ON HOMOLOGY CLASSES DETERMINED BY REAL POINTS OF A REAL ALGEBRAIC

VARIETY

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H3B. AKan. HayK CCCP Math. USSR IzvestiyaCep. MaTeM. TOM 55 (1991), Jfc 2 Vol. 38 (1992), No. 2

ON HOMOLOGY CLASSES DETERMINED BY REAL POINTSOF A REAL ALGEBRAIC VARIETY

UDC 513.6+517.6

V. A. KRASNOV

ABSTRACT. For a nonsingular «-dimensional real projective algebraic variety X theset X(R) of its real points is the union of connected components X\ U · · • U Xm .Those components give rise to homology classes [Χι], . . . , [Xm\ G Hn(X(C, F2).In this paper a bound on the number of relations between those homology classes isobtained.

I N T R O D U C T I O N

I n th i s p a p e r we o b t a i n a b o u n d o n t h e n u m b e r of re lat ions be tween t h e homology

classes which arise f rom t h e c o n n e c t e d c o m p o n e n t s of t h e set of real po int s of a real

algebraic variety, where t h e set of real p o i n t s is cons idered as a subset of t h e set

of complex p o i n t s . T o give a m o r e precise s t a t e m e n t of t h e p r o b l e m we i n t r o d u c e

the following notation. Let X be an η-dimensional real projective algebraic varietywhich we assume to be nonsingular and complete. Then X(R) will denote the set ofreal points and X(C) the set of complex points of X. The set X(R) may consistof several connected components; we denote them X\, ... , Xm . They give rise tohomology classes [X\], ... , [Xm] e Hn(X(C), ¥2) • Let κ(Χ) denote the number ofrelations between those classes. The main goal of this paper is to obtain bounds onκ(Χ).

First we should state the results already known. For curves the result is classical:K(X) < 1. The same holds for surfaces which satisfy the condition Hi(X(C), F2) =0. This is a more recent result, the first published proof of which is due to V. M. Khar-lamov. We should remark that this proof already requires a theory, whereas in thecase of a curve the bound is obtained by elementary arguments. This theory restson the Smith exact sequence. However, to use this sequence for this problem inthe general case would be, in my opinion, rather difficult. With the aid of Galois-Grothendieck cohomology I obtained in [1] the following general bound:

(*) κ{Χ) < Σ dimF2 Hl (G, Hn+«(X(C), F 2 )),

where G = {e, τ} is the group of order two generated by the complex conjugationτ: X(C) —> X(C). Notice that the known bounds for curves and surfaces can bededuced from (*); but, as we will see later, for higher dimensions this formula isclearly imprecise.

Before stating the new results we introduce some more notation. Notice thatsome components of X(R) may be nonorientable. Let X\, ... , Xs be the orientablecomponents of X(R). By κ+(Χ) we shall denote the number of relations between the

1991 Mathematics Subject Classification. Primary 14J99, 55N99; Secondary 14G27.

©1992 American Mathematical Society0025-5726/92 $1.00+ $.25 per page

277

278 V. A. KRASNOV

homology classes [Χι], ... , [Xs] e Hn(X(C), F 2 ) , and by K-{X) the correspondingnumber for the nonorientable components.

We now state the new results. In the first theorem (which is only partially new)we deal with surfaces. More precisely, we have

Theorem. Let X be a surface. Then the following assertions are true:(i) If the homology group H\{X(C), Z) is free, then

K(X)< l+q(X),

where q{X) is the irregularity of the surface.(ii) // degX is odd and H\(X{C), F2) = 0, then κ{Χ) = 0.(iii) // degX is odd and Hi(X(C), Z) is free, then κ(Χ) < q(X).(iv) //Hi(X(C), Z) = 0, then κ_(X) = 0.

In the next theorem we consider complete intersections of arbitrary dimension.

Theorem. Let X be a complete intersection. Then the following assertions are true:(i) κ{Χ) < 1.(ii) // deg^T is odd, then K-{X) = 0.(iii) If ae%X is odd and ώχαΧ is even, then κ{Χ) = 0.

Let

L: H"{X{C), F2) - Hq+2(X(C), F2)

denote the Lefschetz operator. We then have

Theorem. Let X be a variety such that the homomorphisms

L": Hn'q{X{C), F2) -» Hn+q{X{C), F2)

are isomorphisms for 0 < q < n. Then

K(X) < dimHn+l(X(C), F2) + dimHn+2{X(C), F2).

The Lefschetz operator is given by multiplication by the cohomology class corre-sponding to a hyperplane section. We now consider multiplication by the cohomologyclass corresponding to the canonical divisor. Let

K: H"(X(C),F2) -> H"+1{X{C), F 2 )

by the resulting homomorphism. We now have

Theorem. Let X be a GM-variety such that the homomorphisms

Kq. Hn~q(X{C),Y2) -• Hn+q{X{C), F 2 )

are isomorphisms for 0 < q < η. Then κ+(Χ) = 0.

In the next two theorems we use the Hodge structure on X(C).

Theorem. Suppose that the homology groups Hq{X{C), Z) (0 < q < n) are free andk = [(n- l)/2]. Then

κ(Χ)<0<r+s<n

r>s

A more precise bound, although under some additional assumptions, is given bythe following

HOMOLOGY CLASSES AND REAL ALGEBRAIC VARIETIES 279

Theorem. Let Υ be a hyperplane section of X and suppose that the homology groupsHq(X(C) - Y(C), Ζ) (0<q<n) are free. Then

κ{Χ)<n-2<r+s<n-\

r>s

With this we conclude the list of the main results, but we should remark that thispaper contains some other new results. Finally, we remark that we use a continuousnumbering of theorems, propositions, corollaries, and remarks.

§ 1. GENERAL BOUNDS

In this section X will denote an «-dimensional real algebraic variety which maybe singular and noncomplete. The set X(C) of complex points is an «-dimensionalcomplex-analytic space, and the set X(R) of real points is a real-analytic space whichis not necessarily connected. The dimensions of its connected components do notexceed « . Let τ: X{C) —* X{C) denote the complex conjugation and G = {e, τ}the group of order two generated by the involution τ . Let κ(Χ) denote the kerneldimension of the homomorphism

Ha{X(R),V2)-+Hn(X(C),V2),

induced by the embedding X(R) <-> X{C).

1.1. Old bounds. The following bounds were obtained, among other results, in [1]:

(1.1.1) κ(Χ) < £ d i m F 2 / / > ( G , H"+«(X(C), F 2 )),

(1.1.2) κ(Χ) < £ d i m F 2 H°M{G, Hn+"(X(C), Z)),

where e(q) equals 1 when q is odd and 2 when q is even (see [1], Proposition 2.6,3.9).

In this section we shall generalize (1.1.1) and (1.1.2). Before doing that we shouldremark, however, that those bounds have relative versions. Let Υ be a subvariety ofX and let κ(Χ, Υ) denote the kernel dimension of the homomorphism

Hn(X(R), Y(R); F2) - Hn(X(C), Y(C); F2).

We now have the following bounds:

(1.1.3) κ(Χ, Y) < X;dim F 2 // ' (G, H"+"(X(C), Y(C);T2)),q>0

(1.1.4)

Although these results cannot be found in [ 1 ], their proof is identical to that of (1.1.1)and (1.1.2). Notice also that, examining the homomorphism

0 > Hn(X(R)) > HH(X(R), Y(R))

• Hn(X(C)) > Hn(X(C), Y(C))

between the long homology exact sequences of (X(R), Y(R)) and (X(C), Y(C)),one has that

(1.1.5) κ(Χ)<κ(Χ,Υ).

280 V. A. KRASNOV

1.2. New bounds. Let f:X—>Z be a map of real algebraic varieties. Consider thehomomorphism

(1.2.1) Hn(X(R),F2) - Hn(X(C),F2)®Hn(Z(R),F2),

which is the direct sum of the homomorphisms

(1.2.2) Hn(X(R), F2) -> Hn(X(C), F 2 ) ,

and

(1.2.3) Hn(X(R),F2)-*Hn(Z(R),F2),

induced by the maps X(R) -̂» X(C) and X(R) -> Z(R). Let κ ( / ) denote thekernel dimension of the homomorphism (1.2.1) and d(f) the image dimension ofthe homomorphism (1.2.3). We then have that

(1.2.4) K(X)-d(f)<K(f)<K(X).

In some cases one can find a bound on S(f). For example, if Ζ is a projectivespace, then S(f) < 1. Thus if we find a bound on /c(/), then (1.2.4) would yield abound on κ(Χ). Before we formulate the corresponding results we recall some factsabout Galois-Grothendieck cohomology groups HN(X(C), G; F2) and the filtration^ , G; F2)} of them given by the spectral sequence

(1.2.5) / | ' ? ( X ( C ) , G; F 2 ) = / / " ( G , H"{X{C); F 2)) => ffw(I(C),G; F 2 )

(see [1]). All those facts are contained in the following theorem.

Theorem 1. The following assertions are true:(i) If Ν > In, then the homomorphism

HN(X(C) ,G;F2)^ HN(X(R) ,G;¥2),

induced by the embedding X(R)«-+ X(C) is an isomorphism.(ii) There is a canonical isomorphism

Ν

HN(X(R), G; F 2 ) = 0 H"(X(R), F 2 ) .

9=0

(iii) If k < Ν, then there are canonical isomorphisms

kFkHN(X(R), G; F2) = 0 H"(X(R), F2).

?=o

We recall also that the variety X is called a GM-variety if the spectral sequenceI{X{C), G; F2) degenerates.

We now formulate a new result.

Theorem 2. Suppose that Ζ is a GM-variety. Then

K(f) < ^ d i m ^ C o k e r l i / ^ G , Hn+"(Z(C), F2)) ^Hl(G, Hn+"(X(C),F2))].q>0

Proof. Let m = max{dimX, d i m Z } . We will make use of the Galois-Grothen-dieck cohomology group H2m+l(-, G; F 2 ) and the filtration {Fk{-, G; F 2 )} givenby spectral sequence (1.2.5). First we remark that

fl 2 6)

HOMOLOGY CLASSES AND REAL ALGEBRAIC VARIETIES 281

since, by virtue of Theorem 1,

We also have

-• Fn(X{R))/Fn-l(X(R))]

= lm[Fn{X(C)) θ F"(Z(R)) 1

Since

K(f) = aimKer[Hn(X(R)) -» Hn(X(C)) θ

= dimCoker[//"(X(C))e//"(Z(R))

it follows from (1.2.6) and (1.2.7) that

(1.2.8) K(f) <dimCokeT[Fn(X(C))®Fn(Z(R))^Fn(X(R))/F"-l{X{R))].

Consider the commutative diagram

F"(X(C))®F2m+l(Z(C)) • Fn{X{C))®F2m+1(Z(R))

(1-2.9)

F2m+l{X(C)) —+ F2m+l(X(R))/F"-l(X{R)),

where the upper homomorphism equals the sum of the homomorphisms

id: Fn(X(C)) - Fn(X(C)), F2m+l(Z(C)) - F2m+l{Z(R)),

and therefore is an isomorphism by Theorem l(i) since

F2m+l(Z(C)) = H2m+l(Z(C),G;¥2), F2m+l(Z(R)) = H2m+l{Z{R), G; F 2 ) .

The lower homomoφhism in (1.2.9) is, by a similar argument, an epimorphism.Therefore it follows from (1.2.9) that

d i m C o k e r t f ^ C ) ) ®F 2 m + 1 (Z(R)) -> F2m+l(X(R))/F"-l(X(R))]

(1.2.10) < dimCoker[Fn{X{C)) ® F2m+l(Z{C)) ^ F2m+i(X(C))]

= dimCoker[F2 m + 1(Z(C)) -» F2m+l(X(C))/Fn(X(C))].

On the other hand,

dimCoker[Fn{X(C)) φ F2m+l{Z(R)) ^ F2m+l{X(R))/F"-l(X(R))]

= dimCoker[F"(X(C)) ©F"(Z(R)) -• Fn(X(R))/Fn-l(X(R))],

since

F2m+l(X(R)) = φΗ9(Χ(Κ)), F2m+l(X{R))/Fn'l(X(R)) =

9=0and the homomorphism F 2 m + 1 (Z(R)) -» i="2m+1(X(R)) is the direct sum of thehomomorphisms H"{Z{R)) - 77«(JT(R)). It now follows from (1.2.8), (1.2.10),and (1.2.11) that

(1.2.12) K{f) <dimCoker[F2m+l(Z(C)) ^ F2m+l{X{C))/Fn(X{C))].

282 V. A. KRASNOV

We now notice that

dimCoker[F2 ;"+ 1(Z(C)) -»F2m+1(X(C))/Fn(X(C))]

q>n

Furthermore, since Ζ is a GM-variety,

dimCoker[/^m-«+1'«(Z(C)) -f I^-9+l

(1 2 14) " d*mCo1cer[I*m-«+l'*(Z(C)) - /2

2m-«+1'«(X(C))]

= dimCoker[i/2 m-«+ 1(G, H"{Z(C))) -> H2m-q+{{G, Hq{X{C)))}

= dimCoker[7/'(G, H*(Z(C)))

where the last equality follows from the equality

HP(G, Α) = Ατ/{α + τ(α)}, p>0

(here the group A consists of elements of order two; see [1]). It now remains toapply (1.2.12)—(1.2.14), and the theorem is proved.

Before we state another result about «:(/) we recall the necessary facts about thecohomology groups HN{X(C), G;Z) and the filtration {Fk(X(C), G; Z)} of themgiven by the spectral sequence

I%'q(X{C),G;Z) = Hp(G, H"(X(C); Z)) =• HP+«(X(C),G;Z)

(see [1]). All those facts are contained in the following theorem.

Theorem 3. The following assertions are true:(i) If Ν > 2n, then the homomorphism

HN(X(C), G; Z) -> HN(X(R) ,G;Z)

is an isomorphism.(ii) If 2M > η, then there is a canonical isomorphism

Μ

H2M(X(R), G; Z ) - 0 H2"(X(R), F 2 ) ,

9=0

and if 2m > η, then there is a canonical isomorphismΜ

H2M+1(X(R),G; Z ) =

(iii) There are canonical isomorphisms

F2kH2M(X(R),G;Z)= 0 / / 2 « ( I ( R ) , F 2 ) Θ H2k(X(R), Z)/(2), 2M>n,\?=o J

kF2k+lH2M{X(R), G; Z) = 0 H 2 q { X { R ) , F2), 2M>n,

q=Qk-\

F2kH2M+l(X{R), 6 ; Z ) = 0 # + 1 ( I ( R ) , F 2 ) , 2M > η ,

A-i \F2k+lH2M+l(X(R), G; Ζ) = φ H2"+l (X{R), F2) Θ H2k+l(X(R), Z)/(2),

V»=° /2 M > « .

HOMOLOGY CLASSES AND REAL ALGEBRAIC VARIETIES 283

We recall also that the variety X is called a GM Z-variety if the spectral sequenceI(X{C), G; Z) degenerates.

We now state another new result.

Theorem 4. Suppose that Ζ is a GM Z-variety. Then(1.2.15)

, Hn+q{Z(C), Z)) -> He(-"\G, Hn+q{X{C), Z))].

Proo/. We shall use the Galois-Grothendieck cohomology groups H2m+X(·, G; Z) ifη is odd, and H2m+2(-, G\Z) if « is even. When η is odd the argument is almostidentical to that of Theorem 2. When η is even some modifications are required.Since

I^-"+l'n(X(R), G;Z) = H2m-n+l(G, Hn{X{R),Z)) = H"(X(R),Z)2,

where H"{X(R), Z) 2 is the subset of elements of H"(X(R), Z) of order two, weshould replace H2m+l{-, G; Z) with H2m+2{-, G; Z) . We then have that

G;Z) = H2m-n+2{G, Hn(X{R),Z))

= H"(X(R), Z)/(2) = Hn(X(R), F 2 ) .

Now we adapt the proof of Theorem 2 to our situation. We start at the end of itand proceed backwards.

Since Ζ is a GM Z-variety, we have that

dimCoker[/£"-«+ 2 '«(Z(Q) - I^-"+2

(l.z.lo) , . , .< dimCoker[/T(<?)(G, Hq(Z{C))) -» H^\G, H9(X(C)))],

where the cohomology of X(C) and Z(C) is considered with integer coefficients.Notice that we also have

dimCoker[F2 m + 2(Z(C)) -• F2m+2{X{C))/Fn{X(C))]

( L 2 · 1 7 ) < ^dimCoker[/^"-«+ 2-?(Z(C)) -»I^-«+ 2^{X(C))].q>n

Similarly to (1.2.10), one can prove that

2 1 8 ) dimCoker[F"(X(C)) Θ F 2 m + 2 (Z(R)) - F2m+2(X(R))/F"-l(X(R))]

<dimCoker[F2m+2{Z(C))^F2m+2(X(C))/Fn(X(C))].

Instead of (1.2.11), one has

(l 2 19)

= dimCoker[F"(X(C)) e f " + 1 ( Z ( R ) ) -• F"(X(R))/F " " '

i.e., F"(Z(R)) must be replaced by Fn+l(Z(R)), and (1.2.8) must be replaced by

(1.2.20) 1 1

Now (1.2.15) follows from (1.2.20), (1.2.19), (1.2.17), and (1.2.16). The theorem isproved.

§2. APPLICATIONS OF THE GENERAL RESULTS

In this section X will be an η-dimensional real algebraic variety, which is as-sumed to be nonsingular and complete. Under these assumptions, the set X{C) of

284 V. A. KRASNOV

complex points is an «-dimensional compact complex variety, and the set X(R) ofreal points if it is nonempty, is an η-dimensional real-analytic variety which need notbe connected. Let Xlf ... , Xm denote the connected components X(R). They giverise to homology classes [Χι], ... , [Xm] e Hn(X(C), F 2 ) . The number of linearlyindependent relations between them is equal to the number κ(Χ) introduced in §1.

2.1. Curves and surfaces. First we shall re-prove two known results.

Proposition 1. Let X be a curve. Then κ(Χ) < 1 and if κ(Χ) - 1 then the onlyrelation is of the form

(2.1.1) [*i] + " - + [*w] = 0.

Proof. In the case of a curve inequality (1.1.1) yields

K{X) < dim H^G, H2(X(C), F2)) = 1.

Hence only one relation is possible. To show that this relation is indeed of the form(2.1.1) we consider the noncomplete curve X which is obtained from X by deletinga real point. We then have that

K{X) < dim Hl (G, H2(X(C), F2)) = 0,

i.e., K(X) - 0 and the homomorphism H\(X(C), F2) -> Hi(X(C), F2) is an iso-morphism. This means that if we delete a point from X\ then the homologyclasses [X2], ... , [Xm] are linearly independent in H{(X{C), F 2 ) . The propositionis proved.

Proposition 2. Let X be a surface such that Hi(X{C), F2) = 0. Then κ(Χ) < 1,and if κ(Χ) — 1 then the only relation is of the form (2.1.1).

The proof is identical to that of Proposition 1.

Remark 1. Proposition 1 is a classical result. An elementary proof of it can be found,for example, in [2]. Another proof of Proposition 2 can be found in [3].

Proposition3. Let X be a surface such that the homology group Hi(X(C), Z) is free.Then

(2.1.2) K(X) < 1 + q{X),

where q{X) is the irregularity of the surface.

Proof. In the case of a surface inequality (1.1.2) becomes

(2.1.3) K(X)< l+dimF2Hl(G,H3(X(C),Z)).

Since the group H3(X(C), Z) is free,

dimF2 H\G, H\X(C), Z)) = dim¥l H2(G, H3(X(C),Z))

(see [4]). Therefore

dimF2 H\G, H\X(C), Z)) < { rkH\X{C), Z) = q{X),

and the proposition is proved.

Remark 2. The bound (2.1.2) is sharp. To see that, consider the surface X which isthe product of a curve Μ of genus g by the projective line. Then κ(Χ) = g + 1 =

Remark 3. Let A{X) be the Albanese variety of a surface X and let |Λ(ΛΓ)| denotethe number of connected components of the set of real points of A(X). Then

dimF 2//'(<?, H3(X(C),Z)) = log2\A(X)\

HOMOLOGY CLASSES AND REAL ALGEBRAIC VARIETIES 285

(see [4]), and therefore (2.1.3) can be rewritten as

K(X)<l+log2\A(X)\.

Remark 4. If the equality κ(Χ) = 1 + q{X) holds, then log2 \A(X)\ = q{X), whichmeans that A(X) is an M-variety.

Proposition 4. Let X be a projective surface of odd degree such that H\(X(C), F2)= 0. Then κ(Χ) = 0.

Proof. Consider an embedding map f:X—> PN and the commutative diagram

H2(X(R)) • H2(¥N(R))

(2.1.4)

Since X is of odd degree, this diagram shows that f, sends the homology class[X\]+---+[Xm] to a generator of the group H2(PN(C)). Therefore [Χι]+·- +[Xm] Φ0, and the proposition is proved.

Proposition 5. Let X be a projective surface of odd degree such that the groupis free. Then

κ(Χ)<\ο%2\Α{Χ)\, K(X)<q(X).

Proof. Consider an embedding map f\ X -*¥N . It follows from (1.2.15) that

K{f) < dimF2 H\G, H\X(C), Z)) = log2 \A(X)\.

We shall now show that κ(Χ) = /c(/). For this we have to show that if the image ofa homology class a e H2(X(R)) under the map H2(X(R)) -> H2(PN(R)) is differentfrom zero, then it is also nonzero under the map H2(X(R)) -> H2(X(C)). But thelatter can be seen from the diagram (2.1.4). The proposition is proved.

Before we formulate the next proposition we make some remarks. Let X bea surface, Υ a curve on X, and suppose that κ(Υ) = 1 and the cycle Y{R) ishomologous to zero in X(R). We can now form a new topological surface W asfollows. The set Y(R) splits Y(C) into two parts. Choose one of them. On the otherhand, take a part of X(R) whose boundary equals Y(R). Then W is the union ofthe chosen parts of F(C) and X(R) with common boundary Y(R). Notice that Wis not defined uniquely.

Proposition 6. Let X be a surface such that HX{X{C), F2) = 0, Υ c X, and letΥ c X be a nonsingular curve. Then κ(Χ, Υ) < 2 and κ(Χ, Υ) = 2 only whenthe following conditions hold: κ(Χ) = κ(Υ) = 1, the cycle Y(R) is homologous tozero in X(R), and, for some choice of the part of X(R), the topological surface Wis homologous to zero in X(C), where the homology is considered with coefficients inF 2 .

Proof. If the homomorphism

(2.1.5) H2(Y(C)) -> H2(X(C))

is a monomorphism, then it follows from the long homology exact sequence ofthe pair (X(C), Y(C)) that H3(X(C), Y(C)) = 0. Therefore (1.1.3) implies thatK(X, Y) < 1. If the homomorphism (2.1.5) is zero, then 7/3(X(C), Y(C)) = F 2

and similarly to the above we have that κ(Χ, Υ) < 2. Hence we always have that

286 V. A. KRASNOV

κ(Χ, Υ) < 2. Consider now the homomorphism between the long homology ex-act sequences of the pairs (X(R), Y(R)) and (^(C), 7(C)). It gives rise to thecommutative diagram(2.1.6)

0 » H2(X(R)) > H2(X(R),

•·· • F2 • H2(X(C)) > H2{X(C), K(C)) -» HX(Y(O) • 0

Let κ(Χ, Y) = 2. Then it follows immediately from this diagram that κ(Χ) =κ (Υ) = 1. Furthermore, since the only relation between the homology classes[Yi], . . . , [Ym] is of the form [Yl] + ... + [Ym] = [Y(R)] = 0, we have that ifκ(Χ, Υ) = 2 then the curve Y(R) splits X{R) in such a way that for some choiceof the part of X(R) this part is homologous to zero in H2(X(C), Y(C)); all thisfollows from (2.1.6). If we now take the corresponding surface W, then it will behomologous to zero in H2(X(C)). Thus the condition κ(Χ,Υ) — 2 implies theconditions κ{Χ) = κ(Υ) = 1 and W ~ 0. Diagram (2.1.6) shows that the converseis also true. The proposition is proved.

2.2. Complete intersections and double coverings.

Theorem 5. Let X be a nonsingular complete intersection PN. Then the followingassertions are true:

(i) K{X) < 1, and the only relation is of the form [X(R)] = 0.(ii) If degX is odd and dimX = η is even, then κ(Χ) = 0.

Proof. If Υ is a hyperplane section of X, then Hq(X(C), Y(C)) = 0 for η < q <2n . Therefore (1.1.3) implies that

K(X, Y)<dimH1(G,H2n(X(C),

Now repeating the argument with a deleted pointed Xo e X(R)\7(R) from theproof of Proposition 1, we see that the only relation between the relative homologyclasses is of the form

(2.2.1) [X(R)] modr(R) = 0.

After that we consider the homomorphism between the long homology exact se-quences of the pairs (X(R), Y(R)) and {X(Q, Y(C)). We then have that κ(Χ) <κ(Χ, Y) < 1, and the only relation will be, by virtue of (2.2.1), of the form[X(R)] = 0. Thus the first assertion of the theorem is proved. The second as-sertion can be proved by an argument similar to the one in the proof of Proposition4. The theorem is proved.

Theorem 6. Let X be a two-fold covering of a nonsingular complete intersection Ζ cP^ with branching along a regular intersection by a nonsingular hypersurface in PN.Then K(X)<1, and the only relation is of the form [X(R)] = 0.

Proof. Let Υ be the preimage of a general hyperplane section of Ζ under the pro-jection I - » Z . Then the desired result follows from the argument in the proof ofthe first assertion of Theorem 5. The theorem is proved.

2.3. Hyperplane sections.

Proposition 7. Let Ζ be a nonsingular projective GM-variety, X a hyperplane sectionof Ζ such that the operators

L": Hn-"(X{C), F2) -> Hn+i(X{C), F2)

HOMOLOGY CLASSES AND REAL ALGEBRAIC VARIETIES 287

are isomorphisms for 0 < q < n, where L is the Lefschetz operator, and f: X •—> Ζthe embedding map. Then κ ( / ) = 0.

Proof. We shall use the bound from Theorem 2:

K{f) < ^dimCokerLfir'tG, Hn+q(Z(C))) -^Hl(G, Hn+q(X(C)))].q>0

It suffices to show that the homomorphisms in the right-hand side are epimorphisms.To this end consider the commutative diagram

Lq:H"-q(Z(C)) • Hn+q{Z(C))

It gives rise to the

L":

L«:Hn-<l(X(C))

commutative diagram

Hl(G,H"-<i(Z(C))) —

I '

—» Hn+"{X{C)).

-+ Hl(G,Hn+"(Z(C)))

1

L«:Hl(G,Hn-<{X(C)) —^-» H\G, Hn+"{X{£))),

whence the desired assertion. The theorem is proved.

Remark 5. In the above proposition X can be replaced by a complete intersectionof Z .

Theorem 7. Let X be a projective variety such that the operators

Lq. Hn-g(X(C),F2) -» Hn+q{X{C),¥2)

are isomorphisms for 0 < q < η, and Υ a hyperplane section. Then

K{X, Y) < dimHl{G, Ηη~2{X(C) ,F2)) + dimHl(G, Hn~\X{C), F2)).

Proof. By (1.1.3),

κ(Χ, Y) < Y^dimHl(G, Hn+"(X(C), Y(C); F2))

= ^ dimH\G, Hn+q{X{C), Y{C); F2))

n=\= y£dimHl(G,H"(X(C)-Y(C),F2)).

q=0

Now we want to compute the cohomology group Hq(X(C) - Y(C), F2) for q <η - 1. Applying Poincare duality to the long homology exact sequence of the pair(X(C), y(C)), we have the exact sequence

• · · -+ Hq-2{Y{C)) -» W{X{C)) -• Hq{X{C) - Y(C))

Since Hk{Y(C)) = Hk(X(C)) for k < η - 1, it follows from the above sequencethat the sequence

3 tf(X(C)) Λ tf(*(C)) - //(X(C) - Y(C))

\ ±+ Hq+l(X(C))

288 V. A. KRASNOV

is exact for q < n-1. Let Pq{X(C), F 2 ) denote the primitive part of Hq(X(C), F 2 )i.e.,

P"(X{C), F 2 ) = Ker[L"-« + 1 : Hq(X(C), F 2 ) -» H2"-q+l(X(C), F 2 )] .

The condition that Lq: Hn~q(X(C)) - tf"+«(X(C)) are isomorphisms for 0 < q < ηgives rise to the Lefschetz decomposition

[?/2]

(2.3.2) Hq(X{Q) = 0 LkPq-2k(X{Q).

fc=0

Therefore it follows from (2.3.1) that

Hq{X{C) - Y{C)) = Pq{X(C))

for q < η - 1. Hence

n-\ n-\

Y^aimH'iG, Hq(X(C) - Y(C))) = ΣάχπιΗι(Ο, Pq(X(Q)).q=0 q=0

But, according to (2.3.2), this sum is equal to

dimHl(G, Hn-2(X(C))) + dimHl(G, H"-l(X(C))).

The theorem is proved.

Corollary 1. Let X be a projective variety such that the operators

Lq: H"-q(X(C), F2) -• Hn+q(X(C), F2)

are isomorphisms for 0 < q < η . Then

K{X) < dimH l {G, H"-2(X(C))) + dimH l (G, Hn~' (X(Q)).

To prove this, just apply (1.1.5).

2.4. Applications of the Hodge structure.

Theorem 8. Let X be a projective variety such that the groups Hq(X(C),Z){0<q<n) are free and k = [(n- l)/2]. Then

(2.4.1) * ( * ) <0<r+s<n

r>s

where the hr's(X(C)) are the Hodge numbers.

Proof. We shall show that the right-hand side of the inequality

K{X) < Y^dimHe^(G, Hn+q(G, Hn+q(X(C), Z))

(see (1.1.2)) does not exceed the right-hand side of (2.4.1). To this end, note that

, H"+q(X(C), Z)) - d i m / / ' ( G , H"+q(X(C), Z))

< TkHn+i(X(C), Z ) - r * = dime Hn+q{X(C), C)~T'

when q is odd, and

dimHeiq)(G, Hn+q(X(C), Z)) = dimH2(G, Hn+q(X(C),Z))

< T)aHn+q{X{C), Z)T* = dimHn+q(X(C), C)T*

when q is even. Note also that when q is odd the Lefschetz isomorphism

Lq: H"-q(X(C), C)^Hn+<l{X(C), C)

HOMOLOGY CLASSES AND REAL ALGEBRAIC VARIETIES 289

induces the isomorphism

(2.4.4) L": H"-q(X{C), C)T'^ Hn+«(X{C), C)"T*,

because

(2.4.5) LOX* = -T*OL,

and when q is even we have the isomorphism

(2.4.6) L": Hn-"(X(C), C)T* ̂ Hn+"{X(C), C)T*.

It now follows from (2.4.2)-(2.4.4) and (2.4.6) that

(2.4.7) ^ d i m / T ^ C , Hn+q{X{C), Z)) < ^dim7/"-«(X(C), Ζ) τ *.

Consider now the Hodge decomposition

H"-"(X(C),C)= φ Hr's(X(C)).r+s=n—q

Since r*(Hr>s(X(C))) = Hs>r(X(C)), it follows from this decomposition that

(2.4.8) aimHn-q{X{C),Cf = ^ hr's{X{C))r+s=n-q

r>s

when η - q is odd, and

(2.4.9) dimH"-"(X(C),CY' =

r>s

when n — q = 2t.It follows from (1.1.2) and (2.4.7)—(2.4.9) that to finish the proof of (2.4.1) it

remains to show that

(2.4.10) Σ dirnH'^iXiC))'' < Σ hk~2i>k~2'(X{Q).0<t<k i>0

To abbreviate our notation let H'-' = H'''(X(C)), and let P'-' denote the primitivepart of Htl, i.e., the kernel of the operator

τη-21+Ι . fjt,t _^ ijn-t+l ,n-t+l

By virtue of (2.4.5), it follows from the Lefschetz decomposition

//''' = Ρ11 Θ LP'-1·'-1 θ L2P'-2'-2 φ L3P'-3'-3 θ • · ·

thatΗ'+'1 = Pi·'® LP'J1'-1 φ L2P'-2''-2 θ L 3 ^ " 3 · ' " 3 θ · · · ,

which in turn implies that

Replacing t by t — 1 in this equality, we get

h'-1-'-1 = ρ'-1'1'1 + p'S2-'-2 + p'-3'1-3 + p>_-4><-4 +

Adding the two equalities, we have

290 V. A. KRASNOV

which implies, since p'+'{ </?'-', that

Therefore

0<t<k i>0

which is (2.4.10). The theorem is proved.

Remark 6. In the surface case Theorem 8 yields (2.1.2).In the next theorem we will need the following notation. Consider the set 0, 1, . . . ,

η - 1, and count the number of elements of it of the form Ak. If η is odd, subtractone. The number thus obtained is denoted by A(n). Note that

Δ(/ΐ) =

Theorem 9. Let X be a projective variety of odd degree such that the homology groupsHg(X(C), Z) (0 < q < n) are free, and let k = [(n- l)/2]. Then

κ(Χ)<

' η/4

( « -

(η +

1)/4,

2 ) / 4 ,

3 ) / 4 ,

« Ξ Ο

« Ξ 1

Η Ξ 2

« Ξ 3

m o d

m o d

m o d

m o d

4 ,

4 ,

4 ,

4 .

0<r+s<nr>s

Proof. Let / : X —> P^ be the embedding map. Applying (1.2.5), we have

(2.4.11)η

K(ff) < J 2 d i m H ^ \ G , H n + q { X { C ) , Z ) )

9=1

η

- ^ d i m I m [ i / E ^ ( G , Hn+q(T?N{C), Z ) ) - • H e { q ) ( G , H n + q { X { C ) , Z ) ) ] ,

9 = 1

w h e r e t h e right-hand s i d e o f ( 1 . 2 . 5 ) w a s r e w r i t t e n i n a d i f f e r e n t w a y . N o t e n o w t h a t

/ / e ( ? ) ( G , H n + q ( P N { C ) , Z ) ) = F 2

only when η - q = 0 mod 4; otherwise this group is zero. We shall now show that

(2.4.12) dimIm[/r ( < ? )(G, Hn+<1(PN(C), Z)) -» He^{G, Hn+q{X{C), Z))] = 1

whenever « - ? Ξ 0 mod 4. Let α denote a generator of Hl(G, H2(PN(C), Z)).Then the image of a" under the homomorphism

H"{G, H2n(PN(C),Z)) ^Hn(G, H2n(X(C),Z))

is different from zero, since the degree of X is odd. Hence the image of am underthe homomorphism

Hm(G, H2m(PN(C),Z))-^Hm(G, H2m(X(C),Z))

is also different from zero for m = 1, ... , η. This gives us (2.4.12). Finally, weremark that K(f) = κ{Χ) for η even—this can be checked as in the proof of

HOMOLOGY CLASSES AND REAL ALGEBRAIC VARIETIES 291

Proposition 5—and κ(Χ) < κ ( / ) + 1 for η odd. It remains to apply (2.4.11),(2.4.12), and the inequality

',Hn+q(X(C),Z))q=l

0<r+s<n ;>0

obtained in the proof of Theorem 8. The theorem is proved.

Now we state the relative version of Theorem 8.

Theorem 10. Let X be a projective variety and Υ a hyperplane section of X. Supposethat the homology groups Hq(X(C) - Y{C), Z) are free for 0 < q < n, and letk = [(n- l)/2]. Then

K(X,Y)<

n-2<r+s<n-lr>s

Proof. By the Poincare isomorphism, we can identify homology with cohomology.Since the complex conjugation changes the orientation of X{C) only when η is odd,we have that

τ, = (-1)"τ · .

By virtue of (1.1.4), this yields

K{X, Y) < 5 3 d i m / / ^ ( G , Hn+<I(X(C), Y(C);Z))q>0

(2.4.13) = Y.^^H^{G,Hq{X{C)-Y{C),Z))q=0n-l

= 5 3 d i m Η ' { 9 ) ( G > H"{X(C)- Υ•(C).Z)),9=0

where in the last equality we used the freeness of the group Hq(X(C) - Y(C), Ζ),0 < q < η — 1. Now we make several observations. First,

4 dimF2 H^(G, Hq(X(C) - Y(C), Z))

<dimcHq{X(C)-Y(C),C){-l)'T'

Second, for 0 < q < η - 1 the equality

(2.4.15) Hq(X{Q - Y(C), C) = Pq(X(C), C)

holds. And finally

(2.4.16) dimP'i(X(C),C)-T' = Y] pr's + pql2'ql2

r+s=qr>s

when q is odd, and

(2.4.17) dimP"{X(C),Cy' < 5 3 pr-s + p"'2'9'2

r+s=qr>s

when q is even.

292 V. A. KRASNOV

Similar equalities and inequalities have been obtained before, and we omit thedetails. Thus we deduce from (2.4.13)—(2.4.17) that

K(X,Y)< > pr's + > p'·'= > hr's + hk-k.

0<r+s<n 0<i<k n-2<r+s<n-lr>s r>s

The theorem is proved.

Corollary 2. Under the assumptions of Theorem 10,

κ(Χ)<n-2<r+s<n—\

r>s

This follows from (1.15).

§3. ORIENT ABLE AND NONORIENTABLE COMPONENTS

Among the components X\, ... , Xm there can be both orientable and nonori-entable ones. Let X\, ..., Xs be the orientable components, and the rest nonori-entable. The number of relations between the homology classes [X\],..., [Xs] eHn(X(C), ¥2) will be denoted by κ+(Χ), and the number of relations between theremaining classes by K-(X). It is clear that

κ+(Χ) + K.{X) < κ(Χ),

and therefore any bound on κ(Χ) yields bounds on κ+(Χ) and K-(X). In thissection we shall obtain more precise bounds than those derived from the above in-equality. In order to get information about κ+(Χ) we need to recall some facts aboutcomplex line bundles with real structure.

3.1. Characteristic classes. Let L —» X be a line bundle, where L and X are realalgebraic varieties. Then the map L(C) —> X(C) between the sets of complex pointsin a complex-analytic line bundle and the map L(R) —> X(R) is a real-analytic bun-dle. Associated with L(C) there is the Chern class c(L(C)) e H2(X(C), Z) and withL(R)-the Stiefel-Whitney characteristic class w(L(R)) e Hl(X(R), F 2 ) . In this sub-section we shall define a new characteristic class cw(L) e H2(X(C), G; Z(l)) whichis a "mixture" of the characteristic classes c(L(C)) and w(L(R)). Here Z(l) is theconstant sheaf on X(C) with fiber Ζ on which involution τ acts by multiplicationby - 1 . We shall define the characteristic class cw (L) with the aid of the exponentialexact sequence

(3.1.1) o - > z ( i ) ^ ^ ( C ) - > < ? ; ( C ) - > i

of G-sheaves, where the homomorphism Z(l) —> ^ - ( Q sends k to 2nik, and theother homomorphism sends / to exp(/). The group Η' (X(C), G; ^w C J is thegroup of complex-analytic bundles on X(C) with real structure. It coincides withPicA\ From the exact sequence (3.1.1) we obtain the coboundary operator

We now set

(3.1.2) cw(L) = S(L(C)).

We can also define homomorphisms

a: H2(X(C), G; Z(l)) - H2(X(C), Ζ),

β: H2(X(C), G; Z(l)) - Hl(X(R), F 2 ) .

HOMOLOGY CLASSES AND REAL ALGEBRAIC VARIETIES 293

The map a is the composition of the projection onto I^2(X(C), G; Z( l)) and theembedding

^ 2 , G; Z( l ) ) c H2(X(C), Z( l)) = H2(X(C),Z).

Before defining the homomorphism β we examine the spectral sequence

I%'q(X{R), G;Z{\)) = Hp(X{R),%'q(Z{\)))^Hp+i{X{R), G ;

Since

r2, q is odd,

we have the canonical homomorphism

H2(X(R),G;Z(\))^H1(X(R),¥2).

The composition

Η (Λ ( C ) , (J , L\i}) —> π (Λ ( Κ ) , ( J , ZJ(1 )) —» η (Λ [κ.), Γ2)

will be denoted by β .

Proposition 8.

a(cw(L)) = c(L(Q), fi(cw{L)) = w(L(R)).

Proof. First we remark that the exact sequence of sheaves (3.11) gives rise to theexact sequence of Galois-Grothendieck cohomology groups and the exact sequenceof usual cohomology of sheaves. Moreover, there is a homomorphism from the firstsequence into the second one. In particular, we have the commutative diagram

-> H2(X(C),G;Z(l))

0·) — ^ H2(X(C),Z(l)),

which implies the first equality in the proposition. To prove the other equality,consider the exact sequence of G-sheaves

(3.1.3) υ —»^r (υ, (y\x(R) —>Λ ((J, <y )U(R) — > ^ (Cr, ̂ ( i ) ) —> u,

which is obtained from (3.1.1). It coincides with another exact exponential sequence

(3.1.4) o - J ^ ^ i ' ^ F z - O

on X(R), where si is the sheaf of germs of real-valued complex-analytic functionson X{R) and sf* is the sheaf of germs of invertible functions. Thus exact sequences(3.1.3) and (3.1.4) give rise to the commutative diagram

(3.1.5) I I

Hl(X(R),j/*) - ^ Hl(X(R),F2)

Note now that the homomorphism

Hl(X(R), &\G, (9*)) - Hl(X(R), F 2 ) ,

294 V. A. KRASNOV

which is derived from (3.1.5), coincides with the composition

Hl(X(R), jr°(G, &*)) -> Hx(X(R) ,G;@*)

Λ H2(X(R), G; Z(l)) Λ Hl(X(R)F2).

In particular,

(3.1.6) β(δ(Σ(Ο\Χ{η))) = sgn(L(R)).

Sincesgn(L(R)) = w(L(R)), fi(d(L(Q\x<K)) = fi(cw(L)),

the second equality follows from (3.1.6). The proposition is proved.

3.2. Orientable components. First we introduce some notation. Let X+(R) denotethe union of the orientable components of X(R). For brevity we set

FqH2n+l(·, G; F2) = Fq{·),

where FqH2n+l(', G; F2) denotes the filtration obtained from the spectral sequence

/£'«(·, G; F2) = H"(G, H"(-,F2)) => H2n+\-,G; F2).

This abbreviation will be used for the spaces X(C), X(R), and X+(R). Moreover,let

3f€H2(X(C),G;Z(l)), Κ e H2(X(C), Z), k e Hl(X(R), F2)

be the characteristic classes of the canonical bundles on X, X{C), and X(R), re-spectively. Note that Κ = a(Jf), k = β{3?), and k\x+{K) = 0. Also let

(3.2.1) Kq: H"-q{X{C), F2) ^Hn+q(X(C), F2)

denote the homomorphism given by multiplication by Kq .

Theorem 11. Let X be a GM-variety such that the homomorphisms (3.2.1) are iso-morphisms for 0 < q < η . Then κ+(Χ) = 0.

Proof. Consider the homomorphism

(3.2.2) F"(X(C))/F"-l(X(C))^Fn(X+(R))/F"-l(X+(R)).

It coincides with

(3.2.3) H"+l(G, Hn(X(C), F2)) - H"(X+(R), F2),

since X is a GM-variety. In turn, (3.2.3) is determined by the homomorphism

(3.2.4) H"(X(C), F2) - Hn(X+(R), F 2 ) ,

induced by the embedding X+(R) c X(C). We shall prove that (3.2.2) is an epimor-phism. This would imply that (3.2.4) is also an epimorphism, and the latter meansthat κ+(Χ) = 0. To prove that (3.2.2) is an epimorphism we first remark that thehomomorphism

(3.2.5) F2n+1(X(C))/F"-l(X(C)) - F2n+l(X+(R))/F"-l(X

is an epimorphism because the map

X2n+l (X(C), G; F2) - H2n+l(X{R), G; F2)

is an isomorphism and the map

H2n+l(X(R), G; F2) -» H2n+l(X+(R), G; F2)

HOMOLOGY CLASSES AND REAL ALGEBRAIC VARIETIES 295

is an epimorphism. Now we prove by induction on q that

2 6)

where q > 0. Because

the surjectivity of (3.2.2) follows from (3.2.6) and the surjectivity of (3.2.5). Whenq = 0 both sides of (3.2.6) are the same. Assume that this is also true for q -m - \ > 0. We now show that this is true for q = m. To this end, consider thehomomorphism

F"-m(H2"-2m+i(X{C), G; F2))/Fn-m-1

(3.2.7) , ,_U >Fn+m(H2n+l{X(C), G,F2))/Fn+m-1.

Since X is a GM-variety, this homomorphism coincides with

I I Km

G, Hn-m(X{C),F2)) — • H"-m+l(G, Hn+m(X(C), F2)).

It now follows from the hypothesis of the theorem that (3.2.7) is an isomorphism.Therefore

Fn+m(X{C)) = Fn+m-l{X(C))+^m U F"-m(H2n-2m+l(X(C), G; F 2 )) .

The image of 3?m U F"-m{H2"-2m+1(X(C), G;F 2 )) under the homomoφhismFn+m(X(C)) -> Fn+m(X+(R)) is zero because 3? goes to k . Therefore the imagesof Fn+m(X(C)) and Fn+m-\X(C)) coincide. The theorem is proved.

Remark 7. Under the assumptions of Theorem 11 the number K" is odd, andtherefore X{R) contains at least one nonorientable component because in this casekn φθ.

3.3. Nonorientable components. First we obtain two general bounds, and then weapply them to certain types of varieties.

Proposition 9.£ * ( ) , Z)),

where1 q even,2 q odd

Proof. We use the Galois-Grothendieck cohomology groups H2n+2(·, G; Z)when η is odd, and H2n+l(·, G; Z) when η is even. Consider the case of oddη . Note first that

l£2'n(X{R), G;Z) = Hn+2(G, Hn(X(R), Z)) = H"(X(R), Z)2 ,

where Hn(X{R), Z) 2 is the subgroup of Hn(X(R), Ζ), consisting of elements oforder two. On the other hand,

K-(X) < dimCoker[//"(X(C), Ζ)" τ * -> H"(X(R), Z)2]

< dimCoker[//"+2(G, H"(X(C),Z)) -> Hn+2(G, Hn(X{R),Z))]

< dimCoker[/^+ 2-"(^(C), G\Z)-> l£2'n{X(R), G;Z)].

296 V. A. KRASNOV

Since the homomorphism H2n+2(X(C) ,G;Z)-* H2n+2{X(R), G; Z) is an isomor-phism, the homomorphism

H2n+2(X(C), G; Z)/F"-l(X(C), G; Z)

_, H2n+2(X(R), G; Z)/F"-l(X(R), G; Z)

is an epimorphism. Therefore

dimCoker[/^2-"(X(C), G; Z) - InJ2 ·»(X(R), G; Z)]

< dimi/ 2 n + 2 (X(C), G; Z)/Fn(X(C), G; Z)

, Z))

, Z)).

The proposition is proved.

Remark 8. Similar arguments yield the relative version of Proposition 9:

κ_(Χ, Υ) < ΣdimF2 H^\G, H"+"(X(C), Y(C); Z)).

Notice that under the map

Hn(X(C), Z) -• Hn{X(R), Z)

the subgroup Hn(X(C), Ζ)~τ" goes to the subgroup Hn(X(R), Z)2 . Therefore thenumber

= dimCoker[//n(X(C), Ζ)~τ' @ H"{Z{R), Z) 2 -» //"(X(R), Z)2]

can be defined for a map / : X —» Ζ of real algebraic varieties.

Theorem 12. Suppose that Ζ is a GMZ-variety. Then

K-(f) < J ^ d i m ^ C o k e r l t f ^ G , Hn+«{Z{C),Z)) - //^>(G, //n + 9(X(C), Z))].

The proof is similar to that of Theorem 4. One should only use the Galois-Grothendieck cohomology groups H2m+2(- ,G;Z), η is odd, and H2m+l (· ,G;Z),η is even.

We now consider some consequences of our results.

Corollary 3. Let X be a surface such that Hx (X(C), Z) = 0. Then κ_ (X) = 0.

This follows from Proposition 9.

Corollary 4. Let X be a complete intersection of odd degree. Then K-(X) = 0.

Proof. If η is even, then κ(Χ) = 0 by Theorem 5, and therefore K-(X) = 0. If« is odd, then on applying Theorem 12 to the embedding f: X ^->PN we have theequality «_(/) = 0. On the other hand, H"{PN{C), Z) = 0 when « is odd, andtherefore

0 = * _ ( / ) = dimCoker[//"(X(C), Z)"T* -> //"(X(R), Z) 2 ] .

It now follows that Κ-(Λ') = 0. The corollary is proved.

HOMOLOGY CLASSES AND REAL ALGEBRAIC VARIETIES 297

BIBLIOGRAPHY

1. V. A. Krasnov, Harnack-Thom inequalities for mappings of real algebraic varieties, Izv. Akad.Nauk SSSR Sen Mat. 47 (1983), 268-297; English transl. in Math. USSR Izv. 22 (1984).

2. I. R. Shafarevich, Basic algebraic geometry, "Nauka", Moscow 1972; English transl., Springer-Verlag, 1974.

3. V. M. Kharlamov, Topological types of nonsingular surfaces of degree 4 in RP3 , Funktsional.Anal, i Prilozhen. 10 (1976), no. 4, 55-68; English transl. in Functional Anal. Appl. 10 (1976).

4. V. A. Krasnov, The Albanese map for real algebraic varieties, Mat. Zametki 32 (1982), 365-374;English transl. in Math. Notes 32 (1982).

Received 5/MAY/88

Translated by A. MARTSINKOVSKY