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Let (X,L) be a polarized manifold of dimension n. In this paper, by using theith sectional geometric genus and the ith \Delta-genus, we will give anumerical characterization of (X,L) with K_{X}=-(n-i)L for the following cases(i) i=2, (ii) i=3 and n \geq 5, (iii) max{2, dim Bs|L|+2} \leq i \leq n-1.

A numerical characterization of polarized manifolds (X,L) with K_{X}=-(n-i)L by the ith sectional geometric genus and the ith \Delta-genus