On the codegree Turán density of projective geometries

Let $F$ be a $k$-uniform hypergraph, abbreviated as $k$-graph. The codegree Tur\'an density $\gamma(F)$ is the supremum over all $\gamma \in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $\gamma n$ edges. Let $PG_m(q)$ be the projective geometry of dimension $m$ over finite field $\mathbb{F}_q$. In this paper, we prove that $\gamma(PG_m(q)) \ge \frac{1}{p}> 0$ for all $m$ and $q$, where $p$ is the smallest prime divisor of $q+1$. This resolves an open problem proposed by Keevash and Zhao (JCT-B, 2007). Moreover, we determine the exact codegree Tur\'an density of $PG_4(q)$ when $q$ is an odd prime power.