A NOTE ON NON-NEGATIVITY OF THE ith Δ-GENUS OF QUASI-POLARIZED VARIETIES

Abstract

Let (X, L) be a quasi-polarized variety of dimension n. In this paper, we will study non-negativity of the ith Δ-genus Δ_i(X, L). We will prove the following. Assume that X is a normal Gorenstein variety such that the irrational locus of X consists of at most finite points and the dimension of the singular locus of X is less than or equal to the dimension of the base locus of |L|. If i is greater than the dimension of the base locus of |L|, then Δ_i(X, L) is non-negative. We will also give a lower bound for Δ_i(X, L) when (X, L) is a polarized abelian variety.