LEFSCHETZ : THEORY OF ALGEBRAIC MANIFOLDS. 5 



whatever the manifold M. Proceed by induction. The formula 

 is easily verified when 



M = M, = ArA.2 A,. s>r. 



Grant that it holds for s>s_, I say that it is also true for s = s^. 



For /.M,^D = M,S> 



.-. -: Ms^ (1+D)> \ = \ M,^ (i + C) f . (6) 



ButM,^Z)^ = 3/,^D^-\ D. 



Hence from the assumptions made follows : 



jM,^z)^-M.i+z)) ;■ = :m,^d''-Uj+C) ' \ ■,k>i 



This remains true when Ms, is replaced by Mg^ C^ . 



.-. { M.D'' i = j M.Ail^C) ' -1 .)'(, k>l 

 From (6.) follows 



This together with (7) gives finally 



(7) 



\M,^{{{i+c)'' -i+iV +m+D)-/.{i+cy)\. 



.-. j Mjl + D) -M,A1+CV \ =0, 

 as was to be proved. 



5. It remains to be shown that if /.D = Ci — C2 , then 



\_ _ J 



\M{1+D) \ = jM(i+Ci)^(i+C2) ' \. 

 If we set /.D = D' = Ci- C., we have by (5) 



(8) 



\M(1+D) \ = \ MU+D'Y \ 



(9) 



