On the mixed Hodge structure associated to hypersurface singularities

Abstract

Let $f:{\mathbb{C}}^{n+1}\to \mathbb{C}$ be a germ of hypersurface with isolated singularity. One can associate to $f$ a polarized variation of mixed Hodge structure $\mathcal{H}$ 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 ${\Omega }_f$. A mixed Hodge structure can be defined on ${\Omega }_f$ in this way. The polarization on $\mathcal{H}$ 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.

References in corpus (7)

• Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence
• Fourier-Laplace transform of a variation of polarized complex Hodge structure
• Primitive forms via polyvector fields
• Vanishing cycles and Hermitian duality
• Gauss-Manin systems, Brieskorn lattices and Frobenius structures (II)
• On the structure of Brieskorn lattices, II
• Multiplier ideals, b-function, and spectrum of a hypersurface singularity

Cited by in corpus (1)

• Variation of Mixed Hodge Structure and Primitive elements