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 Qunderlies a mixed Hodge module of weight ≤ n on X, with weight n graded piece isomorphic to the intersection cohomology complex IC • X with constant Q coefficients. In this paper, we identify the weight n − 1 graded piece Grin 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 [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, 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 Lê Dũng Tráng regarding the equisingularity of parameterized surfaces in C 3 .

Deformation Formulas for Parameterized Hypersurfaces

“…suggests a sort of "conserved quantity" between the sum of the Lê numbers of f t and the characteristic polar multiplicities of N • V (ft) in one parameter deformations of parameterized hypersurfaces. It is a very interesting question to see how this relates to results in [17] regarding the structure of N • V (f ) as a mixed Hodge module, and the isomorphism N…”

“…Since the category of perverse sheaves is Abelian, this morphism has a kernel, which we define to be N • X . This perverse sheaf, called the comparison complex on X, was first defined by the author and Massey in [18] (where we originally referred to it as the multiple-point complex), and subsequently studied by the author in [16], [17] and Massey in [29]. N • X will play a crucial role in this paper as the cohomological generalization of the function m(p) = |π −1 (p)| − 1 above.…”