Bounds for Genus Zero Gromov-Witten Invariants

Abstract

We give a bound for the norm of each primary genus zero Gromov-Witten invariant in any smooth projective variety. This bound depends only on the maximum degree of its defining polynomials, the norms of the differential forms representing each insertion, the number of marked points and the degree of the curves. These bounds grow factorially in the number of marked points and a multiple of the degree. This implies that the Borel transform of the genus zero primary generating function converges absolutely in a particular translation of the ample cone. To prove these bounds, we start with Siebert's formula, expressing genus zero Gromov-Witten invariants in terms of the Segre class of the normal cone of the underlying moduli space inside the space of curves mapping to projective space and the Chern classes of the bundle whose fiber over a curve is the space of sections of the pulled back normal bundle. We then use this description to bound each primary Gromov-Witten invariant in terms of an intermediate count of curves called the D-volume. A deformation to the normal cone argument is then used to compare such an intermediate count with one in projective space.