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 ${\Delta }^b(X,D)$ that refines the Δ-genus introduced by Fujita for nef and big divisors. We relate ${\Delta }^b(X,D)$ to the Okounkov body of D, and give some examples. In particular, we show that ${\Delta }^b(X,D)$ can be arbitrarily negative already on surfaces.

Keywords:

• Big divisors
• Δ-genus
• volume
• Okounkov body

2010 Mathematics Subject Classification:

• Primary 14C20
• Secondary 52B20

Acknowledgement

The author would like to thank the referee for giving very useful comments and suggestions.

Additional information

Funding

This research was partially supported by Japan Society for the Promotion of Science KAKENHI [grant no. 16K05103].