Generalization of the Virtual Arithmetic Genus of a Smooth Polarized Surface

Abstract

Let (X, L) be a smooth polarized surface, i.e., a pair consisting of a smooth complex projective surface X and an ample line bundle L on X. The virtual arithmetic genus g(X, L) of (X, L) is defined by the formula 2g(X, L)- 2 = (K X + L)L, where K X is the canonical bundle of X. The purpose is to generalize several results on the virtual arithmetic genus to the following cases: (I) X is a smooth complex projective variety of dimension n > 3, and e is an ample vector bundle of rank n - 1 on X; (II) X is a normal complex projective surface, and L is a nef line bundle on X.