A mixed Hodge structure on a CR manifold

The mixed Hodge-Riemann bilinear relations for compact Kahler manifolds

We prove the Hodge-Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kahler manifolds in the mixed situation.

Some mixed Hodge structure on l^2-cohomology of covering of Kähler manifolds

We give methods to compute l^2-cohomology groups of a covering manifolds obtained by removing pullback of a (normal crossing) divisor to a covering of a compact Kähler manifold. We prove that in suitable quotient categories, these groups admit natural mixed Hodge structure whose graded pieces are given by expected Gysin maps.

The log-term of the disc bundle over a homogeneous Hodge manifold

We show the vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kaehler--Einstein manifold of classical type.

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 $Ω_f$. A mixed Hodge structure can be defined on $Ω_…

On the irregular Hodge filtration of exponentially twisted mixed Hodge modules

Given a mixed Hodge module and a meromorphic function f on a complex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module, which extends the construction of arXiv:1302.4537. We show the strictness of the push-forward filtered D-module through any projective morphism, by using the theory of mixed twistor D-modules of T. Mochizuki. We consider the example of the rescaling of a regular function f, which leads to an expression of the irregular Hodge filtration of the Laplace transform of the Gauss-Manin systems of f in ter…

Nonabelian Hodge theory for Fujiki class $\mathcal C$ manifolds

The nonabelian Hodge correspondence (Corlette-Simpson correspondence), between the polystable Higgs bundles with vanishing Chern classes on a compact Kähler manifold $X$ and the completely reducible flat connections on $X$, is extended to the Fujiki class $\mathcal C$ manifolds.

混合霍奇结构的变化及其应用

Variation of mixed Hodge structure and its applications

本文研究复解析空间射影态射诱导的混合霍奇结构变化，并将其用于推广 Kollár 无挠定理与消失定理，为复解析空间射影态射的极小模型理论奠定基础。

Lowest Weights in Cohomology of Variations of Hodge Structure

Let X be a smooth complex projective variety, let $j:U\into X$ an immersion of a Zariski open subset, and let V be a variation of Hodge structure of weight n over U. Then IH^k(X, j_*V) is known to carry a pure Hodge structure of weight k+n, while H^k(U,V) carries a mixed Hodge structure of weight $\ge k+n$. In this note it is shown that the image of the natural map $IH^k(X,j_*V) \to H^k(U,V)$ is the lowest weight part of this mixed Hodge structure. The proof uses Saito's theory of mixed Hodge modules.

Frölicher spectral sequence and Hodge structures on the cohomology of complex parallelisable manifolds

For complex parallelisable manifolds $Γ\backslash G$, with $G$ a solvable or semisimple complex Lie group, the Frölicher spectral sequence degenerates at the second page. In the solvable case, the de-Rham cohomology carries a pure Hodge structure. In contrast, in the semisimple case, purity depends on the lattice, but there is always a direct summand of the de Rham cohomology which does carry a pure Hodge structure and is independent of the lattice.

Categories of complex variations of Hodge structure over compact K"ahler manifolds

We give a complex polarized variation of Hodge structure over a compact K"ahler manifold $M$ which controls all finite-dimensional complex polarized variations of Hodge structure over $M$ and their tensor relations. As a corollary, we obtain the cohomology algebra with values in a local system admitting multiplicative Hodge structures.

Rescalability of integrable mixed twistor $D$-modules

We study the rescalability of integrable mixed twistor $D$-modules. We prove some basic functoriality of the rescalability and the associated irregular Hodge filtration. We also observe that rescalable integrable mixed twistor $D$-modules are equivalent to exponential Hodge modules.

Weights and t-structures: in general triangulated categories, for 1-motives, mixed motives, and for mixed Hodge complexes and modules

We study certain 'weights' for triangulated categories endowed with $t$-structures. Our results axiomatize and describe in detail the relations between the Chow weight structure (introduced in a preceding paper), the (conjectural) motivic t-structure for Voevodsky's motives, and the conjectural weight filtration for them. This picture becomes non-conjectural when restricted to the derived categories of 1-motives (over a smooth base) and of Artin-Tate motives over number fields; other examples include Beilinson's derived category of graded polarizable mixed Hodge complexes and Saito's derived c…