Asymptotic cohomological functions of toric divisors

We study functions on the class group of a toric variety measuring the rates of growth of the cohomology groups of multiples of divisors. We show that these functions are piecewise polynomial with respect to finite polyhedral chamber decompositions. As applications, we express the self-intersection number of a T -Cartier divisor as a linear combination of the volumes of the bounded regions in the corresponding hyperplane arrangement and prove an asymptotic converse to Serre vanishing. Suppose D is an ample divisor on an n-dimensional algebraic variety. The sheaf cohomology of O(D) does not necessarily reflect the positivity of D; O(D) may have few global sections and its higher cohomology groups may not vanish. However, for m ≫ 0, O(mD) is globally generated and all of its higher cohomology groups vanish. Moreover, the rate of growth of the space of global sections of O(mD) as m increases carries information on the positivity of D. Indeed, if we write h(mD) for the dimension of H(X,O(mD)), then by asymptotic Riemann-Roch [La1, Example 1.2.19], (D) = lim m h(mD) mn/n! . In general, when D is not necessarily ample, this limit exists and is called the volume of D. It is written ĥ(D) or vol (D). The regularity of the rate of growth of the cohomology groups of O(mD) for m ≫ 0 contrasts with the subtlety of the behavior of the cohomology of O(D) itself and motivates the study of asymptotic cohomological functions of divisors. Lazarsfeld has shown that the volume of a Cartier divisor depends only on its numerical equivalence class and that the volume function extends to a continuous function on N(X)R [La1, 2.2.C]. The volume function is polynomial on the ample cone, where it agrees with the top intersection form. In some special cases, including for toric varieties, smooth projective surfaces, abelian varieties, and generalized flag varieties, the volume function is piecewise polynomial with respect to a locally finite polyhedral chamber decomposition of the interior of the effective cone. The behavior of the volume function outside the ample cone