Let Y be an (m + 1)-dimensional irreducible smooth complex projective variety em-bedded in a projective space. Let Z be a closed subscheme of Y, and 6 be a positive integer such thatL_(z,y)(δ)is generated by global sections. Fix an integerd >6 +1, and assume the general divisorX ∈|H~0(Y,L_z,Y(d)) |is smooth. Denote byH~m(X; Q)_(⊥Z)~(van)the quotient of H~Q) by the coho-mology of Y and also by the cycle classes of the irreducible components of dimension m of Z. In thepresent paper we prove that the monodromy representation on H~m(X;Q)_(⊥Z)~(van)for the family of smoothdivisors X ∈|H~0 (Y,L_(Z,Y) (d)) |is irreducible.
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