Canonical map of low codimensional subvarieties

Fix integers a⩾1, b and c. We prove that for certain projective varieties V⊂Pr (e.g. certain possibly singular complete intersections), there are only finitely many components of the Hilbert scheme parametrizing irreducible, smooth, projective, low codimensional subvarieties X of V such thath0(X,OX(aKX-bHX))⩽λdε1+c∑1⩽h<ε2pg(X(h)),where d, KX and HX denote the degree, the canonical divisor and the general hyperplane section of X, pg(X(h)) denotes the geometric genus of the general linear section of X of dimension h, and where λ, ε1 and ε2 are suitable positive real numbers depending only on the dimension of X, on a and on the ambient variety V. In particular, except for finitely many families of varieties, the canonical map of any irreducible, smooth, projective, low codimensional subvariety X of V, is birational.