On the mixed Hodge structure associated to hypersurface singularities

2014/03/05 by Rahmati, Mohammad Reza
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1403.1553

Abstract

Let f:ℂⁿ⁺¹ → ℂ be a germ of hypersurface with isolated singularity. One can associate to f a polarized variation of mixed Hodge structure H over the punctured disc, where the Hodge filtration is the limit Hodge filtration of W. Schmid and J. Steenbrink. By the work of M. Saito and P. Deligne the VMHS associated to cohomologies of the fibers of f can be extended over the degenerate point 0 of disc. The new fiber obtained in this way is isomorphic to the module of relative differentials of f denoted Ω_f. A mixed Hodge structure can be defined on Ω_f in this way. The polarization on H deforms to Grothendieck residue pairing modified by a varying sign on the Hodge graded pieces in this process. This also proves the existence of a Riemann-Hodge bilinear relation for Grothendieck pairing and allow to calculate the Hodge signature of Grothendieck pairing.

Related

• Variation of Mixed Hodge Structure and Primitive elements
• Singularities of variations of mixed Hodge structure
• Real Seifert forms and polarizing forms of Steenbrink mixed Hodge\n structures
• Degenerations of Mixed Hodge structure
• Vanishing of Local Cohomology with Applications to Hodge Theory