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MATHEMATICS: S. LEFSCHETZ 
REAL HYPERSURFACES CONTAINED IN ABELIAN VARIETIES 
By S. Lefschetz 
Department of Mathematics, University of Kansas 
Communicated by E. H. Moore, April 29, 1919 
1. In a recent note of these Proceedings (April, 1919), I showed that an 
abelian variety of genus p and rank one, Vp, is birationally transformable 
into a real one if and only if it possesses 2 p independent linear cycles 71, 72, 
• • • yip, with respect to which p integrals of the first kind have a period 
matrix of type 12 = || 03^,1, . . . cohy, io)k,p+u - • • ^^h,2p \\ ; (h = 1, 2, . . . 
p), the (co)'s being real. I propose now to investigate the number p' of 
algebraically distinct real hypersurfaces which Vp, if real, may have. This 
number p' ^ p, Picard number of Vp, may also be defined as the maximum 
number of real hypersurfaces which cannot be logarithmic singularities of a 
simple integral of the third kind. 
2. In a general way Vp be an abelian variety of rank one, real or not, 
with the independent linear cycles 7i, 72, . . • 72^. By associating y,J^ with 
7j, we obtain a superficial cycle (m, p) and any other depends upon those of 
this type. In particular denoting by (A^~^) the two dimensional cycle 
formed by A^~^, curve of intersection of p — 1 algebraic hypersurfaces of 
the same continuous system as a given one A, we have 
2P 
m ^/^.f ('^j integer = — m^^^). (1) 
1 
It may be shown that if no integral of the first kind is constant on A the 
alternate form 
2 w^,,; oCfj^ (2) 
is a principal form of 12 as defined by Scorza (Palermo Rendic, 1916), and 
conversely to a principal form (2) corresponds an algebraic hypersurface A. 
Moreover to algebraically distinct hyersurfaces correspond linearly independ- 
ent principal forms from which follows at once p = 1 + ^, where k is Scorza's 
index of singularity for 12. 
3. Let us now assume Vp real. A real hypersurface A of V p is trans- 
formed into itself by T, transformation of the variety which permutes its 
pairs of conjugate points and this property is characteristic for A. It may 
be shown that there are real curves A^~^ , — let the one of No. 2 be one of 
them, and a its real line (locus of its real points). A small oriented circuit 
tangent to a. in {A^~^) is transformed by T into one of opposite orientation, 
for in the neighborhood of a, T behaves like an ordinary plane symmetry. 
It follows that T transforms the superficial cycle {A^~^) into its opposite. 
Taking into account the fact that this cycle is a two sided manifold and also 
