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arXiv · 2405.16979

Revisiting Gamma conjecture I: counterexamples and modifications

Sergey Galkin·Jianxun Hu·Hiroshi Iritani·Huazhong Ke·Changzheng Li·Zhitong Su
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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.

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Sergey Galkin, Jianxun Hu, Hiroshi Iritani, Huazhong Ke, Changzheng Li, Zhitong Su. 2024-05-27. Revisiting Gamma conjecture I: counterexamples and modifications. https://arxiv.org/abs/2405.16979

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