Gamma conjecture I for blowing up ${\mathbb{P}}^n$ along ${\mathbb{P}}^r$

Abstract

Consider the Fano manifold $X$ formed by blowing up ${\mathbb{P}}^n$ along its linear subspace ${\mathbb{P}}^r$, we check the conifold conditions [3, 1] for its mirror Laurent polynomial $f$, which can imply that $X$ satisfies both Conjecture $\mathcal{O}$ and Gamma conjecture I by Galkin-Golyshev-Iritani [2].