The Weight Filtration on the Constant Sheaf on a Parameterized Space

Abstract

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