The Weight Filtration on the Constant Sheaf on a Parameterized Space

On an $n$-dimensional locally reduced complex analytic space $X$ on which the shifted constant sheaf ${\text{Q}}_X^{\bullet }[n]$ is perverse, it is well-known that, locally, ${\text{Q}}_X^{\bullet }[n]$ underlies a mixed Hodge module of weight $\le n$ on $X$, with weight $n$ graded piece isomorphic to the intersection cohomology complex ${\text{Idot}}_X$ with constant $\text{Q}$ coefficients. In this paper, we identify the weight $n-1$ graded piece ${\text{Gr}}_{n-1}^W{\text{Q}}_X^{\bullet }[n]$ in the case where $X$ is a “parameterized space”, using the comparison complex, a perverse sheaf naturally defined on any space for which the shifted constant sheaf ${\text{Q}}_X^{\bullet }[n]$ is perverse. In the case where $X$ is a parameterized surface, we can completely determine the remaining terms in the weight filtration on ${\text{Q}}_X^{\bullet }[2]$, where we also show that the weight filtration is a local topological invariant of $X$. These examples arise naturally as affine toric surfaces in ${\text{C}}^3$, images of finitely-determined maps from ${\text{C}}^2$ to ${\text{C}}^3$, as well as in a well-known conjecture of L^{e} D~{u}ng Tr'{a}ng regarding the equisingularity of parameterized surfaces in ${\text{C}}^3$.