Revisiting Gamma conjecture I: counterexamples and modifications

Abstract

We continue investigation of asymptotics of quantum differential equation for Fano manifolds, with a special regard to Gamma conjecture I and its underlying Conjecture $\mathcal{O}$. We introduce the A-model conifold value, a symplectic invariant of a Fano manifold, and propose modifications for Gamma conjecture I based on this new definition. We discuss an interplay of birational transformations with an extension of Gamma conjecture I over the K\"ahler moduli space. These heuristics are applied to rigorously identify the principal asymptotic class in the case of $\mathbb{P}^1$-bundles $X_n=\mathbb{P}_{\mathbb{P}^{n}}(\mathcal{O}\oplus\mathcal{O}(n))$. We observe, in particular, that for $X_n$ of dimension at least four, the Conjecture $\mathcal{O}$ holds just for even values of $n$, and in these cases we falsify the original non-modified Gamma conjecture I.