Multigraded Fujita approximation

Abstract

The original Fujita approximation theorem states that the volume of a big divisor D on a projective variety X can always be approximated arbitrarily closely by the self-intersection number of an ample divisor on a birational modification of X. One can also formulate it in terms of graded linear series as follows: Let W • = (W k ) be the complete graded linear series associated to a big divisor D, where For each fixed positive integer p, define W (p) • to be the graded linear sub-series of W • generated by Wp: Then the volume of W (p) • approaches the volume of W • as p → oo. We will show that, under this formulation, the Fujita approximation theorem can be generalized to the case of multigraded linear series.