On the asymptotic enumerativity property for Fano manifolds

TL;DR

This paper investigates the asymptotic enumerativity of Gromov-Witten invariants on Fano manifolds, providing counterexamples and new positive cases, thus advancing understanding of when these invariants are truly enumerative.

Contribution

It offers the first counterexamples to the conjecture that all Fano manifolds satisfy asymptotic enumerativity and identifies new classes of Fano manifolds where it holds.

Findings

Counterexamples include certain hypersurfaces and projective bundles.

Positive examples include many Fano threefolds and specific hypersurfaces.

Provides conditions under which asymptotic enumerativity is satisfied.

Abstract

We study the enumerativity of Gromov-Witten invariants where the domain curve is fixed in moduli and required to pass through the maximum possible number of points. We say a Fano manifold satisfies asymptotic enumerativity if such invariants are enumerative whenever the degree of the curve is sufficiently large. Lian and Pandharipande speculate that every Fano manifold satisfies asymptotic enumerativity. We give the first counterexamples, as well as some new examples where asymptotic enumerativity holds. The negative examples include special hypersurfaces of low Fano index and certain projective bundles, and the new positive examples include many Fano threefolds and all smooth hypersurfaces of degree $d \leq (n+3)/3$ in $\mathbb{P}^n$.