Variation of Mixed Hodge Structure and Primitive elements

We study the asymptotic behaviour of polarization form in the variation of mixed Hodge structure associated to isolated hypersurface singularities. The contribution characterizes a modification of Grothendieck residue as the polarization on the extended fiber in this case. We also provide a discussion on primitive elements to explain conjugation operator in these variations, already existed in the literature.Comment: This article is a brief of my other article On the mixed Hodge structure associated to isolated hypersurface singularities. Its removal is for its content already existed in the aforementioned paper, with more detail

Mixed Hodge Structure on Theta divisor I

We explain the mixed Hodge structure (MHS) on the complement of the generalized theta divisor of a curve in its generalized jacobians. Our approach considers the generalized jacobian via degeneration of smooth jacobians as a determinantal varieties. We make the question if the MHS can be explained by graphs or their degenerations

Toric Structure on Mumford-Tate domains and Characteristic cohomology

We explain a higher structure on Kato-Usui compactification of Mumford-Tate domains as toric stacks. As a motivation the universal characteristic cohomology of Hodge domains can be described as cohomology of stacks which have better behaviour in general.Comment: arXiv admin note: text overlap with arXiv:1107.1906 by other author

Higher residue pairing on Crystalline local systems

We explain a generalization of the K. Saito higher residue pairing for local system of $p$-adic isocrystals

On the mixed Hodge structure associated to hypersurface singularities

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

A Hodge index for Grothendieck residue pairing

In this text we apply the methods of Hodge theory for isolated hypersurface singularities to define a signature for the Grothendieck residue pairing of these singularities

Serre Intersection Multiplicity Conjecture and Hodge theory

We explain intersection multiplicity defined by J. P. Serre, in terms of the Poincare product in Hodge theory by a modification of the chern character map. We also discuss a formulation of the Euler characteristic via the action of correspondences on the Chow groups of projective varieties, assuming the Grothendieck Standard conjectures over $Q_l$

Positivity of Hochster Theta

M. Hochster defines an invariant namely $\Theta(M,N)$ associated to two finitely generated module over a hyper-surface ring $R=P/f$, where $P=k\{x_0,...,x_n\}$ or $k[X_0,...,x_n]$, for $k$ a field and $f$ is a germ of holomorphic function or a polynomial, having isolated singularity at $0$. This invariant can be lifted to the Grothendieck group $G_0(R)_{\mathbb{Q}}$ and is compatible with the chern character and cycle class map, according to the works of W. Moore, G. Piepmeyer, S. Spiroff, M. Walker. They prove that it is semi-definite when $f$ is a homogeneous polynomial, using Hodge theory on Projective varieties. It is a conjecture that the same holds for general isolated singularity $f$. We give a proof of this conjecture using Hodge theory of isolated hyper-surface singularities when $k=\mathbb{C}$. We apply this result to give a positivity criteria for intersection multiplicty of proper intersections in the variety of $f$.Comment: arXiv admin note: text overlap with arXiv:1103.5574 by other author

On Serre Intersection Multiplicity Conjecture

In this short note, we expose some of the works on Serre intersection multiplicity conjecture. I provide a proof of the vanishing of Serre intersection multiplicity in non-proper intersection over a regular ring based on the intersection theory in W. Fulton

de Rham Cohomology of Period Domains

This is a review article discussing the de Rham cohomology of period domains of Hodge structures. We explain it as the de Rham cohomology of differentiable stacks as of a moduli space. We also discuss the cohomology of the partial toroidal compactification of these domains using known formulas on cohomology or Chow rings of toric structures. The text is expository and we have tried to connect some existing ideas that probably their relations not processed in the literature of Hodge theory. We state the significance of ideas as they naturally could be related, probably with not serious mathematical proof. The proofs stated in the text maybe expressed in a more serious context