The Lawson-Yau Formula and its generalization

THE LA WSON-Y A U F ORMULA AND IT S GENER ALIZA TION WENCHUAN HU Abstract. The Euler cha racteristic of Chow v arieties of algebraic cycles of a give n degree in complex pro jectiv e spaces wa s computed b y Blaine Lawson and Stephen Y au by using holomorphic symmetries of cycles spaces. In this paper we compute this in a direct and elemen tary wa y and generalize th is formula to the l -adic Euler- Poincar´ e charac teristic for Chow v arieties o ver any algebraically closed field. M oreo ver, the Euler charact eristic for Chow v arieties with certain group action i s calculated. In particular, we calculate the Euler c haracteristic of the s pace of right quat ernionic cycles of a given dimension and degree in complex pro j ectiv e spaces. Contents 1. Int ro duction 1 2. An ele men tary pr o of of the Lawson-Y au for m ula 2 3. The l -adic Euler -Poincar´ e Characteristic for Chow v arieties 4 4. Algebraic cy c le s with gr oup actio n 6 5. The Euler Cha racteristic for the spa ce of r ight-quaternionic cycles 9 References 11 1. Intr oduction Let P n be the complex pro jective spa ce of dimensio n n and let C p,d ( P n ) b e the space of effective algebra ic p -c ycles of degree d o n P n . A fact proved by Chow and V an der W aerden is that C p,d ( P n ) carr ies the structure of a closed complex alg ebraic v ariet y [1]. Hence it carries the str ucture o f a compact Haus do rff spa c e. Denote by χ ( C p,d ( P n )) the Euler Characteristic of C p,d ( P n ). This num b er w as computed in terms o f p, d and n by Bla ine Lawson a nd Stephen Y au explicitly , i.e., Theorem 1.1 (La wson-Y au, [11]) . (1) χ ( C p,d ( P n )) =  v p,n + d − 1 d  , wher e v p,n = ( n +1 p +1 ) . This equa tion (1) is c alled the La wson-Y au form ula . The o r iginal metho d of calculation is an application o f a fixed point formula for a compact complex analytic space with a weakly holomor phic S 1 -action. Equiv alen tly , if we define Q p,n ( t ) := ∞ X d =0 χ ( C p,d ( P n )) t d , Date : Decem b er 5, 2008. Key wor ds and phr ases. Algebraic cycles; Chow v arieties; Euler c haracteristic, Group action. 1 2 WENCHUAN HU then the L awson-Y au formula may b e restated as (2) Q p,n ( t ) =  1 1 − t  ( n +1 p +1 ) , where χ ( C p, 0 ( P n )) := 1 . A direct and elemen tary pro of o f the Lawson-Y au for m ula is g iven in section 2 by using the tec hnique “ pulling of nor mal co ne” established in the b o ok of F ulton ([3]), which w as used by Lawson in proving his Complex Susp ension Theor em ([8]). As an a pplication o f this elementary metho d, we obtain an l -adic version o f the Lawson-Y au formula: Theorem 1.2. L et C p,d ( P n ) K b e the sp ac e of effe ctive p -cycles of de gr e e d in P n K . F or al l l prime to cha r ( K ) , we have (3) χ ( C p,d ( P n ) K , l ) =  v p,n + d − 1 d  , wher e v p,n = ( n +1 p +1 ) , wher e χ ( X K , l ) denotes the l -adic Euler-Poinc ar´ e Char acteristic of an algebr aic variety X K over K . The detailed explanation of notations in Theorem 1.2 a s w ell as its proo f is given in section 3. W e apply o ur method to the space of a lgebraic cy cles with certain finite g roup action G to obtain the E uler characteristic of the G -in v ariant Chow v ar ieties (see Theorem 4.5). As an application, w e obtain the Euler characteristic of the spac e of righ t quater - nionic c y cles of a giv en dimens io n and degre e in co mplex p ro jective spa c es (see Corollar y 5 .2 ). A cknow le dgements : I would like to gra titude Michael Artin, Eric F riedlander and James McKerna n for their interest a nd enc o uragement as well as helpful advice on the organiz a tion of the pap er. 2. An element ar y proof of the La wson-Y au formula First we list a result we will use in our calculation. See [4], pag e 95 . Lemma 2.1. L et X b e a c omplex algebr aic variety (not ne c essarily smo oth, c om- p act, or irr e ducible) and let Y ⊂ X b e a close d algebr aic set with c omp lement U . Then χ ( X ) = χ ( U ) + χ ( Y ) . Now we give a brief r e v iew of Lawson’s constructio n of the susp ension map a nd some of its pr op erties. The r eader is referred to [8] for details. Fix a h yper plane P n ⊂ P n +1 and a point P = [0 : · · · : 0 : 1] ∈ P n +1 − P n . Let V ⊂ P n be an y closed algebraic subset. The algebraic susp e ns ion o f V with vertex P (i.e., co ne ov er P ) is the set Σ P V := [ { l | l is a pr o jective line through P a nd intersects V } . Extending by linea rity a lg ebraic susp ensio n gives a contin uous homo mo rphism (4) Σ P : C p,d ( P n ) → C p +1 ,d ( P n +1 ) for a n y p ≥ 0. Set T p +1 ,d ( P n +1 ) :=  c = X n i V i ∈ C p +1 ,d ( P n +1 ) | dim( V i ∩ P n ) = p, ∀ i  The La wson-Y au formula and its generalization 3 and B p +1 ,d ( P n +1 ) := C p +1 ,d ( P n +1 ) − T p +1 ,d ( P n +1 ). The following pr op osition proved b y Lawson in [8] is the key p oint in our calcu- lation. An a lgebraic version was g iven by F riedlander in [2]. Prop ositio n 2. 2 . The subset T p +1 ,d ( P n +1 ) ⊂ T p +1 ,d ( P n +1 ) is Zariski op en. Mor e- over, the image Σ P : C p,d ( P n ) → C p +1 ,d ( P n +1 ) is include d in T p +1 ,d ( P n +1 ) and Σ P ( C p,d ( P n )) ⊂ T p +1 ,d ( P n +1 ) is a str ong deformation r etr act. In particular, their Euler characteristics coincide, i.e., we hav e (5) χ ( C p,d ( P n )) = χ ( T p +1 ,d ( P n +1 )) . By P rop osition 2.2, B p +1 ,d ( P n +1 ) is a close d subset of C p +1 ,d ( P n +1 ). F rom the definition, B p +1 ,d ( P n +1 ) = { c = X n i V i ∈ C p +1 ,d ( P n +1 ) | V i ⊂ P n , for s ome i } , i.e., there is at leas t one irreducible comp onent lying in the fixed hyperpla ne P n . Lemma 2. 3. B p +1 ,d ( P n +1 ) = ∐ d i =1 B p +1 ,d ( P n +1 ) i , wher e ∐ me ans disjoint union and B p +1 ,d ( P n +1 ) i =            c ∈ B p +1 ,d ( P n +1 )           c = P n k V k + P m j W j , V k ⊂ P n , ∀ k, dim( W j ∩ P n ) = p, ∀ j deg( P n k V k ) = i, and deg ( P m j W j ) = d − i .            . F or e ach i , B p +1 ,d ( P n +1 ) i = C p +1 ,i ( P n ) × T p +1 ,d − i ( P n +1 ) . Pr o of. Clear from the definition of B p +1 ,d ( P n +1 ) i .  F rom Le mma 2.3, we hav e χ ( B p +1 ,d ( P n +1 ) i ) = χ ( C p +1 ,i ( P n )) · χ ( T p +1 ,d − i ( P n +1 )) . Hence we g et χ ( B p +1 ,d ( P n +1 )) = P d i =1 χ ( B p +1 ,d ( P n +1 ) i ) (b y inclusion- exclusion principle) = P d i =1 χ ( C p +1 ,i ( P n )) · χ ( T p +1 ,d − i ( P n +1 )) = P d i =1 χ ( C p +1 ,i ( P n )) · χ ( C p,d − i ( P n )) , (b y equatio n (5)) Therefore we hav e the following result: Prop ositio n 2. 4. F or any inte ger p ≥ 0 and d ≥ 1 , we have t he fol lowing r e cursive formula (6) χ ( C p +1 ,d ( P n +1 )) = χ ( C p,d ( P n )) + d X i =1 χ ( C p +1 ,i ( P n )) · χ ( C p,d − i ( P n )) , wher e χ ( C q, 0 ( P N )) = 1 for inte gers N ≥ q ≥ 0 . In p articular, when d = 1 , e quation (6) is just the c ombinatorial identity ( n +2 p +2 ) = ( n +1 p +1 ) + ( n +1 p +2 ) . T o co mpute χ ( C p,d ( P n )), it is enoug h to identify the initial v alues. Lemma 2.5. χ ( C 0 ,d ( P n )) = ( n + d d ) . The equality is a spec ial cas e of MacDonald formula ([12]). 4 WENCHUAN HU Pr o of of L emm a 2.5. Now w e give a n indep e nden t pr o of for MacDonald formula in this sp ecial ca se. W e ca n write C 0 ,d ( P n +1 ) = C 0 ,d ( C n +1 ) a B 0 ,d ( P n +1 ) , where C 0 ,d ( C n +1 ) ⊂ C 0 ,d ( P n +1 ) c o nt ains effectiv e 0-cycles c o f deg r ee d such that no points in c lying in the fixed h yp erplane P n and B 0 ,d ( P n +1 ) is the complement of C 0 ,d ( C n +1 ) in C 0 ,d ( P n +1 ). It is ea sy to see that C 0 ,d ( C n +1 ) is co n tractible. W e can write B 0 ,d ( P n +1 ) = ` d i =1 B 0 ,d ( P n +1 ) i as in Lemma 2 .3, where B 0 ,d ( P n +1 ) i contains 0-cycles c of degree d on P n +1 in which there are ex a ct i po in ts (count m ultiplic- ities) lying in P n , hence B 0 ,d ( P n +1 ) i = C 0 ,i ( P n ) × C 0 ,d − i ( C n +1 ). In par ticular, χ ( B 0 ,d ( P n +1 )) = P d i =1 χ ( C 0 ,i ( P n )). Therefore, we hav e χ ( C 0 ,d ( P n +1 )) = 1 + d X i =1 χ ( C 0 ,i ( P n )) . The firs t fo rmula in the lemma follows from this by induction.  Pr o of of The or em 1.2. Equation (6) together with Lemma 2.5 is equiv alent to the following re cursive functional eq uation with initial v alues (7) Q p +1 ,n +1 ( t ) = Q p +1 ,n ( t ) · Q p,n ( t ) , Q 0 ,m ( t ) = ( 1 1 − t ) m +1 . F rom this, we get the equation (2) by induction on n and hence the Lawson-Y au formula (1).  Example 2.6. F or divisors of de gr e e d in P n , we have the formula χ ( C p,d ( P p +1 )) = ( p + d +1 d ) . F rom eq uation 7 , w e have Q p,p +1 ( t ) = Q p,p ( t ) · Q p − 1 ,p ( t ) = 1 1 − t · Q p − 1 ,p ( t ) since C p,d ( P p ) contains exactly one degree d cycle and so χ ( C p,d ( P p )) = 1, i.e, Q p,p ( t ) = 1 1 − t . By the fact that Q 0 , 1 ( t ) = ( 1 1 − t ) 2 and induction on p , we g e t Q p,p +1 ( t ) = ( 1 1 − t ) p +2 . Hence χ ( C p,d ( P p +1 )) = ( p + d +1 d ). Alternatively , this formula follows directly from the fact that C p,d ( P p +1 ) is the mo duli space of hype r surfaces of degr ee d in P p +1 and hence it is a co mplex pro jec- tive space o f dimension ( p + d +1 d ) − 1. T o see this, we choose a basis for the mo nomials of degree d in p + 2 v ariables a nd then a sso ciate a p oint in this pro jective spa ce to the hypersurface whose defining eq uation is g iven by the co or dinates of tha t p oint (cf. [2]). 3. The l -adic Euler-Poincar ´ e Characte ristic for Chow v arieties . In the se ction, the Lawson-Y au formula is generalized to an algebraically closed field K w ith arbitrar y c haracteristic char ( K ) ≥ 0. Let l b e a p ositive in teger prime to char ( K ). F or a v ariety X ov er K , let H i ( X, Z l ) b e the l -adic coho mo logy group of X . Set H i ( X, Q l ) := H i ( X, Z l ) ⊗ Z l Q l . Denote by β i ( X, l ) := dim Q l H i ( X, Q l ) the i -th l -adic B e tti num b er o f X . The l -a dic Euler Cha r acteristic is defined b y χ ( X , l ) := The La wson-Y au formula and its generalization 5 P i ( − 1) i β i ( X, l ). Similarly , let H i c ( X, Z l ) b e the l - adic cohomology gr oup o f X with compact suppo rt. Set β i c ( X, l ) := dim Q l H i c ( X, Q l ) the i -th l - a dic Be tti num ber of X with c o mpact supp ort and χ c ( X, l ) := P i ( − 1) i β i c ( X, l ) the l -adic E uler-Poincar´ e Character is tic with compact supp ort. Note that χ c ( X, l ) is independent of the choice of l prime to char ( K ) (See, e.g., [6] o r [5]). By using the metho d in the last section we will deduce the l -adic version of Lawson-Y au formula (see Theo rem 1.2). F rom the pro of of equation (2) ab ov e, w e need similar results for Lemma 2.1-2.5, Prop osition 2.2 and the homotopy inv a r iance of l -adic Euler-Poincar´ e Char a cteris- tics. As we stated b efore, an algebraic version of Pro po sition 2.2 o ver a ny algebra ic ally closed field K w as proved b y F riedla nder (cf. [2], Pr op.3.2). Lemma 2.3 is a purely algebraic result and hence it holds for an y algebraically clo sed field K . An a lgebraic version of the first statement in Lemma 2.5 follows fro m a co r resp onding result of Lemma 2 .1. An algebr aic version of the seco nd statement in Lemma 2.5 ho lds over any a lgebraica lly closed field. Ther e fo re, the key par t to prov e Theor em 1.2 is the following algebraic version of Lemma 2.1 and the homo topy in v ariance of l -a dic cohomolog y . Lemma 3 . 1. L et X b e an algebr aic varie ty (n ot ne c essarily smo oth, c omp act, or irr e ducible) over an algebr aic al ly close d file d K and let Y ⊂ X b e a close d algebr aic set with c omplement U . Then χ ( X, l ) = χ ( U, l ) + χ ( Y , l ) for any p ositive inte ger l prime to char ( K ) . Pr o of. This lemma follows from the long lo calizatio n exact sequence for l - adic coho- mology and the following result pro ved by Laumon (independently b y Gabber).  Prop ositio n 3.2 ([7]) . F or any algebr aic variety X over an algebr ai c al ly close d field K and inte ger l prime to cha r ( K ) , we have χ ( X, l ) = χ c ( X, l ) . T o pr ov e T he o rem 1 .2, we also need the following definition (cf. [2], pa ge 6 1). Definition 1. A pr op er morphism g : X ′ → X of lo c al ly no etherian schemes is said to b e a bic onti nuous al gebr aic morphi sm if it is a set the or etic bije ction and if for every x ∈ X the asso cia te d map of re sidue fi elds K ( x ) → K ( g − 1 ( x )) is pur ely insep ar able . A c onti nuous algebr aic m ap f : X → Y is a p air ( g : X ′ → X, f ′ : X → Y ) in which g is a bic ontinuous algebr aic morphism (and f is a morphism). Lemma 3.3. L et F : X × A 1 K → Y b e a c ont inuous algebr aic map such that F ( − , 0) = f ( − ) and F ( − , 1) = g ( − ) . Then the pul lb ack f ∗ : H i ( Y , Z l ) → H i ( X, Z l ) is e qual to g ∗ : H i ( Y , Z l ) → H i ( X, Z l ) . Pr o of. It is essen tially prov ed in [2], Pr op. 2.1. The map F : X × A 1 K → Y induces a ma p in co homology F ∗ : H i ( Y , Z l ) → H i ( X × A 1 K , Z l ). By the us ua l homotopy inv ar iance of etale cohomology , one has the homotopy inv aria nce of l - adic co homology , i.e., H i ( X × A 1 K , Z l ) ∼ = H i ( X, Z l ). Therefore, we get the equa lit y of f and g by the restr ic tion of F to X × 0 and X × 1 .  Corollary 3.4. L et i : Y ⊂ X b e an algebr aic al ly close d su bset. L et F : X × A 1 K → X b e a c ontinuous alge br aic map such that F ( − , 0) = id X , F ( x, 1 ) ◦ i = i d Y and F ( y , t ) = y for y ∈ Y . Then i ∗ : H i ( X, Z l ) → H i ( Y , Z l ) is an isomorphism. 6 WENCHUAN HU The detailed computation is given b elow. Pr o of of The or em 1.2. Set e Q p,n ( t ) := P ∞ d =0 χ ( C p,d ( P n )) K t d , T p +1 ,d ( P n +1 ) K :=  c = X n i V i ∈ C p +1 ,d ( P n +1 ) K | dim( V i ∩ P n K ) = p, ∀ i  and B p +1 ,d ( P n +1 ) K :=  c = X n i V i ∈ C p +1 ,d ( P n +1 ) K | V i ⊂ P n K , for some i  By Coro llary 3.4 and the a lgebraic version o f Pro po sition 2.2, we hav e (8) χ ( C p,d ( P n ) K ) = χ ( T p +1 ,d ( P n +1 ) K ) . F rom the K¨ unneth formula for l -adic cohomolo g y (cf. [13]), the algebraic v ersio n of Lemma 2 .3 and Equa tion (8), we get (9) χ ( C p +1 ,d ( P n +1 ) K ) = χ ( C p,d ( P n ) K ) + d X i =1 χ ( C p +1 ,i ( P n ) K ) · χ ( C p,d − i ( P n ) K ) , for integers p ≥ 0 and d ≥ 1. By Coro llary 3 .4, w e hav e χ ( C 0 ,d ( A n K )) = 1, wher e A n K is the n -dimensiona l affine spa ce over K . Hence the co mputation in the pro of o f Lemma 2.5 works when C is replac ed by any algebra ically closed field K and we get the formula for χ ( C 0 ,d ( P n ) K ): (10) χ ( C 0 ,d ( P n ) K ) = ( n +1 d +1 ) . F rom the fact that C p,d ( P p +1 ) K is the mo duli space of hypers urfaces of degree d in P p +1 K and hence it is a pro jective space ov er K of dimension ( p + d +1 d ) − 1. (11) χ ( C p,d ( P p +1 ) K ) = ( p + d +1 d ) . F rom the definitio n of e Q p,n ( t ) a nd Equa tion (9)-(11), we get (12) e Q p +1 ,n +1 ( t ) = e Q p +1 ,n ( t ) · e Q p,n ( t ) , e Q 0 ,m ( t ) = ( 1 1 − t ) m +1 , e Q q,q +1 ( t ) = ( 1 1 − t ) q +2 . F rom E quation (12), we complete the pr o of of Theo rem 1.2 by induction on n .  4. Algebraic cycles with group action In this section, w e apply our method to the space of a lgebraic cycles with c ertain finite group action G to obtain the Euler c haracteris tic of the G -in v ariant Chow v arieties. Let G be a finite group of the automorphism of P n . By pass ing to an appropria te group extension we ca n always assume that ρ : G → U n +1 and that its action o n P n comes from the linear action of U n +1 on homogeneous coor dinates. The actio n of G on P n induces a c tio ns on the Chow v arieties C p,d ( P n ). Denote by C p,d ( P n ) G := { c ∈ C p,d ( P n ) : g ∗ c = c, ∀ g ∈ G } The La wson-Y au formula and its generalization 7 the G -inv ar ia nt subse t of C p,d ( P n ). Since G is a finite g roup o f the automorphism of P n , it induces a n automor phism of C p,d ( P n ). Hence C p,d ( P n ) G is a clos e d c o mplex subv ariety o f C p,d ( P n ). Cho ose homog eneous co ordinates C n +2 = C n +1 ⊕ C for P n +1 = Σ P n and extend the fixed linear repres en tation ρ : G → U n +1 to a representation ˜ ρ : G → U n +2 by setting ˜ ρ = ρ ⊕ λ · id C , where λ ∈ C ∗ is a fixed complex num ber. The co ns truction was given in [9], where λ is chosen to b e 1. Set T p +1 ,d ( P n +1 ) G :=  c = P n i V i ∈ C p +1 ,d ( P n +1 ) G | dim( V i ∩ P n ) = p, ∀ i  and B p +1 ,d ( P n +1 ) G = C p +1 ,d ( P n +1 ) G − T p +1 ,d ( P n +1 ) G . The following pr o po sition pr ov ed by Lawson and Michelsohn in [9] will be used in our calcula tion. Prop ositio n 4.1 ([9]) . F or e ach p ≥ 0 , T p +1 ,d ( P n +1 ) G ⊂ C p +1 ,d ( P n +1 ) G is Zariski op en . Mor e over, the image Σ : C p,d ( P n ) G → C p +1 ,d ( P n +1 ) G is include d in T p +1 ,d ( P n +1 ) G and Σ( C p,d ( P n ) G ) ⊂ T p +1 ,d ( P n +1 ) G is a str ong defor- mation re tr act. In particular, their Euler characteristics coincide, i.e., we hav e χ ( C p,d ( P n ) G ) = χ ( T p +1 ,d ( P n +1 ) G ) . Remark 4. 2. In [9] , L awson and Michelsoh n c onsid er t he extension ˜ ρ = ρ ⊕ λ · id C only for the c ase λ = 1 . However, their pr o of works for al l λ ∈ C ∗ without changing anything exc ept that 1 is r epla c e d by λ in ˜ ρ . F rom the definitio n, B p +1 ,d ( P n +1 ) G =  c = X n i V i ∈ C p +1 ,d ( P n +1 ) G | V i ⊂ P n , for some i  , i.e., there is at least one irreducible comp onent lying in the fixed G -inv ar iant hy- per plane P n . Lemma 4. 3. B p +1 ,d ( P n +1 ) G = ∐ d i =1 B p +1 ,d ( P n +1 ) G i , wher e ∐ me ans disjoint union and B p +1 ,d ( P n +1 ) G i =            c ∈ B p +1 ,d ( P n +1 ) G           c = P n k V k + P m j W j , V k ⊂ P n , ∀ k , dim( W j ∩ P n ) = p, ∀ j deg( P n k V k ) = i, and deg ( P m j W j ) = d − i .            . F or e ach i , B p +1 ,d ( P n +1 ) G i = C p +1 ,i ( P n ) G × T p +1 ,d − i ( P n +1 ) G . Pr o of. An algebr aic cycle c ∈ B p +1 ,d ( P n +1 ) G i may be written a s c = P n k V k + P m j W j as the formal sum of irre ducible v ar ieties, wher e V k ⊂ P n and dim( W j ∩ P n ) = p + 1 . Since c is G -inv aria nt , we hav e P n k V k and P m j W j are G -inv aria nt. T o see this, recall that G is iden tified with the subgroup o f the unitar y gro up U n +1 . Supp ose g ∗ ( V i ) = W j for some g ∈ G a nd i, j . Since P n is G -inv a riant, W e ha ve P n = g ∗ ( P n ) ⊂ g ∗ ( V i ) = W j . This contradicts to the a ssumption that dim( W j ∩ P n ) = p + 1. Similar fo r P m j W j . Now the lemma follo ws fro m the definitio n of B p +1 ,d ( P n +1 ) G i .  8 WENCHUAN HU F rom Le mma 4.3, we hav e χ ( B p +1 ,d ( P n +1 ) G i ) = χ ( C p +1 ,i ( P n ) G ) · χ ( T p +1 ,d − i ( P n +1 ) G ) . Hence we g et χ ( B p +1 ,d ( P n +1 )) = P d i =1 χ ( B p +1 ,d ( P n +1 ) G i ) (b y inclusion- exclusion principle) = P d i =1 χ ( C p +1 ,i ( P n ) G ) · χ ( T p +1 ,d − i ( P n +1 ) G ) = P d i =1 χ ( C p +1 ,i ( P n ) G ) · χ ( C p,d − i ( P n ) G ) . Therefore we hav e the following result: Prop ositio n 4.4 . F or any inte ger p ≥ 0 and d ≥ 1 , we have t he fol lowing formula (13) χ ( C p +1 ,d ( P n +1 ) G ) = χ ( C p,d ( P n ) G ) + d X i =1 χ ( C p +1 ,i ( P n ) G ) · χ ( C p,d − i ( P n ) G ) , wher e χ ( C q, 0 ( P N ) G ) = 1 for inte gers N ≥ q ≥ 0 . F rom our constr uction, we know that if the representation ρ : G ⊂ U n +1 is diag- onalizable , i.e., up to a linea r transfor mation, ρ = ⊕ n +1 i =1 λ i · id C , then equatio n (13) gives us a recursive formula. In these cases, the E uler ch ara cteristic is calculated explicitly a s follows. Theorem 4 . 5. L et ρ : G ⊂ U n +1 b e a diagonali zable r epr esent ation. The Euler char acteristic of Chow variety of G -invariant cycles χ ( C p,d ( P n ) G ) is given by the formula χ ( C p,d ( P n ) G ) =  v p,n + d − 1 d  , wher e v p,n = ( n +1 p +1 ) . Pr o of. The theorem fo llows from Prop os itio n 4 .4 and the following initial v alue s ident ities: (14) χ ( C 0 ,d ( P n ) G ) = ( n + d d ) As b e fore, we can wr ite C 0 ,d ( P n +1 ) G = C 0 ,d ( C n +1 ) G a B 0 ,d ( P n +1 ) G , where C 0 ,d ( C n +1 ) G ⊂ C 0 ,d ( P n +1 ) G contains effective G -in v ariant 0 -cycles c of de- gree d such tha t no p oints in c lying in the fix ed hype rplane P n and B 0 ,d ( P n +1 ) G is the complement of C 0 ,d ( C n +1 ) G in C 0 ,d ( P n +1 ) G . W e claim that C 0 ,d ( C n +1 ) G is contractible. T o see this, note that P n +1 − P n = C n +1 and Let φ t : C n +1 → C n +1 denote s calar multiplication by t ∈ C . The family of maps φ t induces a family of maps φ t ∗ : C 0 ,d ( C n +1 ) → C 0 ,d ( C n +1 ) since the multiplication by t ∈ C is G - inv ariant. F rom the definition, the map φ 1 ∗ = id and φ 0 ∗ is a constant map. W e c an write B 0 ,d ( P n +1 ) G = d a i =1 B 0 ,d ( P n +1 ) G i as in Lemma 4.3, where B 0 ,d ( P n +1 ) G i contains G -inv ariant 0-cycle s c of degr ee d on P n +1 in which there a re exact i p oints (count m ultiplicities) lying in P n , hence B 0 ,d ( P n +1 ) G i = C 0 ,i ( P n ) G × C 0 ,d − i ( C n +1 ) G . In particula r , χ ( B 0 ,d ( P n +1 ) G ) = P d i =1 χ ( C 0 ,i ( P n ) G ). The La wson-Y au formula and its generalization 9 Therefore, we hav e χ ( C 0 ,d ( P n +1 ) G ) = 1 + d X i =1 χ ( C 0 ,i ( P n ) G ) . Now the for m ula in the lemma follo ws fro m this by induction.  Remark 4. 6. By a c ar eful ly che cking the pr o of of The or em 4.6 in [9] and the pr o of of The or em 4.5 ab ove , we observe that if the line ar r epr esentation ρ : G → U n +1 is diago nalizable, then the c onclusion in The or em 4.5 holds even if ther e is no assumption of fi niteness of G . Mor e pr e cisel y, let T n +1 ⊂ U n +1 b e the maximal torus and let G b e any sub gr oup of T n +1 . Then The Euler char ac teristic of Chow variety of G -invariant cycles χ ( C p,d ( P n ) G ) is given by the formula χ ( C p,d ( P n ) G ) =  v p,n + d − 1 d  , wher e v p,n = ( n +1 p +1 ) . This explains The o r em 4.1, one of the m ain r esults in [11] , on the invarianc e of Euler char acteristic of a c omp act c omplex analytic s p ac e and t hat of the fixe d-p oint set under a ho lomorphic S 1 -action in the imp ortant c ase for Chow varieties over C . 5. The Euler Characte ristic f or the sp a ce o f right-qua ternionic cycles Let H denote the qua ternions with standa r d basis 1 , i , j , k , and le t C 2 ∼ = → H b e the canonical isomo rphism given by ( u, v ) 7→ u + v j . This gives us a ca nonical complex is omorphism C 2 n ∼ = → H n . Under this iden tification right scalar m ultiplication b y j in H n bec omes the complex line a r ma p J : C 2 n → C 2 n , J ( u 1 , ..., u n , v 1 , ..., v n ) = ( − v 1 , ..., − v n , u 1 , ..., u n ) . This induces a holomo rphic map ¯ J : P 2 n − 1 → P 2 n − 1 with ¯ J = I d . Note that the fixed p oint set of ¯ J is a pair of disjoint P n − 1 . The inv olution ¯ J c arries algebr aic subv arieties o f P 2 n − 1 to themselves and induces a holomorph ic inv olution (15) ¯ J ∗ : C p,d ( P 2 n − 1 ) → C p,d ( P 2 n − 1 ) for a ll p and d . Remark 5.1. The c onstruction is an analo g to the one given in [10] , wher e a left sc alar multiplic atio n by j on H n was che cke d in detail. We c onsider the right multiplic atio n her e s inc e the induc e d map ¯ J ∗ is a holomorphic. Let C p,d ( n ) ⊂ C p,d ( P 2 n − 1 ) denote the ¯ J ∗ -fixed p oint set, i.e., the set of ¯ J ∗ - inv ariant alge braic p -cycles. An element c ∈ C p,d ( n ) is called a right quater nioni c cycle . Since ¯ J ∗ is a holomo rphic inv olution, C p,d ( n ) is a c losed complex a lg ebraic set. Since J is diag o nalizable, in fact, J ∼ diag   √ − 1 0 0 − √ − 1 ,  , · · · ,  √ − 1 0 0 − √ − 1   . As an a pplication o f Theo rem 4.5, we hav e the following result. 10 WENCHUAN HU Corollary 5.2. F or any p ≥ 0 , we have χ ( C p,d ( n )) = χ ( C p,d ( P 2 n − 1 )) =  v p, 2 n − 1 + d − 1 d  , wher e v p, 2 n − 1 = ( 2 n p +1 ) . Example 5.3 . F or p = 0 , we have χ ( C p,d ( n )) = ( 2 n + d − 1 d ) . Alternatively , this can b e s een in the following wa y . The set C 0 ,d ( n ) can b e decomp osed into the disjo in t union of quasi-pro jective a lgebraic v ar ie ties accord- ing to the num ber of fixed p oints of ¯ J ly ing in P n − 1 ∐ P n − 1 , i.e., C 0 ,d ( n ) = ∐ d i =0 SP i ( P n − 1 ∐ P n − 1 ) × SP 1 2 ( d − i ) ( G ( n )), where G ( n ) = P 2 n − 1 − ( P n − 1 ∐ P n − 1 ) is a bundle over P n − 1 with fib ers C n − { 0 } . Hence we have χ ( C 0 ,d ( n )) = d X i =0 χ (SP i ( P n − 1 ∐ P n − 1 ) · χ (SP 1 2 ( d − i ) ( G ( n )) . Since the Euler c haracter istic of G ( n ) is zero, we get χ (SP m ( G ( n ))) = 0 for a ll m > 0 by MacDo na ld formula (cf. [12]). Therefore , χ ( C 0 ,d ( n )) = χ (SP d ( P n − 1 ∐ P n − 1 )) . Note that (16) SP d ( P n − 1 ∐ P n − 1 ) = ∐ d i =0 SP i ( P n − 1 ) × SP d − i ( P n − 1 ) , where a 0-cycle c = P n i P i + P m j P ′ j ∈ SP d ( P n − 1 ∐ P n − 1 ) o f deg ree d is written as the sum of tw o 0-cycle s such that P i is in the first co p y of P n − 1 but P ′ j is in the second copy of P n − 1 . By e quation (16) and the fact that χ ( C 0 ,i ( P n − 1 )) = ( n + i − 1 i ), we ge t χ (SP d ( P n − 1 ∐ P n − 1 )) = P d i =0 χ (SP i ( P n − 1 )) · χ (SP d − i ( P n − 1 )) = P d i =0 ( n + i − 1 i ) · ( n + d − i − 1 d − i ) = ( 2 n + d − 1 d ) , where the last equality is o btained by comparing the co efficients of t d in the T aylor series o f 1 (1 − t ) 2 n = 1 (1 − t ) n · 1 (1 − t ) n . Example 5.4 . F or d = 1 , we have χ ( C p, 1 ( n )) = ( 2 n p +1 ) . Alternatively , this can b e seen in the follo wing w ay . The eigenv alues of J are ± √ − 1, each of them is of mult iplicity n . Let { e i } 1 ≤ i ≤ n be the eigenv ectors o f the eigenv alue √ − 1 and { f i } 1 ≤ i ≤ n be the eigenv ectors of the eig en v alue − √ − 1. The J - inv ariant ( p + 1)-co mplex vector space is s panned by i eig env ectors from { e i } 1 ≤ i ≤ n and p + 1 − i eig env ectors fr om { f i } 1 ≤ i ≤ n . Therefore, C p, 1 ( n ) = C p, 1 ( P 2 n − 1 ) ¯ J ∗ = a 1 ≤ i ≤ p +1 G ( i, n ) × G ( p + 1 − i, n ) , where G ( i, n ) := G ( i, C n ) is the Gra s smannian o f i -dimensional complex linear subspaces in C n . Therefore, χ ( C p, 1 ( n )) = χ ( C p, 1 ( P 2 n − 1 ) ¯ J ∗ ) = P p +1 i =1 χ ( G ( i, n )) · χ ( G ( p + 1 − i, n )) , = P p +1 i =1 ( n i ) · ( n p +1 − i ) = ( 2 n p +1 ) , where the last eq uality is obtained by co mpa ring the co efficients o f t p +1 in the binomial ex pansion of (1 + t ) 2 n = (1 + t ) n · (1 + t ) n . The La wson-Y au formula and its generalization 11 References [1] W-L. Chow and B. L. v an der W aerden, Zur alge braischen Geometrie. IX. (German) Math. Ann. 113 (1937) , no. 1, 692–704. [2] E. F r iedlander, Algebraic cycles, Chow v arieties, and Lawson homology . Comp ositio Math. 77 (1991 ), no. 1, 55–93. [3] W. F ulton, Intersect ion theory . Second edition, Spri nger-V erlag, Berlin, 1998. [4] W. F ulton, Int ro duction to to ric v arieties. 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Michelsohn, On equiv ariant algebraic susp ension. (English summary) J. Algebraic Geom. 7 (1998), no. 4, 627–650 . [11] H. B. Lawson and Stephen S. T. Y au, Holomorphic symm etries. Ann. Sci. ´ Ecole Norm. Sup. (4) 20 (1987 ), no. 4, 557–577. [12] I. G. Macdonald, The Poincar ´ e p olynomial of a symmetric pro duct. Proc. Cambridge Philos. Soc. 58 1962 563–568. [13] J. Milne, ´ Etale cohomology . Princeton Mathematical Series, 33. Pr inceton Unive rsity Press, Princeton, N.J., 1980. xiii+323 pp. ISBN: 0-691-08238 -3 Dep ar tment of M a thema tics, MIT, Room 2 -363B, 77 M a ssachusett s A venue, Cam- bridge, MA 02139, USA E-mail addr ess : wen chuan@math. mit.edu