$(\mathbb{RP}^{2n-1},\xi_{std})$ is not exactly fillable for $n\ne 2^k$

Abstract

We prove that $(\mathbb{RP}^{2n-1},\xi_{std})$ is not exactly fillable for any $n\ne 2^k$ and there exist strongly fillable but not exactly fillable contact manifolds for all dimension $\ge 5$.