Do Hodge $*$-operators glue?

Question

For ${\Bbb P}_{\Bbb C}^2 = \underset{i = 0, 1, 2}{\bigcup} {\Bbb A}_{i, \Bbb C}^2$, we have Hodge $*$-operators on each affine open ${\Bbb A}_{i,{\Bbb C}}^2$. For example $z_1 = X_0/X_2, z_2 = X_1/X_2$ on ${\Bbb A}_{0,\Bbb C}^2$. For example, $* \colon dz_1 \mapsto dz_2$. Do such definitions in terms of local corrdinates glue over global manifold? I am awkward that it could collapse over the intersection, say ${\Bbb A}_{0, {\Bbb C}} \, \cap\, {\Bbb A}_{1, {\Bbb C}}$.