A generalization of Δ-genus for big divisors on projective varieties

Abstract

Let X be a complex projective variety, and let D be a big Cartier divisor on X. For this data, we define an invariant that refines the Δ-genus introduced by Fujita for nef and big divisors. We relate to the Okounkov body of D, and give some examples. In particular, we show that can be arbitrarily negative already on surfaces.