-one correspondence with the foowing set: F(n) {(P 0, P 1, P ) P(n 0 ) P(n 1 ) P(n ) : n 0 + n 1 + n n}. For each (P 0, P 1, P ) F n, denote by F P0,P 1,P the corresponding fixed point. It is cear that a neighborhood of F P0,P 1,P in CP [n] can be identified with the product of some neighborhoods of F Pi in U [ni] i. Hence we have a decomposition By.3 and (9) - (11), we have (1) T FP0 U [n ] n where T FP0,P 1,P CP [n] (λ a(e) 1 e E(P t) 0 µ (e) T FP0 U [n ] n. + λ a(e) µ (e) 1) ). (13) (14) (15) λ 0, µ 0 t, λ 1 1/, µ 1 t /, λ 1/t, µ /t.

6 6 KEFENG LIU, CATHERINE YAN, AND JIAN ZHOU 3.. Appication of the ocaization formua. By (4) and (7), we have where χ y (CP [n] ) F P0,P 1,P χ (0) y(p 0 )χ (1) y(p 1 )χ () y(p ), (16) χ () y(p ) 1 yλ e B(P t) 1 λ a(e)+1 a(e)+1 µ (e) µ (e) 1 yλ a(e) 1 λ a(e) µ (e)+1 µ (e)+1. Now we have χ y (CP [n] )qn n 0 0 (P 0,P 1,P ) F(n) (P 0,P 1,P ) F(n) P P(n ) χ y (CP [n] ) F P0,P 1,P q n χ (0) y(p 0 )χ (1) y(p 1 )χ () y(p )q n y(p )q n. χ () I.e., (17) χ y (CP [n] )qn n 0 0 P P(n ) y(p )q n. χ () Notice that the eft hand side does not depend on and t. As we wi show beow, on the right hand side we can take the its for 0 then t 0, or we can first et then t. So we have χ y (CP [n] )pn n 0 n t 0 t χ() 1 0 P P(n ) t t χ() 1 P P(n ) y(p ) y(p ) The 0 case. From (16) and (13), for any P P(n) we have χ (0) y(p ) e E(P t ) 1 yt a(e)+1 1 t (e) 1 t a(e)+1 1 t (e) 1 yt a(e) 1 t a(e). Since a(e) 0, it is straightforward to see that 1 yt a(e)+1 1 t (e) 1, 0 1 t a(e)+1 1 t (e) 1 yt a(e)+1 1 t (e) y. 1 t a(e)+1 1 t (e)

7 HIRZEBRUCH χ y GENERA OF THE HILBERT SCHEMES 7 When a(e) > 0, when a(e) 0, since (e) 0, we have 1 yt a(e) y; 0 1 t a(e) 1 yt a(e) 1; 1 t a(e) 1 yt a(e) 1 t a(e) 1 yt(e)+1, 1 t (e)+1 1 yt a(e) 1; t t a(e) 1 yt a(e) y. t 1 t a(e) Counting terms with y as the doube its and use (5), we get t 0 t χ(0) 1 0 y(p ) y n r(p t), 0 t χ(1) 0 0 t t χ(0) 1 for a P P(n). To summarize, we get (18) y(p ) (19) t χ(1) 0 y(p ) y(p ) y n+r(p t), y n r(p t) y n+r(p t) y n r(p ), y n+r(p ). Remark 3.1. Furthermore, each term in the product which goes to 1 under the doube it t 0 t1 0 goes to y under the doube it t t1, and vice versa. This sti hods for 1 or The case. Simiary, we have We have hence and χ () y(p ) e B(P t ) 1 y( 1 t ) a(e)+1 ( t1 t ) (e) 1 ( 1 1 y( 1 t ) a(e) ( t1 t ) (e)+1 t ) a(e)+1 ( t1 t ) (e) 1 ( 1 t ) a(e) ( t1 t ). (e)+1 1 y( 1 t ) a(e)+1 ( t1 t ) (e) 0 1 ( 1 t ) a(e)+1 ( t1 t ) (e) { y, (e) > 0, t a(e)+1 y t a(e) y( 1 t ) a(e)+1 ( t1 t ) (e) t ( 1 y, t ) a(e)+1 ( t1 t ) (e) 1 y( 1 t ) a(e) ( t1 0 1 ( 1 1. t ) a(e) ( t1 t ) (e)+1 t ) (e)+1, (e) 0,

8 8 KEFENG LIU, CATHERINE YAN, AND JIAN ZHOU By (5) we have (0) Simiary we have (1) 0 t χ() 0 0 t χ() 0 y(p ) y(p ) y n. y n The 1 case. For any P P(n), by (16) and (14), we have We have hence Simiary, and so Set χ (0) y(p ) e B(P t ) e B(P t ) 1 y( 1 ) a(e)+1 ( t ) (e) 1 ( 1 1 y( 1 ) a(e) ( t ) (e)+1 ) a(e)+1 ( t ) (e) 1 ( 1 ) a(e) ( t ) (e)+1 1 yt (e) a(e) 1 1 t (e) 1 t (e) a(e) 1 1 t (e) 1 yt (e) a(e) 1 1 t (e) 0 1 t (e) a(e) 1 1 t (e) 1 yt (e) a(e) 1 1 t (e) t t (e) a(e) 1 1 t (e) 1 yt a(e) (e) t a(e) (e) 1 1 yta(e) (e) 1 1 t a(e) (e) 1 y, a(e) + 1 > (e), t (e) y t (e) 1, a(e) + 1 (e)(> 0), 1, a(e) + 1 < (e), { y, a(e) + 1 (e), 1, a(e) + 1 < (e). 1, a(e) > (e) + 1, 1 yt (e)+1, a(e) (e) + 1, 1 t (e)+1 y, a(e) < (e) + 1, 1 y( t1 t 0 ) a(e) ( t t 0 ) (e) 1 { 1, a(e) (e) + 1, 0 t ( t1 t 0 ) a(e) ( t t 0 ) (e) 1 y, a(e) < (e) + 1. then it is cear that To summarize, we have () Simiary we have (3) s(p ) card{e E(P t ) : (e) 1 a(e) (e)} card{e E(P t ) : a(e) (e) a(e) + 1}, 0 t χ(0) 0 0 y(p ) y n s(p ). 0 t χ(0) 0 0 P P(N) t χ(0) 0 y(p ) y(p ) P P(N) y n s(p ). y n+s(p )..

9 HIRZEBRUCH χ y GENERA OF THE HILBERT SCHEMES Putting pieces together. Now we have χ y (CP [n] ) y n y n y n+r(p ) y n r(p ) y n s(p ) y n+s(p ). Lemma 3.1. We have y r(p ) y s(p ). Proof. For n > 0, write r n (y) y r(p ), s n (y) y s(p ). Aso write r 0 (y) s 0 (y) 1. Suppose that we have shown that r j (y) s j (y) for j 1,, n. From the ove formua we have (py) n r n (y) (py) n s n (y 1 ) (py) n r n (y 1 ) (py) n s n (y). Comparing the coefficients of (py) n+1 one gets r n+1 (y) + n r j (y)s j (y 1 ) + s n+1 (y 1 ) r n+1 (y 1 ) + j1 By the induction hypothesis, one gets n r j (y 1 )s j (y) + s n+1 (y). j1 r n+1 (y) + s n+1 (y 1 ) r n+1 (y 1 ) + s n+1 (y). Now r n+1 (y) and s n+1 are poynomias in y, and r n+1 has no constant terms, so we must have r n+1 (y) s n+1 (y). The proof is compete. As a coraary, we have χ y (CP [n] ) 0 y n y n+( 1)r(P ). y n+r(p ) y n r(p )

10 10 KEFENG LIU, CATHERINE YAN, AND JIAN ZHOU On the other hand, we aso have ( ) ( χ y n(cp ) exp n 1 (yp) n exp n exp n (1 + yn + y n ) m 0(yp) nm ) 1 + y n + y n 1 (yp) n m 0 exp m 0 m 0 m 1 ( ) 1 n ((ym p m+1 ) n + (y m+1 p m+1 ) n + (y m+ p m+1 ) n ) exp ( n(1 y m p m+1 ) n(1 y m+1 p m+1 ) n(1 y m+ p m+1 ) ) 1 (1 y m p m+1 )(1 y m+1 p m+1 )(1 y m+ p m+1 ) 1 (1 y m 1 p m )(1 y m q m )(1 y m+1 p m ). For 0, 1,, 1 1 y m+ 1 p m (y m+ 1 p m ) Nm m 1 m 1 N m 0 y (mn m+( 1)N m) p mn m and so mnmn y n+( 1)r(P ), ( χ y (CP [n] )pn exp n mnmn ) χ y n(cp ) 1 (yp) n. y n+( 1) N m 3.7. The case CP 1 CP 1 case. Consider the foowing T -action on CP 1 CP 1 : (, t ) ([z 0 : z 1 ], [w 0 : w 1 ]) ([z 0 : z 1 ], [w 0 : t w 1 ]), where, t C, [z 0 : z 1 ], [w 0 : w 1 ] CP 1. It has four fixed points: P 00 ([1 : 0], [1 : 0]), P 01 ([1 : 0], [0 : 1]), P 10 ([0 : 1], [1 : 0]), P 11 ([0 : 1], [0 : 1]). Simiar to the discussion in 3.1, the fixed point set of the induced action on the n- th Hibert scheme of CP 1 CP 1 is in one-to-one correspondence with the foowing set {(P 00, P 01, P 10, P 11 ) P(n 00 ) P(n 01 ) P(n 10 ) P(n 11 ) : n 00 +n 01 +n 10 +n 11 n}. Furthermore the tangent space at the fixed point corresponding to the quadrupe of partitions (P 00, P 01, P 10, P 11 ) has the foowing decomposition: T T 00 T 01 T 10 T 11,

11 HIRZEBRUCH χ y GENERA OF THE HILBERT SCHEMES 11 where in the representation ring of T, T where As in 3. we get (λ a(e) 1 e E(P ) χ y ((CP 1 CP 1 ) [n] )q n where χ () y (P ) µ (e) + λ a(e) µ (e) 1 ), λ t ( 1)a 1, µ t ( 1)b,. 1 a,b0 1 yλ e E(P t ) 1 λ a(e) 1 n 0 P P(n ) a(e) 1 µ (e) µ (e) y (P )q n, χ () 1 yλa(e) 1 λ a(e) µ (e) 1 µ (e) 1 After taking the it as 0 then taking the it as t 0, one obtains χ y ((CP 1 CP 1 ) [n] ) y n y n+r(p ). 1 (1 y m p m+1 )(1 y m+1 p m+1 )(1 y m+ p m+1 ) m 0 ( ) ( ) 1 + y n + y n χ y n(cp 1 CP 1 ) exp n 1 (yp) n exp n 1 (yp) n Since χ y is a compex genus, by 1.3, we have competed the proof of (3). References y n r(p ) [1] M.F. Atiyah, R. Bott, The moment map and equivariant cohomoogy, Topoogy 3 (1984), 1-8. [] J.B. Carre, D.I. Lieberman, Hoomorphic vector fieds and Kaeher manifods, Invent. Math. 1 (1973), [3] J. Cheah, On the cohomoogy of Hibert schemes of points, J. Agebraic Geom. 5 (1996), no. 3, [4] R. Dijkgraaf, G. Moore, E. Verinde, H. Verinde, Eiptic genera of symmetric products and second quantized strings, Comm. Math. Phys. 185 (1997), no. 1, [5] G. Eingsrud, L. Göttsche, M. Lehn, On the Cobordism Cass of the Hibert Scheme of a Surface, preprint, math.ag/ [6] G. Eingsrud, A. Strømme, On the cohomoogy of the Hibert scheme of points in the pane Invent. Math. 87 (1987), [7] J. Fogarty, Agebraic famiies on agebraic surfaces, Amer. J. Math. 10 (1968), [8] H. Göttsche, W. Soerge, Perverse sheaves and the cohomoogy of Hibert schemes of smooth agebraic surfaces, Math. Ann. 96 (1993), no., [9] M. Haiman, t, q-cataan numbers and the Hibert scheme. Seected papers in honor of Adriano Garsia (Taormina, 1994). Discrete Math. 193 (1998), no. 1-3, [10] F. Hirzebruch, Topoogica ethods in agebraic geometry, Grund. der math. Wiss. 131, Springer Verag, Berin, [11] J. Minor, O the cobordism ring Ω and a compex anaogue, Amer. J. Math. 8 (1960),

12 1 KEFENG LIU, CATHERINE YAN, AND JIAN ZHOU [1] J. Zhou, Deocaized equivariant cohomoogy of symmetric products, preprint, math.dg/ Department of Mathematics, Stanford University, Stanford, CA E-mai address: Department of Mathematics, Texas A&M University, Coege Station, TX E-mai address: Department of Mathematics, Texas A&M University, Coege Station, TX E-mai address: