Gamma conjecture I and the underlying Conjecture $\mathcal{O}$ for Fano manifolds were proposed by Galkin, Golyshev and Iritani recently. We show that both conjectures hold for all two-dimensional Fano manifolds. We prove Conjecture $\mathcal{O}$ by deriving a generalized Perron-Frobenius theorem on eigenvalues of real matrices and a vanishing result of certain Gromov-Witten invariants for del Pezzo surfaces. We prove Gamma conjecture I by applying mirror techniques proposed by Galkin-Iritani together with the study of Gamma conjecture I for weighted projective spaces. We also provide applications of our generalized Perron-Frobenius theorem on Conjecture $\mathcal{O}$ for two Fano manifolds of higher dimensions.

Gamma 猜想 I 和基础猜想 $\mathcal{哦}$最近，Galkin、Golyshev 和 Iritani 提出了 Fano 流形。我们证明这两个猜想都适用于所有二维 Fano 流形。我们证明猜想$\mathcal{哦}$通过推导出关于实矩阵的特征值的广义 Perron-Frobenius 定理和 del Pezzo 曲面的某些 Gromov-Witten 不变量的消失结果。我们通过应用 Galkin-Iritani 提出的镜像技术以及对加权投影空间的 Gamma 猜想 I 的研究来证明 Gamma 猜想 I。我们还提供了我们的广义 Perron-Frobenius 定理在猜想上的应用$\mathcal{哦}$ 对于两个更高维的 Fano 流形。