Canonical map of low codimensional subvarieties

Fix integers agreater-or-equal, slanted1, b and c. We prove that for certain projective varieties Vsubset ofPr (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 thatimagewhere 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.