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(2) 2 K. ToKi. Ci*･t･*C. == d･u4. Especially, when Ci = ･･･ =:C.=C, we write Ci*･･･*C. ==:C. With considering t,hat gh.e Pontrjagin product *,on.,an abelian variety is a: ddition of cycleS'and c6mmuta' tive, we can see the following properties" from'the definiti'bn .t ttof l-numbers of curves on u4.. associative, dist' ribtitive'for' the'. PRopERTIEs. (i) Fo.r any curve Con u4, the value l(C) is integer. And if C is positive, then we have l(C)>=O and the inequality holds if and only if the product n*C is not zero. (ii) .Let C, C/ be two curves on u4. If they. are numerically equivalent to each other (we shall denote this property by C6IijCi), we have l(C)='l(C/). Consequently, l-numbers of curves on abelian varieties are numerical invariants. (iii) Let C be any curve on ,-jZ and m be. any.integer. Thenwehavel(mC)=mn･l(C). (iv) Forrgiven1-dimensional. JtT. /'. i } ffl･. g. ･･(. -. r .. subvarieties C,, ･･･-, C.'on 'u4, we put F(m,, ･･･.,m.) =l(Z'miCi). Then the. t t /t. .i==1 is,･a ' valued function F(ml,･･･,m.) of integral vectors (mi,･･･,m.) integer homogeneous polynomial of degree n on m,, ･･･,m. with integer coefficients and the coefficient of every term m: is equal to the value l(CD (ISi.S r). (v) For any'curve C on 2-dimension,al abelian variety, we hqv.e l(C) = I.(C2)/2 !,. where ,the left hand means the 'l-number of curve C and the value I.(C2) in. the 'right hand means the 2-fold intersection number of divisor C. Consequently, i･n the case of dim u4 =･ 2, when a curve C on u4 is positive nondegenerate as divisor,･,we can see that its l-number determines the dimension of eomplete linear system ICI defined by･the divisor C,･from ･the Poihcar6-. ' Nishi'sformula. '' ' ･,･･ ,r'-' where PRooF. Let C be a curve on u4. We decompose C=,E.1,m,Ci, Ci, ･･･,C. are distinct components of the 1-cycle 'Ctt and every mi is integer.. Then we hqve , '. '. deg('""C)",,.1.Z.,.,..b,!"..'.!p.!mfi･･･merdgg[(P*iC,)*･･･*(P*r,ci?] (1.1). plZ-O,･･･,pr>=O ' '. '. Let le be a common field of definition of the abelian variety u4 and its subvarieties Ci,･･･,C.. Let xi,･･･,xp,;･･･;yp･･･,,yp. be independent generic. Pl PT. "i''. p6ints of Cl;;,:-[':- ;b･･･, C,;･･･; CIIII,1:-:-;Z)･･･, C. over le. ,,Then we 47.ow deg [(P*iCi)* ･･･ *(P*r' C.)]. ==O Or [k(xi,'",Xpi;''';yi,･･･,yp.):fe(Zxi+t･･+2yD] if this degree is. i･=1 j'--1. finite (cf. Lang [3]). In the latter case, since the pgsitive integer [k(x,,･･･);. ;' .1. '. .f,.f;j .i.
(3) Certain Numerical Values of Cqrves on Abelian Varieties 3 ife(2 xi+ ･･･)] is divisible by the integer p,!･･･ p,!3', the value l(C) is an integer.. And, when C is positive, since every milO and every deg[(;iC,)*･･･*(S't).)]. IOin (1.1), we have. I(C)IO and moreover, when C is positive, from the e.X,Pr,e,Si&O."d,"nCthi ".,!g,i,(tCigfic-g/s. We have *"'CtO if and oniy if i(c)>o. Thus･. Let C, Ci be two curves on .rz numerically equivalent to each other. 'Then, since we know that the product * preserves the equivalen'ce ij{ki (cf. [3], IV,' tw '. 1 i. ･g3), we have *nC4{lijxeCi. Therefore, when P, is a point of u4, we have I･[(*"C)･P,]==I.[(ZC)･P,]. Therefore, in expressions :C='d･u4 and "*C'. =di･u4, since Ir(UZ･P,)==1, we have d==d'. Thus we have l(C)=l(C/). i,e.x3,w=hfi￡･,･,c,jf .a2y ,c.uJ,ve,en,flll,. zn.g ,y,,･ls fting,,:'kts,ffe'ri,,sip,cev. p,, //ggg. 1-dimensional subvarieties of.,,u4. Thens.from the equality (1.1), we h.ave F(M.i, ''･･mr)==,,..]Z. ].,.=. deg [("pC, !i).:'b'.:(*Cr)] miPi ･･･..7ner. Th.us. ,we,, obt,ai,p as-,. 'sertions (ii), (iii) and (iv).. Now we consider the case of dim u4==2. We suppose that UZ,is defined o,ver. a field le and a curve C on .4 is fational over le. Theri we have deg(*C) =I.(C･C.-), where u means a generic point of di over le and C- means the, transfprm of C by the endomorphism xF--> -x of u4 (xE u4).and C.- means the translation of C- by u (cf. [3], I, g2, Prop. 6). And, for any divisor Y o,n an abelian varietY, we know Ydlij Y- (cf. Lang [3]). Therefore we have ,ii,(,C,i,Ci,uMR=7.i)'.(C2)･ ,COnSilefing C as a divis,or ,on -jZ･ IF,hys,, ,we,obt/r.nEyhDe.. ' ' a divisor. g2. I-numbers of curves defined by the self-interseetion.of ',,i :. ' ttt For every divisor X on a n-dimensional abelian va;iety u4, we denote the translation of X by a point a of .;Z by X., a,nd when the intersection. :. X.,･･･ X..-, is defined, we shall call that the curve･X.,･･･Xa..i on u4 is of, "T, -.. iv l. .tyPe Xn--i. ,. Then, since the numerical equivalence of cy61es on ,-jZ is preserved by the. intersection(cf.[3]),weobtainthefollowingfactfrom(ii)ofgl. , ' 3) We denote the group of le-automorphisms' of le(xi,････,yp ;･･)'obtained from] permutations of xi's,･･･, permutqtions of yJ･'s respectively by Spi,,.,,p.. Then we know that the field le(xi,･･･,yi,'･･･) is a finite Galois extension of the field le((xi,･･･),,･･･ ,(yi, ･･･),) with the Galois group Spi,,,,,,o. and we have [fe(xi, ･･･,yi,･･･): fe((xp･･･)s, ･･' ,(Y" ･")s)] = pi!･･･ pr!f where le((xi, ･･･)s, ･･･,(yb ･･･)s) == <6 Ek(rpi･, ･･･,yi, ･･･)1ga = 6 for. all aeSp,, ,,o.>. On the other harid,'the O-cycle i (xi)+ ･･･ +4(yj･) on u4 is rational. over le((xi, ･･･)s, ･･･,(yi, ･･･)s). Therefore the"point 4 xi+ ･･･ +;yj･ of u4 is rational over k((xi, ･･･),, ･･･,(yi, ･･･)s) and then w'e have le(7 xi+ ･-･ +Z ptj･) c le((xi, ･･･)s, ･･･,(yi, ･･･)s)･. ts. ' , Thus the degree [le(xi, ･･･,yi, ･･･): k(ep. xi+ ･･･ +>?. y,･)] is divisible, by the degreq [le(Xi,"',Yi,'"): le((Xb''')'s,r''.(Yi,,''')s)]'iPi.!'n''',Dr!･ i , '. '･･ , ,' '･ . i':･. ' ll. I. i.
(4) 4 K. ToKi. Let X, X! be two divisors on u4 such that X;{lijX'. And let C, Ci be curves. on u4 szach thatare of tyPe X"""i, X/"-i resPectively. Then we have l(C)=l(Ci).. Therefore l-numbers of curves of type N"-i for a divisor X are equal to each other. So we may denote the l-number of curve of type X"-i by l(Xn-i). Moreover we have the following proposition. PRoposlTIoN 1. Let .1 be a n-dimensional abelian variety over C Let X be a Positive non-degenerate divisor on u4 and C be a curve on u4 of tyPe X"-i.. Then we have. l(C)==rl(Xn-i)=={(n-1)!}n･l(X)n-i (2.1)t PRooF. Let the expression of ,.rz by a complex torus be E/G, where E means a complex n-dimensional vector space and G means a lattice in the･ additive group E. Then we have a positive divisor X== div (0) for a holo-. 'i. morphic thata-function 0 relative to the torus E/G of reduced type (￠x, Hx> in the sense of Wei! [8], that is, 0= 0(x) means a holomorphic function on.. E and we have the formula. 0(x+g)=0(x)･ipx(g)･e[2vi.--1HLr(g,x)+4vt-i-Hlr(g,g)] (VgGG), where Hx(x, y) meansahermitian form on ExE, ￠x means a semi-character of G and we denote e2nV=i[] by e[ ]. Since the divisor X is positive non-degenerate, the hermitian form Hx(x, y> is positive definite (cf. S3, Rem.) and. then the imaginary part Az(x,y) of Hz(x,y) is the Riemann form on UZ associated to divisor X, i.e. Ai(x, y) is a R-valued R-bilinear form on ExE such that (i) A.(x, y) = -Ax(y, x) (Vx, Vy E E), (ii) Ax(x, ･vi-1y) == A.(y, ･v/-' 1 x>. (Vx, Vy E E) and Ax(x, V-1 x) >O (O # Vx G E), (iii) Ax(g, g') E Z (Vg, VgiE G).. where we denote the integer domain by Z and the real number field by R: (cf. [8]). Here we remark Ax(x, ･vi=ly) =the real part of HLr(x, y).. For viewing the value l(C), according to Lieberman, we shall compute･ the Pontrjagin product * in the cohomology ring of u4. We denote the addi-･ tive group of algebraic cycles on u4 by %(u4), the cohomology ring of u4 over R (or Q) by ff*(u4, R or Q) the homology ring of di over R (or Q) by HLk(,"jZ,,. ,J. R or Q) and the duality of Poincar6-de Rham by D. Then we consider the commutative diagram by Borel-Haefliger (cf. [1], [4]):. '. ' Q)ara(cy4, R) ' H*(c-, h. %(,";z). ,. 'Dls DIS tttt. i, clt,. , H*(u4, Q)aH*(di, R). l. where the map h means the homomorphism regarding algebraic cycles as topological cycles and the map cl means the composition Doh.. l I. i [. I it.
(5) Certain Numerical Values of Curves on Abelian Varieties 5 We know cl(X･Y)== cl(X)Acl(Y) for X, YG %(u4), whenever the inter-. section X･Y is defined (cf. [1]). (2.2) (n-1)-time Therefore, since cl(X)=cl(X.) for VaE(-;Z, we have cl(C)='cl(X)･･･cl(X) when Cis of type X"-i. So, in this case, if we put va =cl(C), ui=:cl(X) r-time and uk= uxA ･･･ Aux then we have vc= za&-i.. -. ,. On the other hand, according to Lieberman [3], when we denote the Pontrjagin product naturally defined in the homology ring H*(cl,R), using the homomorphism s*:H*(u4xu4,R)-H*(u4,R) induced by the map s:u4xu4 -->u4 such that s(x, y)=x+y (x,yE UZ) by the same symbol * as in. .. S(u4) and the product induced in the cohomology ring H*(,.rz, R) by the iso-. morphism D from the product * in H*(,-;Z, R) by the symbol V, we know. h(X*Y)=h(X)*h(Y) for VX,VYES(u4) and ipv￠=D[Dmi(ip)*D-i(￠)] for 'v ip, v￠ e H*(J, R).. So we have. cl(X*Y)==cl(X)vcl(Y) for VX, VY(iii %(.rz) (2.3) Therefore we have. n sutlMeS n. cl(* C) = V vc= vc V ･･･ V va= deg(* C)･1 (cl(u4)= 1). n. '. '. from the equality *"C= deg ("* C)･u4. Here we use the Lieberman's formula: ip V ￠ = Ilf(Ai) *xi[(*i ￠) A (*x ￠)] for Vip, "￠ E H*(u4, R),. where R7C(Ai) means the Pfaflian of skew-symmetric form Ax(x, y) respect to G and *i`) means the star-operator on H*(UZ, R) associated to positive definite. hermitian form HLr(x, y) (cf. [4]). Then we have the equality. n-times. -" Y vc = l]11`(Ax)n"i･*Ii[(*zvc)A ･･･ A(*xvc)]. t"'. ti. Now we extend the star-operator *x to a C-linear operator on the cohomology ring H*(u4, C). And let {4,, ･･･,C.} be a base of C-linear forms on E. Then we know that the cohomology ring H*((YZ, C) is generated by 1, dCi, ･･･ ,dCn, d4i, ･･･,d4n with the exterior product A over C, where g". means the complex conjugate of C.. So, from the positive-definiteness of the hermitian form Hx, we choose such a base {C,, ･･･,4.} of C-linear forms on E that we tri. 4) We know that, when a positive definite hermitian form H(x,y) is given on Ex E, the star-operator depends only on H(x, y) and does not depend on the choice of a system {zi, ･･･,2.> of C-linear forms on E such that H(x, y) = Xn2.(x)z.(y) (cf. [8], I) and the hermitian form Hx is well determined by x. so we dea n=oite the star-operator. in here by *x･.
(6) '6 K. ToKi. may write H>r(x, y) ==.2.l,4M.(x)･C.(y)･ Then we have ux i= Y2-1' .2n..,c.A gl .5' and. we know that the star-oPerator *x on H*(UZ, C) follows the Weil's formula: *.[4A A C-B A (le).] == (-1)v(M)÷P(P2+i) (-2･vl-1 )p-n(.vi-1 )d-b[4. A g-B A a)M,],.. where. A; i,<･･･<i.(ore), B; j'i<･･･<1'b(or/) ' ' ({ip･･･,i.},{1'p･･･,1'b}aredisjointsubsetsof{1,･･･,n}), ' ' ' ' 4A==<iiA"'ACia, CB==4jzA"'･ACjb, 4￠==4￠==1, (DM==ll(CcrACa) aEM (Mc{1, ''',n} s･ t･ {ii, '",ia}AM={]'" '",1'b}AM= /),. :s. ,. M/={1,''',n}-[{ii,''',ia}V{1'p''',7'b}VM], tu￠=1 and 2･v(M)= the degree of form teM, P =the degree of f6rm CAAC"-BAtuif. =a+b+2v(M)(cf.[8]).Since ･･ '. , vq ,. uyi=(n-1)!･( 'Vi5 ,1 )n-ltr.,(,U.4rA 4-r),'. we have *iva=(n-1)!･zax by theaboveWeil's formula. Therefore we have vvc==Rf(Acb"-i･{(n---1)!}n*2[za&]. Since the operator *zo*x==(-1)P bn all n. P-covectors on the complex vector space E (cf. [8])) we have *x` == *i on all. ,a' even degree covectors. Therefore we have *2i[uk]==*x[zak]==n!･1 by the 'Weil's formula. Therefore we have deg(ZC)=n!･{(n-1)!}n･17e(A.)"-i. ,,, Y,%w,,.we,ev.,kh,:,t ,1,e8ag.gh,9%S,e X"C:.%'e.al&a,fZiS/?;ILgcz,i. EO/; Fn,tPi6)I. where every di Ei! Z s. t.,O < dild,1･･･14..6) Moreover, from the non-degeneracy. of positive hermitian form Hx, we see that the system {g,, ･･･,g.} is a base. of the complex vector space E. Let a system {6i, ･･･,62.} be the base of R-valued R-linear forms on E dual to the base {gi, ･･･,g,.}. Then we have. '/. ns. Ai(x, y) == Z d.･{g.(x)6.+.( y)-g.( y)6.+.(x)}･. '･, .･ a==1, . , .. So,whenweputcrx±. t-. .]Iil)i,d.',(g.A6.+.),wehave, ' . ,.,. '''. {J. ' nl!ak==di'''dn'(6iAgn+i)'A'''A(8nA62n)' ''''.,'I . ti･. e. Therefore we have IZf(Ax) == di ･･･ d. (=the maximal number of linearly independent holomorphic th6ta-functions over C belonging to the Riemann form 5) By Borel-Haefliger [1], 4.12, Prop., we have ux =the Chern class ci([X])･ of the line bundle [X]. -On the other hand, by Weil [8], VI, Prop. 4, we･know. s. i.･i<[iiig.=･,.,V2i.i,'g.fiit¥lh.",PdgC,P.(),d,CI.ii,,Il,he,",..",g(ig.Y),,==z,i-i.hlp4cr(X)'4p(y) (-h6pT'hpa').;. 6) cf. C.L. Siegel, Analytic functions of several complex variables (Princeton)..
(7) Certain Nurnerical Values of Curves on Abelian Varieties 7. Ax)=l(X). Thusweobtainaformula(2.1). ･, . .QED. Let{g"･･･,g,.}besuchabaseofGasabove.Whenweput, .･ (gn+i,''',g2n)==(gi,''',gn)'Px (PxEEM...(C)7)),. 1 tt. ;,Z.08tg,i"'i,h8,t,V e.ng,ti'Ii,,T,X I-,Ii,g P,1 ZekOg",ff9 .102 )the <Z"S,rZ'IZ8.d ¥,eg8.i ,ia,kfg,. t''. the complex torus E/G by means of complex coorainates with respect to a complex base {diig,, ･･･,d.-ig.} of E, we have u4 == Cn/[ax, Tx], where Tx is symmetric apd the imaginary part of Tx is positiy,e definite.. So, we have the following corollary. . ,.. s-･. CoRoLLARy 1. For a Positive non-degenerate divisor X on the comPlex torus d,. Cn/ o. o. T .s dn. (every dj E Z; O < d,i･･･ld.; `T == T;. '. '. Im.T is. Posztzve dofnite)･. '' d,. o -''･･belonging to the Riemann form. 'dn. - d,. tae have l(Xn-i)... '･. Q - dn {(n-1)!}"(d, ･･･ d.)n-i.. (ii) of g1, we have the following Also, from corollary. 2. Let u4abelian be a n-dimensional COROLLARy ,variety over C, and X, Y. be two Positive non-degenerate divisors on cv4.. ILIf a'cu'rv2 to' of type X"-･i is numerically equivalent. a curve of type Yn-i,. then we have l(X)==l(Y).. abelianembedded variety u4 is inaprojective When the. section ofu4, and the divisorXisageneric hyperplane. space P over C. ,.we shall express the. l-number of curve type of byXn-i means of the degree of u4 in P. According to Nishi [5], when we put the Hilbert characteristic function of u4 by X(u4 , nl), ti. == 8o! ･mn for sufificiently we know l(mX)=X(u4, m) larg m ･(ao== the degree. ,the Prop.,1 andequality the. l(m of LjZ in P). from Therefore,. ti. X) = mn･l(X). nln(n-1).I(Xn-1)= (m>O), we have {(n-1)!}n･x(.4, m)n-i.. Thus we have the folloWing corollary.. 3. Let .jZ,Xbeasabove. CoRomARy Then we have. nn-i･l(X"-i)=(n-1)!. (the degree of ,YZ in P)n-i.. Next,inan abelian varietyu4ofwhichgroundfieldmaybecharacteristic. we consider a curve C of type poSitive or zero, Xn-i ,"jZ. And we assume that ' degenerateX ondivisor mtersections' X.･Care for a, positive non-. '. /. ,.. 'n-squareof matnces 7) We denote the totality of overCby EMnxn(C)･.
(8) '. Ldofned in the sense of the foot-note8). For the canonical homomorphism gx ' from u4 onto its Picard variety u4" defined by the non-degenerate divisor X i.e. spx(x) =cl(X.-X)=the linear equivalence class of divisQr X.-X (x (i di), we consider a rational map ￠c: .j"Z -->di such that ￠c(gx(x))=S[(X.-X)･C]9'. <xE u4). About the map ￠c, we havethe following remark by Prof. Koizumi. REMARK of KoizumiiO). Under theabove sitzaations, we have (n･1di)o￠cogx. =I.(X")･1thwher21.emeanstheidentitymaPonclandn==dimu4. (2.4) ,PRooF. Let k be a field over which .jZ is defined and X is rational. We. ,;. n-times ･consider a rational map Ti from the abelian variety u4× ･･･ xu4 to the abelian variety u4 such that Ti(yp ･･･,y.) == S[Xyv･･･,Xy.], where Yi, '",Yn. .d･. are independent generl.,c points of a over le. Since ￠oogi(x) =S[X.･C]. -S[X･C], we have '. ￠c09x(X)=Tx(Xi,''',Xn-i,.X)mWI(Xi,"',Xn-i,O) (a). 'when xG UZ and C= Xxi, ･･',Xx.-i･. Now we know that there exist unique homomorphisms ￠i:u7 (=the i-th component of JZx･･･xu4)->u4 (1$i.Sn) and constant c, such that n. ZY' x( yi, ･･･ ,yn)= i.;ll=,￠i( yt) -+ co (P)･ JSince the function Ti(y. ･･･,y.) is symmetric respect to yi, ･･･,y., considering. the equality Tx(yi,･･･,yi,･･･,y.)==Tx(Ni,･･･,yi,･･･,y.) we have ￠i=:'" =￠n 'from the uniqueness of ipi's. So we put ￠=￠i. Therefore, from (cr) and (P),. we have. ￠,oqx(x)=￠(x) (xEc]) (r)･ Moreover let y,, ･･･,y.-,, z be independent generic points of u4 over le. Then. we have ZIT'x(Yi+Z,''',Yn+2)-ZP'z(Yi,'",Yn) =={t/,ill=,￠(Yi+2)+co}-{t/.ll,￠(yi)+co}==n･￠(2) .(S).. On the other hand, we have a. ZP'z(yi+x,･･･,y.+z)-ZPrx(yi,･･･,y.) == S[Xbi+3, ''', Xlrn÷z]-S[Xleti, ''', Xyn] s,. =={S[Xyi,''',XY.]+I･(X")'X}'S[XYi,"',Xlr.]. =I. (X n) ･z (e). 8) For every irreducible component C/ of C, the intersection .Xle･C/･ is defined and every component of .2Yle･C/ is simple on Ci.. 9) For every O-cycle a= 2]ni(Pi) on UZ (Pi(EI.4), we put S[a]=ZniPi==the :sum-up as points of u4.. ii. 10) This is a fact which Prof. S. Koizumi talked in his seminar at Tokyo Univ.. of Education. ' '.
(9) l :. Certain Numerical Values of Curves on Abelian Varieties 9 Therefore, from (r), (6) and (E), we obtain the equality (2.4). QED.. i. Now'i', when a curve C of type X"-i is given (X is as above), considering. i. a generic translation of divisor X, our assumption is satisfied. We denote a generic translation of divisor X by X/ and our rational map from f to u4. l. f. for the divisor XX and the curve C by ￠6. Let k be a field over which u4 is. defined and X is rational. If C==X..･･･,X...p we take two independent r. I. generic points u, v of u4 over k(nc, ･･･,x.-,). By the square's theorem, we .. i. have cl(X.-X)==cl(X.+.-X.). So, when we put Xi =X., we have gx=n,. on u4 and '. r ,. ￠6(gz･(v)) = S [(X6 - Xi)･ c] = gP'i(xi, ･･･, xn-i, u+v)-ZIT'x(xi, ･･･, x.-i, u). '. ±￠(v). , '. Since moreover we have n･￠(v)==I.(Xn)･v from equalities (S) and (e), we have (n･1.n)oip6ogx= I.(X")･1di. Thus the degree v(￠6) of rational map ￠6 does not depend on the choice of generic translations of X, and so we denote this degree by v(￠c). Then we have the fo!lowing proposition. PROPoslTIoN 2. Let ,v4 be a n-dimensional abelian variety of zvhich ground field may b2 characteristic Positive or zero, X be a Positive non-deg2nerate divisor. on cv4 and C be a curve on ,] of type X"-i.. Then we have. v(￠a) == {(n-1!}2"･l(X)2n-2 (2.5) PRooF. Considering degrees of rational maps in both hands of (2.4), we have n2"･v(￠,)･v(￠.) ={I.(X")}2". By Nishi, we know l(X) =I.(X")/n!= Vv(gi) (cf. [5], [6]). Therefore we have n2n･v(ip.)･l(X)2==(n!)2n･l(X)2n. Thus we. obtain the equality (2.5). QED. Consequently, we may write y(￠a)==v(￠xn-). Thus, for a given positive. non-degenerate divisor X on .4, we have following relations among four numerical values l(X), I.(X"), v(gx) and v(ipxn-i):. (I.(Xn)/n!)2n-2 =v(qx)n-i== v(￠xn-)/{(n-1)!}2n =l(X)2n'-2. "... In particular, in the case of u4 over C, we have the following theorem summarizing (2,1) and (2,5).. THEoREM A. Let ,-jZ be a n-dimensional abelian variety over C, X be a u.. Positive non-degenerate divisor on ,] and C be a curve on u4 of tyPe X"-i.. Then we have. l(C)= ･Vv(￠,) ={(n-1)!}n･l(x)n-i. , When u4 is the jacobian variety 1 of a complete non-singular curve of genus g over C, since l(e) =1 (cf. g4, Cor. 6), we have l(eg-i)={(g-1)!}g, where e means the canonical divisor on J. 11) This discussion is due to a suggestion of Prof. Nishi..
(10) IO K.ToKI' t. j. I. tt C. t ' of curves on a polarized abelian varietY over g3. I-numbers. Throughout this g, let u4 be a n-dimensional abelian variety over C (n ) 2). In following, we shall fix a positive polar divisor X on LfZ.･. We first remark. the following fact in order to see the equivalency of the non-degeneracy of a divisor on di and the non-degenetacy of hermitian form associated to this divisor (cf. g2). We put ,-jZ =E/G as a complex torus as in g2. REMARK. M/e denote the totality of divisors on c.jZ szach that algebraically. 1 i. I. 4i. equivalent to 2ero by 9.(u4), the totality of divisors on JZ such that linea,rly. L･. eqzaivalent to zero by 9i(u4), and identij2y the Pica･rd variety ,-f"Z' and th2 Picard. grouP Pic((-jZ)== .ED.((YZ)/9i(..jZ). Let E be the anti-dual sPace to E (i.e. E={f:. .. E -> Cif(x+y) == f(x)+f(y) for Vx, "y E E and f(cx) =i･f(x) for Vc E C, Vx E E}). A -L and we Put G=:{fEEIIm.f(g)EZfor VgGG}.i2' Mbreover we denote the. i. character grouP of discrete abelian grozaP G by To. Let Y be any divisor on u4.. Then, in a diagram:. ' SDy .･ di= E/G - a=Pic (di). AA. qflyl '' ?lc ･ ' E/G " l"o e. where gH.; the homonzorphism from the n-dimensional conzPlex torus E/G to its dual. AA A. comPlex torzas E/G inducedi3' by a homomozPhism ylt>HY(,y):E-->E. q.; lfhh,eZ..ll; .(i.2.Yl AS,.th,e.h,e,rpMhl.ii,a,nfrfO,r.M,SSStO,CZL[:le.d,t,O,,diZ.VtZidOrt,Y yOn.i:.T:I,. (Cf' [8]),r. 'gy(x)=cl(Y.-Y)(xEuiZ), ' '. E; the homomorPhism from the complex torus E/G to the group Z-', inducecl. byahomomorPhismf-e[-Im.f]:El->l",, . c; the canonical maP in the sense of Weil [7]i`),. tt. ttt tttt t tttt t AA '･ wehaveE/G-iI"oandEogHy=:co9y･ '' , ' '. ". PRooF. Let f be any element of Ker(e). Then we have e[-Im.f(g)]=1. for VgEG. Thereforewehave ---Im.f(g)EZforVgGG. Thereforewehave fE G. And, tt since e(e) == {1}, we see that e is ' injective. Nextly, let X be any. '. 12) We denote the imaginary part of [ ] by Im.[ ].. 13) We know, from the th6ta-function theorye that Im.Hy(,g) :r-Ay(,g) is integer valued on G(VgE! G) (cf. [8]). 14) For Ze 9.(di), when we put Z == div (0z) for a th6ta-function 0z of reduced. type (￠z, Hz), we know Hz =O and then ￠zGIo. We define cz =￠z. Now, let x be. any element of To. Then we have x= e[Ai(,a)], for some aGE. When we put Z== Xd-.X) we have cz=x. And we obtain that cz=1 if and only if ZE9i(.4). Thus we have Pic (oiZ) > Po, if we put c(cl(Z)) = cz for ZE 9a(u4)･. ",.
(11) Certain Numerical Values of Curves on Abelian Varieties 11 element of- I"o. Since G is a free group of rank 2n, we have 1-',=t(R/Z)2n.. Now let a system {gi, ･･･,g,.} be such a base of G choiced for the polar divisor' X as in S2. When Z(gi) == eA/::-ilei (O ;SOi<2T), we put f(X) ==j-S, (' 2.0."dj')Hx(x, gn+j)+ tr.lll, ( 20.n."dj ,. )Hx(x, gLi). '. '. (xe E, di =Ax(gi, g.+i))･. tt t g.i (1;;li.S2n). Therefore we ha've Thep we have' fEE and -Im.f(gi)==. .. E(f)==X. Thus we have E/a ;I",.. 'Now we shall show the commutativity of the above diagram. We put Y=div(0y(x)) for a thata-function 0y(x) respect to E/G of reduced type. .. (￠y, HY). Let u be any point of u4. Then we have za=a modG for some. aEE and gy(u) :cl(Y.-Y). When we put'8y,.(x)=0y(x-a), we have Y.,-Y=='div(8y,.(x)･0y(x)-i). WeputFy,.(x)==8y,.(x)･0y(x)-!. Thenwehave Fy,a(x+g)Fy,.(x)-i=[0y((x-a)+g)･0y(x-a)-i]･[0.(x'+g)･0.(x)-i]-i. =±e[Vil,Hy(g,a)] (vgGG). ,･ ' ' ' ' tt 't tt s6 w'e pu't' ￠.,.(g)==e[ 'Visi tH.(g,q)](gGG). since ViiHy(g,a)== tt. ' logl￠y,.(g)l == -TAy('g, V:1 a) V2=l 't/Y(g, V-17a)--12-Ay(gj ,a), we have. (VgEG), where Ay(x,y)==Im.HY(x,y). We put Ly,.(x)=-nAy(x,V-la) ･+zV,-1Av(V-1x, V-1a) (xE E) and X(g)== ￠y,.(g)/eLY･a(g' (gE G). Then, since ￠y,.(g+g")=￠y,.(g)･ipy,.(g() and Ly,d(x) is a C-linear form on E, we have X(g+g'!)t X(g)･X(gi) (g,pgiEi G). Moreover,'since ￠y,.(g)=e-nAy(g･"ia). ･e-n"iAy(g･a) and A.(V-lx, V-la)=Ay(x, a), we have X(g)=e[Ay(a,g)] '(gEG).'Th'erefore we have XG'To.. On the other hand, when we put Z== Y.-Y, we obtain c(cl'(Z)) =X. 'In･fact, when we ` put ￠(x) =Fy,.(x)/0z(x) ,t. (0z(x) is such a th6ta-function of reduced type (￠z,'O) that Z== div (0z)), since div(di(x)) = O, the meromorphic function op(x) is a holomorphic multiplicative. lr). function respect 'to G on E without zero points. Therefore we have ￠(x) =,eL(x)+c for some C-linear form L(x) on E andsomecopstant c. So we have v ib. ￠y,a(g)=Fy,.(x+g)Fy,.(x)-i= eL(g'･￠z(g), and then,1￠y,.(g)[==leL(g)1 (g ci G). Thus, 'since the real part o'f Ly,.(g) =logig)y,.(g)1 (g (ii G), two C-linear forms. L(x) and Ly,.(x) on E have same real parts. Therefore we have Ly,.(x) = L(x).. Consequently we have ￠z(g)=￠yr.(g)/eLY･a(g'=X(g) (gEiG). Thus, sin6e c(cl(Z)) = gb., we have c(cl(Z)) == X. We also have E(spH.(u)) == e[-Ay( ,a)] == X.. Thergforethe.abovediagramiscommutative. .. , QED. ''. ･ From t'his remark, we have the 'following corollary.. CoRoLLARy of remark. For a divisor Y on an abelian varietN u4'over C, the non-degeneracy of Y is'eqzaival2nt'to t`he non-degen'er'acy of hermiiian form.
(12) 12 ' K.ToKi. -HY associated to Y. Moreover, when Y is Positive, the non-degeneracy of Y is. equivalenttothePositive-dofnitenessofHy. ･ PROoF. The non-degeneracy of Hy is equivalent to the homomorphism "aeHy(,a) (aG E)" being an isomorphism,and we can easily see that the latter is equivalent to the finiteness of Ker(qll.). And, since Ker(qy) =Ker(gHy) from the above remark, we obtain that the non-degeneracy of divisor Y is equivalent to the non-degeneracy of hermitian form Hy. Now let Y be positive. Then we have Hy(x, x) >=O (VxG E) and then. .. {xG EiHy(x, x) =O} == {xe E1Hy(x, y) ==O for all yE E} (== Ker (Hy)) <cf. [8], VI, Lem. 3, Th. 1). Therefore, when Y is positive, we obtain that i. "KergHy)={O}oHy(x, x)>O for OtVxEE". Thus we conclude our asserNow we shall consider how to express the value l(C) of curve C on u4 by means of the value l(X) for the fixed polar divisor X. By the positive. nnon-degeneracy of divisor X, we can write Hi(x, y) == Z4.,(x)･C.( y) by means of a=1. compJex coordinates 4,, ･･･,C. with respect to some complex base {A,･･･,jl,} of E, where the system {C,, ･･･,s".} is the base of C-linear forms on E dual to {A,･･･,]1,}. Let C be any curve on .jZ, We know that the real (2n-2)-. covector cl(C) is of type (n-1,n-1) (cf. Hodge [2]). So, when we put vc == cl(c), we can put vc=( 'V"ll 1 )"'''i･.,#=.,c.p･9.p- (c-..,3 = cp.,), where 9.p-=. CpACcrAtuM.,p (a4P), S2na =a)M., Mlrp=={1, ･･･,n}-{cr,P} (atP) and Mlr.= M}v ={1, ･･･,n}-{cr}. Here we remark that 9.,p-=(--1)"-i･9p. (may be cr =P. or crIP). We put Ha=the n-square matrix (c.p).,p. We remark that a matrix Hc for curve C depends on choices of a polar divisor X and such a system {Ci, ･･･,C.} of C-linear forms as above. Then we have the following proposltlon.. PROPoslTIoN 3. Let X, C, 4.., c.,p and Hc be as above. And we .fix a Polar divisor X and such a system {C. ･･･,C.} as above on u4. Then we have equalities. .Y,Vc=("-1)!'l(X)n-2'V2-1.t/,ill=,Acrp'[CpA4-a･] , (3･1). ･). l(C) == det H, ･l(X)n-i (3.2) , where A.,p == the co.IZactor of c.p in the matrix Ha (1 ; a, PS n), (Consequently,. when we fix a Polar divisor X on u4, detHc is a rational number determined fbr C and does not dePend on the choice of {4i, ･･･,Cn}･). In order to show that above equalities hold, we shall verify equalities in. the following lemma. Here, for ￠, ￠EH*(,-jZ, C), when we put ￠=ip,+V-1 ip, and ￠= ipi+V-1 ￠2 (￠,, ￠, (or ip2, ￠2) are real (or imaginary) parts of ￠, ￠ respectively), we define. ipV￠==[(￠iV￠i)-(ip2V￠2)]+Vml[(ipiVip2)+(￠,Vipi)]. Then we also have. a di.
(13) '. '. l. Certain Numerical Values of Curves on Abelian Varieties 13 ip V￠= l(X)*xi[(*xip)A (*x￠)]. Therefore ￠V￠ is C-linear respect to ￠ or ￠.. Moreover, since (P-covector ip)A(q-covector ￠)= (-1)Pq･￠A ip, the product V satisfies commutativity on even-covectors, associativity and distributivity en any degree covectors. LEMMA. Let Cds, 9.p-'s be as above. Then zve have. '. I j. j 11･ ,･. (i) *x9..p-=(-2･vi-1)n-2･[C./xCp] (may be a= P or crtP), (ii) *x[C.ACp]=(-2･vi-1)2-"･9.p- (may be a==P or atP), (iii) 9.p-v[CpAC-.]= ( v2-1 )"･t(X);1 (may be cr=p or atp), (iv) S2.p-v9.,p,==O, if cr=cri or P= pi.. ". Moreov2r, 9.p- V 9.tp, = 9.,p-･ V 9.p-,. (v) 9.tp-･V[4pA4.,]=O, ifa4cr' or P 7E P' (may be a==P or crtP), (Vi) Let {a"･･･,a.-i}, {Pi,･･･,P.-i} be subsets of {1,･･･,n} andwe assume cri< ''' < crn-i, Pit Pj (1 Si, 7' <,.. n-1). We Put {a}== {1, ･･･,n}-{a,,.. ･･･,cy.-,} and {P}={1,･t･,n}-{P" ･･･,B.-i}. !. a) Jn case of a== P, we have. i. 9crip-iV "' V9crn-ip-n-i. ' = (-1)n2･sgn (pcr,ij 111l ap."--l) ･(2V-1 )"("H2)･l(X)"-2･[C. A 4-.] b) In case of a 7E P, we have S2a p-iV "' V9a'n-ip-n-i. ==(-1)n2va+P･sgn(lp',jLlr.P.';'p';-Z)･(2･vi-1)"("-2'･l(X)"-2･[4pAC-.].. where the symbol v means "excluding ". ' (Verification) From the Weil's formula in S 2, we obtain directly equalities. (i), ･(ii). From (i), (ii) and the Lieberman's formula, we have 9a･p-V [k A <-ar] == l(X)'*Ii [(*i9a'p-) A ("x[Cp A 4-cr])]. == l(X)･*x [C. A Cp A 9pct] = l(X)･*.[.I["[=,(C. A C".)] -･. == ( .v,llil )n･l(x).1. e. Thus we obtain the equality (iii). Since S?.p-'s are even degree, we have: 9.pV9.ip･ =9.tptv9.p. Also, since we have (*i 9a･p") A (*x9crtp-t) = COnSt･[(4. A 4p) A (4.rA 4pt)],. we obtain the former in (iv). From equalities (i) and (ii), we have (*x `S?a"p-t) A (*i [Cp A Ccr]) =:COnSt･ [(4crtA Cp･) A (4cr A k A (OM.,p)]. (may be a=P or ev#P). Therefore we obtain the equality (v). About (vi). we first remark that the equality.
(14) :. 14 K. ToKi, ,t. [. (4"cr'iAC-pi)A'''A.(4.,-iAC-p.-,)=(-1)a'"Pesgn(}',j':.l.E".'i'kA-",)9,,i"' (3.3).. 1. ' holds (may bea==Por ev4P).., ･ ･r.,/･. From(i),Lieberman'sandWeil'sformulas,wehave ･ ･. 9cripHiV ''' V 9crn-ip-n-i. =l(X)n-2･*1 [(*x9.,p-i) A ''' A (*x S2crn-ip-n-i)]. '. ..(.1)(n-i)(n-2).(2.vl-1)(n-i)(n-2)･l(S¥)"-2･*i[lij,l=ii(C.,,AC-p,)]. ' '. 1-. :. L. ' Y =(-b(n-i)(n-2)･c2V-1)(n-i)(n-2)･l(X)"-2･(-1)a'1'P･Sgn(lp',j`:.:.P.';'p'￡-Z)･(*x9p.-). E. t F. ;. t. v'. ..(Tl)n2･(2･viny)n(n-2)･l(x)n-2･(-1)cr+P･sgn(>',j'il.E!l'k:-n,)･[4pAC-cr] i. ･ '(may be cr'=:P･ or a;P).. '. L i. Therefore we have (vi) a) and b). Thus we have verified the above lemma. Next, we shall prove the prop. 3. From (iv) and (vi) in the abOve lemma,. we have. tt. ''. .v.iva = ( 'ViS 1 )("-i)2'(n-1)!'.i<,,Z...-iccripi'''Ccrn-ipn-i(9crip-iV'''V9a'nLip'n-i). PaliEPj , ･. . '..(V2-1)("-i'2.(n-l)! , , ,,' '. ' /t. '. '. '{ al<..2<a.-1 CcrIPI'''Ccrn-IPn-1(IQa'.lp-IV･''VS2crn-lp-.-.1). {cr1,"',an-1}={Pl,"-,Pn-1} ･ '. ' + ' al<･tta..1 CcrIPI'"Ccrn-IPn-1(9crIP-IV"'VS2crn-lp-."1)} {cr1,''',an-1}7L{P1,J",Pn-1}. ,.,(-1)n2.(n"1)!.( Vil )("Hi)2.(2v'ml)n(n-2).I(x)n-2. ' ･12･. ' {cri<･･llk:an-i [(,ei,･･;lpl.-,) Sg" (Sil lll l S#:l) ' Ca'pi ''' ccrn-ipn-i ] ' (Ccr A 4-.). (a=P) 15) Because we can see that we have the equality (4.iAC-pl)A･･･A(4a･.-iAgp.--i)=sgn(pP'lllllIkll)･[(CcriAgpi)A'''A(4an-iA4ph-i)],. where {Pl,･･･,Ph-i}=<1,･r,n>-<P> and Pi<P2<･･･<Ph-i (may be cr==P or cr EP)･. Moreover we have ･. 1.. <4azA4pi)A'"A(Ccr.-iACph-,)==[(4iA4i)A"'A(gcr-iA4cr-i)]A[(Ccr+iACcr)A"'A(CpACfi-i)]. A[(4p+iA4p+i)A"'A(4nACn)] (ifcr<P) =(-1)cr+P9pct. ,.'. ･. ;.
(15) I i. Certain Numerical Values of Curves on Abelian Varieties 15. i. +cri<'-;<an-i[(pi,･･?pB.-i)(-1)cr+P. (a 7EP). tt v '' sgn (}',i'ilA'i'k￡-ag)'caipi ''' ccrn-ipn-i]'(CpAC-cr)} i'''. 2. -(n-i)l･i(x5"-2･(i!t(s:i,)･{.z"...,h,au(ccrAc-'cr)+.m,Aap(epAecr)}. =(n-1)!･l(x)"-2･('V!il1)'.,=),Acrp'[CbAC-cr]･'tt.' Thus'we obtain the' equality (3.1). '. .. '. ･ L From (iii) and (v) in the above lemma and the equality (3,1), we have. VVa=Va V(V n,=[(Vil)"-i･.,:)=ic.p'9crR] n-1 ' '. Va) '. ' ' ' v[(n-1)!･l(X)n-2･ V2-1 '.,#=),Acrp(4pA C-cr)] == (n--1)!･l(X)"H2･( V2-1 )n･.￡p=,c.pA.p'(9crp- V[k'A C` ev]). tt. ,･#･ (?-1)!･l(x)n-i･[tr=,(tT.,c.pA.p)]･1. '== n!Ll(X)n-i･ det Ha ･1. '. Therefore we obtain the equality (3.2). . QED. Especially, when c is of type x"-i, since v, = zami= ( 'Vill 1 )n-i･ .2"=,(n-1)!. t9. respect to {Ci, ･･･,C.}, we have det Hc = {"(n-1)!}". Therefore we obtain the Prop. 1 as a special case of the Prop. 3.. ' Andmoreoverwehavethefollowingcorollary. CoRoLLARy 4. Let X be a Positive Polar divisor on an abelian variety u4 over C, and C be a curve of tyPe X"-i on ,-jZ (n == dim ,-).. Then we have a numerical eqzaivalence for divisors: l. '. '. ,* n--1 C K9 {(n - 1)-! }n ･l(x)n-2,･ x. <Conseqzaently, the divisor "*-'C also is algebraically eqzaivalent to the divisor }L. {(n-1)!}n･l(X)n-2.x16)). .,. pRooF. since v, =( V2-1 )"-i･(n-1)!･.zn=-,s2an respe6t to {4i, ･･･,C.}, we. have .Y,y,c : V2' .1 ･i(n--1)!}n･l(X)"-2･t,E.i,[C.ACny.] from (3.1). Moreover. ' 16) Concerning divisors on an abelian variety, we know thqt the numerically equivalence, the squarely equivalence and the a,lgebraically equivalence are equivalent to each others (cf. Lang [3])..
(16) 16 K. ToKi. SinCe .Y,Va=Ungi.from (2･3) and zai =-V2-1 ･.￡.,[C.A C-.], we have. *c. u.-,={(n-1)!}"･l(X)n-2･u.. (3.4>. n-1 Therefore we have h(*C)==h({n-1)!}"･l(X)"m2･X). Thus topological (2n-2)-cycles n*Mic and {(n-1)!}"･l(x)"-2･x are homolog6us over Q to each. other. Therefore we obtain that the divisorn-1 *C is numerically equivalent. to the divisor {(n-1)!}"･l(X)"H2･X.i7) QED.. v. Let Ybe any divisor on u4, and {4.･･･,C.} be as above. Then we. fe[Mcpajl(kc-.t]hliLpW=hhep".,),W,,e,, PhU.t.,re.ai,i(.i'il3-,COfV,e,Ci::ri.Ci(,Y()3itiYy= V2-i '.,#=),ha'3 .. zanfi,,.(n-1)!.( V2-1 )"-i. :ll) H.p.g.,-, where HZ,p=the cofactor of h.p in the n-square matrix (h.p).,p. In fact, we have un.-i,= ( V2-1 )"-i･p,<.,S.p.-,(n-1)!. ai )kaj･ ' hcripi ''' ha'n-ipn-i[(4pi A Ccri) A ''' A (4pn-i A 4an-i)] '. Xl8ehn..W,e PUt {a} = {1, ''',n}m{cr" "',crn-i} and {P} -- {1, ･･･,n}-{p,, ･･･,p.-,}, u"y-i == (n-1)!.( 'v!i'lli- )nHi. ''. ' aSpl-i ((.,,,.;li.l.-,)('1)a'"P ' Sgn (h',IIil.4i lr' ;mZ) ' hcripi ･'' hcrn-ipn-i) '9a･p-. ' n from the equality (3.3). Since the term( ) of ￡ ( )in the righthand of cr,P=1 this equlity is equal to the cofactor H.p of h.p, we obtain the above formula.. g4. I-numbers of curves generating the ambient abelian variety.. ,t. For a positive curve C on a n-dimensional abelian variety u4, when *CtO (i.e. I(C)>O), we shall call that C generates u4. In following, we n. consider the case of .-;Z over C.. Let C be any curve on u4. Then we have C=Ci-C2 for some positive curves Ci (i=1, 2) generating v4. In fact, when we put C= C'-C" (Ci and CM are positive), using a positive curve C, of type X"Hi for some positive non-degenerate divisor X on u4, we have C==(C,+C')-(C,+C"). Since l(Co) 17) For an algebraic P-cycle Z on u4 over C, if the topological 2P-cycle Z is homologous to zero over Q then p･Z (p; some positive integer) is homologous to zero.. Thereforewehavep･ZdYO,andthenzijlkgO. , '. ;.
(17) Certain Numerical Values of Curves on Abelian Varieties 17. ={(n-1)!}n･l(X)"-i from (2.1)'and l(X)>O, we have l(Co)>O. SO PUt Ci== Co+Ci and C2 =Co+C". Because, since C, and 0 (or C") are posttive, we see l(C,) (or l(C,)) =l(C,)+l(C/) (or l(C"))+(non-negative integer) (cf. prop.,. (iv) of g1). Sincel(C,)>Oandl(Ci) (or l(C")) >= O, we have l(Ci)>O (i=1, 2)･ Therefore we conclude the above assertion. PROPOSITIoN 4. Let C be a Positive curve on a n-dimensional abelian variety u4 over C (n lll 2). Th2n we obtain that Cgenerates u4 ifandonly ijthePositive divisorn-1 * C is non-degen2rate. l. PROOF. We assume that C generates ,-jZ. When we consider equalities (3.1) and (3.2) for some positive non-degenerate divisor X on u4, the hermitian. n-. fOrM Hn.-i,(Xs Y) == ,,i=,(n-1)!･l(X)nm2A.pC.(x)Cp(y) on ExE associated to posit. tive divisorn-1 *C is positive (cf. [8], VI, Th. 1) and, since l(C)>O and l(X)> O, wehave det Hc> o, where ux= VStl-T-.2n..,4.,AcM... Moreover wehave Ha't(A.p).p =det Hc･1., where 1. means theidentity matrix of degree n (4.1). Therefore we have det (A.p) = (det Hc)"-i i O. Therefore, since rank (A.･p) == n. and l(X)> O, the hermitian form H...,.(x, y) is positive definite. Thus, from. the Cor. of Rem. in g3, the positive divisorn-*i c is non-degenerate.. ' ･ n-1 Conversely, we assume that the positive divisor *C is non-degenerate. So, weput u..-i.= 'V" ll. 1 .2"=,caA. 4-'. for some base {c{,･･･,4a} of c-lin6ir'forms. on E. In equalities (3.1) and (3.2), we denote coefficients c.,p, A.,p, the matrix Ha 'and cov,ectors 9.p- respect to {4(, ･･･,4a} by ckp-, A'.p, H6 and 9ap- respec-. tively. Then we' have ' ' v, ,= ( 'v!{l i )"-i･.,;l.).,cap･S?kB,. '. .t' tt. .v-',vo == ( '"i5 1 >･(n-1)!･l("*-iC)n-2･.]piil..,Aap(Ck A 4-'ct'),. ' ' n-1 l(C)==detHli･l(*C)"-i and n-1 l(*C)>O. `. '. Since, from (2.3), we have u.-,= v vc. Therefore we'have. *C n-1. Aa.== 1.-, (a=1,･･･,n)andA.',e=O(a#P). (n-1)!･l( * c)n-2 .. Moreover, since C is positive, we have l(C)). O (cf. Prop. (i) of g 1). Therefore. n-1 we have detH610 from l(*C)>O. With (4.1), we obtain H6=(n-1)!, n-1 ･l(*C)"-2･detH6･1.. Therefore we have. ca.=(n-1)!･l(n*-'c)"-2･detHi(a==1,･･･,n) and ckp==O(a4P). (4･2)･. t. .. When we put ci=ca. since c/llO and Aa. =ci"-i' >O (a ==1, '･･･]n), we have.
(18) 18 '.-. ,K.ToKi ci>O. Therefore we have detH6 == ci"> O. Thus, from (3.2), we have l(C)>O.. ThereforeCgeneratesu4. ' '.'' QED. When we fix a positive polar divisor X on u4 and we put ux= 'Vill 1 ll=},4.A 4-., we aiso see, from equalities (3.2) and (4.1), that a bositiv6. curve C generates u4 if and only if the hermitian matrix Hc' respect to {4. ･･･,4.} associated to positive curve C as in g3 is positive definite.. Now, we shall state some properties about values l(C) and numerical classes of P-cycles ZC for positive curves C's generating .,4, or numerical L,. classes of P-cyc!es of type Xn-P for positive non-degenerates divisors X on u4.. We first have the following proposition about cohomology classes vc and ux･. ". PRoposlTioN 5. For a Positive curve C generating cy4 and a Positive nondegenerate divisor Xon a n-dimensional abelian varie'ty u4 over C, when we put vc==cl(C) and ui == cl(X), we have equalities, for 1:$P .<.- n-1,. [(n-p)!.I("iic)].(vv,)=[p!･{(n-1)!}P･l(C)P]･(u.-,)"-P (4.3). p *c. ' [p!.I(xndi)]･(u&-p)==[(n-p)'!･{(n-1)!}"-P･l(X)"-P]･(Vv.n-i). p ' '.･'n-1. (4.4). ' PRooF. By the Prop. 4, the divisor *Cis positive non-degenerate. We put u4= E/G as a gomplex tgrus. So, when' we put vc=( V2-1 )"-i･."lp=,cap.gap. respect to such a base {41,･･･,CA} of C-linear forms on E that u.-i =. . . ., , .*o. V2-1 ･.Zn=,4'. A C-'., we have cap =O (cr i P) and ca. = ('n-1)!･l(n*-'c)nr2. det Hli. .. (n.1)!.I(c)/l(n*"c) a$ cr $ n) (cf. (4.2) and (3.2)). Therefore, when we put == ca. we have va =( V2-1 )"-i･ c'･ill=l,9ct.-, where 9da =:;ltIa(G A Cnyi)･ Thereci. fore, (i) and (iv) in the Lem. of g3 and the Lieberman's formula, we have ¥ va = ( Vil )P("-i)･c'P･P!'is..i<?,Il,<.ps.9aidiiV '" V 9apudp. ' = ( V2-1 )P(n-i).cip.p!･l(n*-b)p-i･.,<1?,l].<.:.--ii.[(*n.-i.S2aidi)A'"A(*ngi.S2apap)] ' &. ..( vil )p(n-i).( v2-1 )p(n72).c,p.p!. n-1 - -. 'l( ".C)P-i'.,.Z.,..."n.-i.[(4ai.A C'ai) A '" A (Ca, A Ca,)].. Therefore, from the Weil's formula, we have '･ ' yvc=( V2'1 )ndP.p!.{(n-1)!}p. tt. '. ' n--p)] ' ' 'l(C)P/i("*MiC)',s.p,<,,ll<l)p.-...[(4fei A C-Z i) A/t'" <N (4kn-p A 4-B'. '. `.
(19) 1. Certain Numerical Values of Curves on Abelian Varieties 19 On the other hand, we have. '. '. '(Ungi.)"-P == ( V2'1 )"-P'(n-P)!',..p,<,,2tllB.-,...[(CkiA 4-B' i) A ''' A (4kn-p A C-B' n-p)]'. Thus we obtain the equality (4.3). ･. t'. Next, let Co be a positive curve of Xn-'. Then, since Co generates u4 from l(X)>O and (2.1), we havetheformula (4.3) for curve Co. On the other hand, from (3.4), we have u.-, ={(n-1)!}"･l(X)n-2･ux. Moreover, from the * oo Cor. 4 of g 3, we have l("*-'c,) =l({(n-1!}n･l(x)n""2.x) ,= {(n ny1!}n2･l(x)(n--i)2. Therefore, from (4.3), we have. [(n-p)!･{(n-1)!}"2･l(X)(n-i'2]･(vvc,). ". ,p ' ' ==[P!･{(n-1)!}P･l(C,)P]･[{(n-1)!}n･l(x)n-2]n-p.(ux)n=p. Therefore we have [p!･i(co)](ux)n-p == {(..(i")l3,).!.ii-.(.",-. ii()c!,i",2i i((iX(il()n,-.i:i,,.-., (¥ vc,),. IIn this e.quality, since l(C,) =={(n-1)!}"･l(X)"-' from (2.1), the coeMcient of. 'right hand is equal to the value (n-P)!;{(n--1)!}n-P･l(X)"-P. Thus we. robtain the equality (4.4). QED. Thus we have the following theorem. THEoREM B. Let C be a Positive curve generating u4 and X be a nondegenerate divisor on a n-dimensional abelian variety u4 over C. Moreover let c. be a p-cycle of type ("iic)n-p and .x, be a p-cycle of type x"-p on u4. Then we have. equalities qbout l-numbers or numerically equivalences:. tt t: n-1 '. . I(*C) ={(n-1)!}n･l(C)n-i ,(4.5). Forl=<PSn-1, ' ''. t ' p!･ c. iglij '[(n-p)!･{(n-1)!}n-p･l(c)n-i-p]･(* c) (4:6). ' (n-P) !･(P* X,) ijILe [P !･ {(n-1) !}P ･l(X)P-i]･X. (4.7) `. PRooF. Fromthepositivenon-degeneracy of divisorn-1 *C by the Prop. 4, in a proof of Prop. 5, we have showed that ., .. ( 'Vi{l 1 )"-i.(n-1)!･l(c)/l(n*-'c)･.#,9dn･. v. Therefore we have ' det Hl5 = {(n-1)!}n･l(C)n/l( n-1 * c)n.. n-1 C)n-i from (2.2) and l(C)> O, we obtain the equlity Since, l(C)== det H6･l(* <4.5). Moreover, in equalities (4.3) and (4.4), from l(X)> O, (4.5) and (2.1), we. have.
(20) 20 ' K. ToK･I･･ ･ [(n-p)!･{(n-1)!}nmP･l(C)"-i"P]･(Vva)==P!･(za..,)"-P (4.3">. p *c. [p!･ {(n-1)!}P･l(X)P"i]･(zaYP) == (n-P)!･(¥, vi,) (4･4'>. pp. Since be' va == cl(*C), (u.gic)"-P == cl(Cp), ¥ vx, == cl(* Xi) and uyP == cl(Xp) from. p (2.2) and (2.3), we have h([(n-P)!･{(n-1)!}"-P･l(C)"-i'P]･*C)=h(p!･C,) p from (4.3!) and h([P!･{(n-1)!}pu･l(X)P-i]･X,) =h((n-P)!*X,) from (4.4/). Thus,duetothefoot-notei7',weobtainequivalences(4.6)and(4.7). QED. From Th. B, we have the following corollary. CoRoLLARy 5. Notations are as Th. B. Let C, C' be two Positive curves generating u4. Then we have. n-1 n-1. .-. ". i) C;{ltO, if and only if *C61￠ *Ci, esPecially. ii) when C, C/ are oftyPe X""i, X'n-i resP.forsomePositive non-deg2nerate divisors X, X' on u4,. Cfg} C/, if and only if X;{l: X/.. PRooF. We know that the Pontrjagin and intersection products are preserved by the numerically, equivalence (cf. [3]). So, i,f C4{liCi then. *Cij{Y *C'. Conversely, if n-1 *CdY *C/ then C,i{IijC{. Ontheotherhand,from n-1 n-1 n-1 (4.6), we have C,ijll:[{(n-1)!}n･l(C)n-2]･C and C{;{l:[{(n-1)!}n･l(o)n-2].ci.. n-1 n-1 n-1 n-1. When *Cf{le *C/, since l(*C)=l( *C'), we have l(C)==l(Ci) from (4.5). Therefore, when we assume X'c;llij"*-io, we obtain c;{l; o from l(c)=l(c/)>o (cf. the foot-notei7)). Thus we have i). Next, if Xf{lijX/ then X,i{lijX{. Conversely, if X,61;xl then nfi*ix,ij{!g'ZR'x{.. On the other hand, from (4.7), we have n-*'X,f{lij[{(n-1)!}"･l(X)"'2]･x and. ".'ix(,{l,[{(.ml)!}n.I(x/)n-2]･x/. when X,f{lijX(, we have l(X)==l(Xi) from. the Cor. 2. Therefore, when we assume Xiij{liXl, we obtain Xi{:￠X/ from. g(X) ==l(Xi)>O (cf. the foot-notei7'). Thus we have ii). QED. Here we consider the case of ,y4 being a jacobian variety over C. Let J be the jacobian variety of a complete non-singular curve C over C of genus. g and f be the canonical map from C to J. We identify f(C) and C. And let P,, ･･･,P. be independent generic points of C over C (1 $P$g). We denote. `. p pu. the locus of ZP, over C by M7,. Then we know *C =P!･W. (1$PSg) (cf. i=1 [3], II, g2). Therefore, when we put @= VV,-,, since l(C)==1 and C.= {(g-1)!}g-P･e. (e, means a p-cycle of type eg'P), we have e, ij{Ii(g-p)!･VV.. (ISP$g-1) from (4.6) of Th. B. Since we obtain (g-P)!･cl(W.)=(ue)g-P (1Spgg-1), when weexpress ue= Vil .2g)=,4.AC-. by means of some base {Co ･･･,4.} of complex linear forms on E, we have Cl(VVp)= ( V2-1 )g-P',...,<.,Z..,-,...(CcriAag-cri)A'"'A(Ca'g-pA4-a'g-p) (1:llP;:lg-1),-. '. ;.
(21) Certain Numerical Values･ of Curves on Abelian Varieties 21: MoreOver, from Th. B and, the Poincar6-Nishi,'s, formula, we have the. .follQwing known fact. - ･ CoRoLLARy 6. Let JL C,e be as above. Then we have '. eg-!fllij(g-1)!･･c, l(e)=1, L(eg)==g!.. PRooF. In the above numerical equivalence, when we put P=1, we have Le,ille(g-1)!･VV,. Therefore xKie have eg-if{lt(g-1)!･C. From (4.5) of Th. B, i. 4. we have l((g-1)!e) ={(g-1)!}g usingl(C)=1. Since l((g-1)!e)={(g-1)!}g ･l(e), we have l(e)==1. Moreover, from the formula l(@)=I.(eg)/g!, we. hqve l.(eg) == g!. ,. QED. Thus, we can list up main formulas in S2, g3 and S4 as follows: When. we put Lrz; n-dimensional abelian variety over C (n ). 2), .E{Oo(u4)=={X; divisor on u41X is positive and non-degenerate} 0o(di)={C; curve on u41C is positive and generating ,y4}. :21P(u4),={Z; algebraic P-cycle on' u41Z is positive} ' Zo == {positive integer},. in a diagram. .g. ,-pt' zp(a), ･ (n - 1). . Xl>. seo(JZ) - Co(Ljt) *(n-1) 4. lt. where the map "･q" means "itself q-times intersection product" the map "*q" means "itself q-tim･es Pontrjagin product". and the map "l" means "taking l-number". - -,1. we have, for VX, VXi, VX" E 9,(u4); VC, VCi, VC" G e,(cv4) and 1:SP :.{ nTl, ;. i. {:::,- i'i i,(E..Z'3 umIm i,(::l,' ,! ;." .' f,(,X,' :,i :- .',"i;":,￠f-";'. *c. ' (PP.-i): (n-P) !. *P(Xn-') ;glijp!･{(n-1)!}p･l(x)p-i.xn-p. "t. II'. I(.-,p,),p!("Xc)n-p,,{lij(n-p)!.{(n-1)!}"-p･l(c)"-i-p･*Pc -. t. ,. ' L) 7,i IX, L) 7,.i5ec),X.1,X;,℃]i)" iii. I[l],- i) 1. 11 re.".e.i. ,g,-kXi,':,ll'li･ h%"9, i`,Z(F,. **** ', 'J/ t. ' ' .From now on, ･we shall･state some remarks on numerical classes of curves and divisors on an abelian varie,ty v4 over C. A,ccording to the Lang.[3], ･･.
(22) 22 -' ' K. ToK'i we denote the graded module of classes of cycles on u4 modulo numerically equiva･lence by N(u4) and its P-component by Nlo(u4). Then we know that N(.IZ) with the intersection product or the Pontrjagin product make ringstructures and N.(u4) is the submodqle of P-cycles modulo numerically equivalence (cf. [3], IV, g 3).. In following, we consider the case of u4 over C. We first remark that. the additive group 9(u4) of divisors on u4 is generated by 9,(.;Z) over Z and 'i'. the'additive group e(u4) of curves on c-jZ is generated by e,(,v4). In fact, when. Y is any divivisor on u4 and Y= Yi-Y" (Y'>-O, Y">O), we put Y==(X,+Yi)+ --. (.X,+Y") for some positive non-degenerate divisor i¥o on di. Since Hz,+yi,. (or Hx,+ym)==Hx,+Hy, (or Hx,+Hxn), divisors Xo+Y', X,+Y" are positive non-degenerate from the Cor. of Rem. of g3. Thus we obtain the former. And, from the beginning of this g, we obtain the latter. We denote these. ". facts by 9(u4)= [9,(di)] or e(cy4)==[0,(u4)].. Moreover we define some･notations. We denote the submodule of Allo(u4) generated over Z by the totality of (n-P)-times intersections of classes of' divisors by [N.-i(cv4)]n-P, especially the submodule of IV,(u4) generated over Z by the totality of (n-P)-times intersections of classes of positive nondegenerate divisors by [N..i(LjZ)o]"-P and moreover the submodule of Alfo(.4> generated over Z by the totality of itself (n--P)-times intersections of classes of positive non-degenerate divisors by ([N.-i(u4)]"-P)o. We also denote the above three ,#otations for Pontrjagin products of classes of curves instead of intersectioh products of classes of divisors by P*[All(u4)], "*[All(JZ),] and (;[N,(,v4)]), respectively, where replace " positive non-degenerate" by "positive. and generating u4".. Then we have, for 1;;IP$n-1,. ･pp. 10 [N...,(u4)]"-P==[N.-i(o4)b]"'", *[Ni(UZ)]=*[Ni(u4)o]･ ' ' ' we have, for And, from the above list I, II, 1$P$n-1,. ' 20 ([N.-i(LjZ)]n-P)oXQ==(*[2Vi(v4)])oXQ. s. especially when P == 1, from 10, we have 3e IV,(.iZ)([i3)Q=([N},-,(.4)]n-i),XQ.i8) 18) This fact shows that every curve on a n･dimensional abelian variety u4 over C is expressed modulo numerical equivalence as a linear combination over Q of itself (n-1)-times intersections of positive non-degenerate divisors. Here we add that we know the following result due to Prof. S. Koizumi who showed in his lecture for the graduated course of Tokyo'Univ. of Education:. "Let v be any integral class of type (n-1, n-l) on u4. Then, we can express v= :ri (uAi) (finite sum) for some rational numbers ri, where every uAi -means. i. 2-covectors associated to a Riemann form Ai on .jZ.". a.
(23) l. l. Certain Numerical Values of Curves on Abelian Varieties 23 Here we consider the following question.: '9) l i. "For every 4GIVlo(u4), (m･1.a)-i(C) ==m2(n-P'{4 from mEl?, where n== dim u4 and (m･1.A)-i means a ring-homomorphism from the ring. l. i. {IV(u4), ･} (･;intersection) to itself induced by the endomorphism m･1di of ,-;Z". I v. 1. WhenP= n, this is trivially affirmative for any m. When P==O, since v(m･1di)=m2n and any two points of u4 are numerically equivalent to each. other, this is also affirmative for any m When P=n-1, we know that ,. this question is affirmative for any m (cf. [3]). Consequently, when 4G [IV}v-i(u4)]"-P,.from the ring-homomorphism of (m･la)-i on {Ai(,YZ), ･}, we can see that this question is affi")mative for any m. So, nextly We consider this. question about 4G N,(u4) for .4 over C. Then, from the above 30, we have c･4=Zai･e,"･-i (finite sum, 6iEAi},.,(v4)) for some positive integer c and i. l. some integers ai's. Therefore we have (m･1.A)-i(c･C) :c･(m･1.e)-i(4> =Zai･[(m･1di)-i(6i)]"'i=m2(n-i)･(2ai･6y-i)==m2(n-i)c･C. Since c>O, we. have (m･1di)Hi(C)= m2("-i)･4 for VmEZ. Thus, whenever P=1 and u4 is defined over C, this question is affirmative for every integer m. ' For anotherP for .4 over C, when 4E(Z[N,(,-IZ)])o, we see tihat the above. questionisafiftrmativefrom20. '. ,･ ' g5. FdndaMental classes of polar divisors on an abelian variety. In this g, we shall remark that the class ux of a positive polar divisor X on' an abelian variety .4 over C is expressed by means of coefficients of the Riemann form associated to X, i. e. 1?PThen we Put u4 == E/G as a complex torzas (dim u4 == n), by means of the dual base {6,, ･･･,g,.} tb such a real base {g,, ･･･,g,.} of E that is a base of. the lattice Gand its subsystenz {g,,･･･,g.} is a comPlex base of E, we can e tpress `. ux= Z Ax(g,,gi)･[6iA6j],20) " ls-i<j'-s2n. v. where Ax(x, y) means the Riemann form associated to X. From the following easy lemma, we can verify the above fact. LEMMA. Let V be a q-dimensionalvector sPace over 'a commutative .f7eld k, and we Put `a=(a. ･･･,a,), `b=(P,, ･･･,P,), where every crj, Pj Ei the sPace. Homk(V,k)ofk-valuedle-linearformsonV.' ' ･ '･' ,･ Then we,have, for any q-square matrix M with coEu7icients of le,'. i) ta･MAb=`aAM･b, 19) In case of m== -1, this problem is one of Weil. For another case of m, the. authorwasannouncedbyProf.S.Koizumi. .,,･'''' 20) We denote dgi by simply gi.. .'. , ..
(24) ,24 K. ToKi ii) (`aAM･b)+(`bA`M･a)=O,. iii) if M is symmetric, then tesAM･a=O. ,intheexterioralgebraonVwiththeexteriorProductA. ' (Check) i): We put M=(m.)i,, (m,,Ele). Then wehave `a･M==(･･･,2cr,mi,,･･･) :qnd then `es･MAb ==.,t9.,[(t9.,a2mw) A Pj] = ,tl.=,crzA MtJ'Pj =,S.],[crzA (t9.,Mtj "' PJ )]. == `a A M･b･ ' .,, ii): (taAM･b)+(tbA`M･a)==,II.;i,(t9.,mtj･[azAPj])+,#.l],(t9,Mji'[PzAaj]). i. t l{] {mij･([cri A Pj]+[Pj A cri])} = O･ ''. i,j'=1. ･. ". iii): taAM･a=,tLi'l),(t9.,mij･[criAcrj])==L',,l.ll.j(m,j-mj,)[criAcrj]=O,iftM=-M.. Thus we have checked the above lemma. .. So, we shall verify (5.1). Let {2,, ･･･,2.} be the base of Homc(E, C) dual. td'complex base {gi, ･･･,g.} of E, and we put (gn÷i, ･･･,g2.) =(gp '",gn)'Q ･<QEEM.k' .(C)). Then we have (zi(x),･･･,z.(x))=(6,(x),･･･,6,.(x))･(}Q") (xEE)"' .:.:,1,,vy %eip('i,./isilh) es,lr,ni: xal :,!;r illl,g,x,zs! ,2c,`.a,tspa i.?y, ."x.`xi ,l' Vfi 'ii'j gl',l. 'where Hx= (Hx(g., gp)).,p (1 S a, P$ n). Thus we have H.(x, y) =- (s,(x), ････g2n(x))･(iZoXii., ,Q-HHig)'(S(.iy)))'. ',6.-ZX.QQ' ) and AiF the imaginarY part ' we pu' t kx=the reai part of (IQi(Xi}.). of this matrix, Then we have the (i,]')-coefficient (Bi),,,==the real part of. H.(gi,gj)andthe(i,1')-coefficient(Ai)i,j=Ax(gi,gj). .･ on the other hand, since we can put ui= V2-1 lt/k=,Hx(g., gp)'[2pA2"-.] by m6ans of a base {2,, ･･･,2.} of Homc (E, C), we can express ui == "'. ,zn)AtHz'. V -1 2 (Zi,. s-. (iiI)･ By the ' equality *), vee, d6t'aih. . '..,z･ ul-=(6i･'':･62n)A(lii5.'., tQl"Hlg-)'(i'I.). from i) of the above lemma, Since Hi is hermitian, the matrix part of right hand is equai to the eonjugate 'matrix of (lloZXfr., ,Q-HHX.QQ)･ Thgrefore we. have ,. "' u.=V2-1(ki,''',62n)A(BxmV-1AX)(1'l,i.)'. ,.
(25) '. 1. ' Certain Numerical Values of Curves on Abelian 'Varieties 25. 1. '. i l. Since Bx is symmetric, we obtain. 1. '. "i = -S- (6i' ''''62n) A AX(i'l.). '. == l-,,#,li=,aij･[6i A e] =2i-i<j(a,j-aji)･[e, A e]. t. .. = 2aij･[ei A 6j]･(aij = Ax(gt, gj)) i<j'. by iii) of the above lemma. Thus we have verified (5.1). Consequently, since we can choose such a system {gi, ･･･,g2.} depending on X as aij=o (]'#n+i) when i<1' (cf. g2), we can express ui == Sl)Ax(gi, g.+i) i=1 ･[6iA6.+i] by means of the dual {ep ･･･,6,.} to such a real base of E as being a base of the lattice G. ' '. References [1] A. Borel and A. Haefliger, La classe d'homologie fondamentale d'un espace analytique, Bull. Soc. Math. France, 89 (1961).. [2] W.V.D. Hodge, The topological invariants of algebraic varieties, Proc. Int. Congr. Math., 1 (1950). [3] S. Lang, Abelian varieties, Interscience tracts in pureand applied math., No.7 (1959). [4] D.I. Lieberman, Numerical equivalences and homological equivalencesin Hodge. manifolds, Amer. J. Math. Vol. 90 (1968). ' -. [5] M. Nishi, Some'resuks on abelian varieties, Natur. Sci. Rep. of the Ochanomizu Univ., Vol. 9, No. 1 (1958).. [6] M. Nishi, The Frobenius theorem andthedualitytheorem on an abelian variety, Mem. Coll. Sci. Univ. Kyoto, Ser. A. Vol. XXXII, No. 2 (1959). [7] A. Weil, On Picard varieties, Amer. J, Math., Vol. 74 (1952). [8] A. Weil, Introduction a 1'6tude des vari6t6s kah16rienne, Actualit6s Sci. Ind. 1267, Hermann Paris (1958).. `. y,. ..
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