MATHEMATICS: S. LEFSCHETZ 
105 
II ^h,h ^h,2, .... oih,p\ i(^h,p+h ^^h,P+2, . . . ic»)h^2p II, (2) 
= 1, 2, . . . 2p) 
the (aj)'s being real. 
Remark. — In all this Vp could be replaced by any real irreducible algebraic 
variety of irregularity p. 
3. Conversely if to il corresponds a Vp of rank one with 2p linear cycles 
7i, T2, . . . y2p, such that the cycles miji + W2T2 +••.. + w^py^p include 
the double of any other, while p independent integrals of the first kind Wi, 
U2, . . . Up, have with respect to them a period matrix of type (2), is bi- 
rationally equivalent to a real abelian variety. Indeed as a consequence of 
the assumptions made, if the equations (1) represent Vp, and if (coi,a;2, . . . o)p) 
are a set of periods of the (/)'s so are (coi, 0^2, .. . cop). Moreover the (/)'s 
are real meromorphic functions of the (w)'s and of a finite number of con- 
stants. If we replace these constants by their conjugates we obtain new 
functions f(ui, U2, . . . Up) with the same periods as the (/)'s the equations 
x'j =fj+fj, = - i (fj - fj), U =1,2,. . .n) 
represent in an S2n a real abelian variety birationally equivalent to Vp. This 
real abelian variety has co p real points. 
4. Assuming then Vp real with real folds, to determine their number we 
remark that at two conjuguate points the integrals of No. 2 take conjuguate 
values Uh + iujl, ui — iu'h, modulo fi. At the real folds the (i/)'s are given 
p 
by + i 2* ^h,p+ixy (uft arbitrary, = 0 or 1; /f = 1, 2, . . . There 
1 
are in general p — s periods of this type such that no linear combination of 
them with integral coefficients is of the form 
p 
2} ^^'f" ^^'^ ^h,p+fx), (h = 1,2, . . . p). 
1 
Hence p — s oi the integers r can be made equal to zero. Taking the others 
equal to zero or one yields 2' distinct real folds. — That, there are varieties 
having that number of real folds whatever 5 ^ ^ is shown by the 
canonical matrix 
{a^v = aj^fj^ = i m^j;-\- ia'^i^', ntf^y integers 
forming a matrix of rank s modulo 2; the 
quadratic form S aJ^V ^/x is definite, 
positive.) 
Hence a real abelian variety can have any number of real folds given by 
2^ 0 ^ ^ ^ ^. 
5. Any real fold of the variety can be transformed into any other by a bira- 
tional transformation belonging to the continuous group {ui,, + Ch). Hence 
1, 0 . . .0, an, ai2, . . . aip 
0, 1 . . .0, fl21, 022, .. . (hp 
0, 0 . . . l,api, app 
