I am working on some basic of Gromov-Witten theory and stuck in understanding obstruction bundle. A perfect obstruction theory on a scheme or stack $M$ due to Behrend and Fantechi is the data $\phi:\mathcal{E}\rightarrow \tau_{\ge -1}L_M$ in $D^b(M)$ satisfying some conditions. Taking the first cohomology of the dual of the complex $\mathcal{E}$, we get so-called obscturction sheaf $Ob=h^1(\mathcal{E}^{\vee})$.

Assume $M=\overline{M}_{g,n}(X)$ for some smooth variety $X$ with the usual perfect obstruction theory on it. My question is, why the obstruction sheaf forms a vector bundle on $M$ when the moduli space $M$ is non-singular?

I know that the modul space $M$ is intuitively obtained by cutting out the deformation space by $\dim Ob$ many equations, but I am not really convinced by this argument.

I am working on some basic of Gromov-Witten theory and stuck in understanding obstruction bundle. Recall that a perfect obstruction theory on a scheme or stack $M$ due to Behrend and Fantechi is a moprhism $\phi:\mathcal{E}\rightarrow \tau_{\ge -1}L_M$ in $D^{[-1,0]}(M)$ satisfying some conditions. Taking the first cohomology of the dual of the two-term complex $\mathcal{E}$, we get so-called obscturction sheaf $Ob=h^1(\mathcal{E}^{\vee})$.

Assume now that $M=\overline{M}_{g,n}(X)$ for some smooth variety $X$ with the usual perfect obstruction theory on it. My question is, why the obstruction sheaf forms a vector bundle on $M$ when the moduli space $M$ is non-singular?

I know that the modul space $M$ is intuitively obtained by cutting out the deformation space by $\dim Ob$ many equations, but I am not really convinced by this argument.