Continuous and Discrete Asymptotic Behaviours of the J-function of a Fano Manifold

Abstract

In this paper, we propose a condition on the coefficients of a cohomology-valued power series, which we call ``asymptotically Mittag-Leffler''. We show that if the $J$-function of a Fano manifold is asymptotically Mittag-Leffler, then it has the exponential growth as $t\to +\infty$. This provides an alternative method to compute the principal asymptotic class of a Fano manifold using the coefficients of $J$-function. We also verify that the $J$-function of the projective space is asymptotically Mittag-Leffler, and the property of having an asymptotically Mittag-Leffler $J$-function is preserved when taking product and hypersurface.