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

TL;DR

This paper verifies the Gamma conjecture I for a class of Fano manifolds obtained by blowing up projective space along a linear subspace, using mirror symmetry techniques to check conifold conditions.

Contribution

It demonstrates that the blown-up projective space satisfies Gamma conjecture I by analyzing its mirror Laurent polynomial and conifold conditions.

Findings

Gamma conjecture I holds for the blown-up projective space along a linear subspace

Mirror Laurent polynomial satisfies conifold conditions

Supports the conjecture that certain blow-ups meet Gamma conjecture I

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].