Conjecture O for Fano manifolds

Let $X$ be a Fano manifold. Write $H(X)=H^{even}(X)$ , let $\rho$ be its Fano index, and let $\boldsymbol{c_1(X)\star_0}$ denote the linear operator on the unital commutative ring $(H(X),\star_0)$ given by small quantum multiplication by $c_1(X)$ . Define

T=\sup\left\{|u|:u\in\operatorname{Spec}(c_1(X)\star_0)\right\}.

Thus $T$ is the spectral radius of $c_1(X)\star_0$ . Conjecture O. The number $T$ is an eigenvalue of $c_1(X)\star_0$ with multiplicity one, and every eigenvalue $u$ with $|u|=T$ has the form $u=\zeta T$ for some $\zeta$ satisfying $\zeta^\rho=1$ . This is one of the spectral conditions underlying Gamma conjecture I; the source presents it as a conjecture of Galkin, Golyshev, and Iritani, and no resolution is supplied here.

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Conjecture O for Fano manifolds
Let $X$ be a Fano manifold. Let $E=c_1(X)$ and let $T$ denote the maximal norm of the eigenvalues of quantum multiplication by $E$ at $\tau=0$ . Conjecture O. The number $T$ is a simple eigenvalue of $c_1(X)\star_0$ , meaning that its multiplicity in the characteristic polynomial is one. The conjecture is a spectral condition on quantum multiplication; the source notes that it follows from the Perron–Frobenius theorem in suitable cases and was proved for homogeneous spaces $G/P$ .
source: Hiroshi Iritani, “Gamma classes and quantum cohomology”, arXiv:2307.15938 (2023).
Conjecture O for Fano manifolds
Let $X$ be a Fano manifold. Its quantum cohomology defines a linear operator $\hat c_1$ on the even cohomology, and Property $\mathcal O$ requires that the spectral radius of $\hat c_1$ is a simple eigenvalue and that every eigenvalue of maximal modulus differs from it by the prescribed roots of unity determined by the Fano index. Conjecture O. Every Fano manifold satisfies Property $\mathcal O$ . This is the spectral assumption underlying Gamma conjecture I; it is stated here as a conjecture, with no resolution supplied in the source.
source: Sergey Galkin, Jianxun Hu, Hiroshi Iritani, Huazhong Ke, Changzheng Li and Zhitong Su, “Revisiting Gamma conjecture I: counterexamples and modifications”, arXiv:2405.16979 (2025).

Sources & referencesView supporting material

Primary source

Zongrui Yang, “Gamma conjecture I for blowing up P^n along P^r”, arXiv:2202.04234 (2022).

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Conjecture O for Fano manifolds

Equivalent formulations 2