2018/11/11 by Brian Hepler, Hepler, Brian
Mathematics · #32S25 #32S35 #32S60 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1811.04328
openalex publication_date 2018/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
On an n-dimensional locally reduced complex analytic space X on which the shifted constant sheaf \QX^\bullet[n] is perverse, it is well-known that, locally, \QX^\bullet[n] underlies a mixed Hodge module of weight ≤ n on X, with weight n graded piece isomorphic to the intersection cohomology complex \IdotX with constant \Q coefficients. In this paper, we identify the weight n-1 graded piece \Grn-1W \QX^\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 \QX^\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 \QX^\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 \C3, images of finitely-determined maps from \C2 to \C3, as well as in a well-known conjecture of Lê Dũng Tráng regarding the equisingularity of parameterized surfaces in \C3.