Asymptotic Cohomological Functions on Projective Varieties

Our purpose here is to consider certain cohomological invariants associated to complete linear systems on projective varieties. These invariants — called asymptotic cohomological functions — are higher degree analogues of the volume of a divisor. We establish the continuity of asymptotic cohomological functions on the real Néron–Severi space and describe several interesting connections which link them to classical phenomena, for example Zariski decompositions of divisors, or Mumford’s index theorem for the cohomology of line bundles on abelian varieties. Our concepts have their origins in the Riemann–Roch problem. The classical version asking how h (X,OX(mD)) changes as a function of m (where X is an irreducible complex projective variety, and D is a Cartier divisor on X), has only been answered in dimensions up to two, by Riemann and Roch for curves, and by Zariski [24], and Cutkosky and Srinivas [5] for surfaces. The lack of a satisfactory answer in higher dimensions makes it important to look at the question from an asymptotic point of view. For ample divisors, the by now classical asymptotic Riemann–Roch theorem of Kleiman [17] and Snapper [23] settles the issue. For arbitrary divisors, however, the question has only surfaced recently in the form of the volume of a divisor, ie. the asymptotic rate of growth of the number of global sections of its multiples. The notion of the volume first arose implicitly in Cutkosky’s work [4], where he used asymptotic computations to establish the non-existence of rational Zariski decompositions on a certain threefold. It was then studied subsequently by Demailly, Ein, Fujita, Lazarsfeld, and others, while pioneering efforts regarding other asymptotic invariants of linear systems were made by Nakayama [21] and Tsuji. In this process the properties of the volume were more fully explored, and instead of thinking of the volume as an invariant linked to a single divisor, one started to consider it as a function on the Néron–Severi space, thus as an intrinsic invariant of the underlying variety X . More precisely, the volume of a Cartier divisor D on an irreducible projective variety X of dimension n is defined to be