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 Kähler 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.

Figures (9)

• Figure 1: A logical relationship among various conditions.
• Figure 2: The divisor diagram for $X_n$ (for $n=4$) and the two GIT chambers.
• Figure 3: The partially compactified Kähler moduli space $\overline{\mathcal{M}}_{X_n}$ together with the (dashed) infinity line $\{{\mathfrak{q}}_1=0\}=\{q_1=\infty\}$ which is identified with $\overline{\mathcal{M}}_{Y_n}$.
• Figure 4: Bifurcation of the leading exponent (valuation) of the critical values $u$ (for $n=4$).
• Figure 5: Critical values of $f_{q_1,q_2}$ for $n=4$. We let $q_1,q_2$ be positive real numbers and vary $q_2/q_1^2$; the critical values marked by the same number correspond to each other.

Theorems & Definitions (111)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Conjecture 1.4
• Conjecture 1.5: Modified Gamma conjecture I: weak form
• Remark 1.6
• Remark 1.7
• Theorem 1.8: Theorem \ref{['thm:Lefschetz_A']}
• Theorem 1.9: Theorems \ref{['XnasympGamma']}, \ref{['thm:A_for_Twrong']}
• Definition 2.1: Property ${\mathcal{O}}$