involution acting on Kaehler sheaf

Question

Let $f \circlearrowright X$ act on a smooth projective variety over $\mathbf{C}$ such that $f^2=id$. As this defines a $\mathbf{Z}/2\mathbf{Z}$ action on $X$,lets say there is a closed fixed locus $X^f \subset X$ (assumed to be smooth). My question: Can I say that for the sheaf of differentials $\Omega_X$, the restriction $\Omega_X|_{X^f}\cong\Omega_{X^f}\oplus N$ splits into the $\pm 1$ eigenspaces where $df$ acts trivially (+1) and as $-1$ respectively (which should be the normal bundle N)? Simply because any involution on a locally free thing should split it into $\pm 1$ eigenspaces? Thanks!