Simple obstructions and cone reduction

Let $X$ be a Deligne-Mumford stack locally of finite type over an algebraically closed field $k$ of characteristic zero. We show that the intrinsic normal cone $C_X$ of $X$ is supported in the subcone $\mathbb{V}(\Omega_X[-1])$ ($h^1/h^0((\Omega^1_X)^\vee)$) of its intrinsic normal sheaf $N_X$. This leads to an alternative proof of cone reduction by cosections for $C_X$. We also discuss vanishing of simple obstructions under the Buchweitz-Flenner semiregularity map for sheaves.