Conjecture O for Fano manifolds

About 4 years old · traced to 2022

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 2Other wordings

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).

References

Primary source

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

Posted 1 month ago

Conjecture O for Fano manifolds