Related papers: The Hodge theory of algebraic maps

We develop a model for the cohomology of the complement of a hypersurface arrangement inside a smooth projective complex variety. This generalizes the case of normal crossing divisors, discovered by P. Deligne in the context of the mixed…

Building on the nonabelian Hodge theory in positive characteristic developed by Ogus, Vologodsky, and Schepler, we propose a generalization of the decomposition theorem of Deligne and Illusie from the perspective of mixed Hodge modules.…

Motivated by classical Alexander invariants of affine hypersurface complements, we endow certain finite dimensional quotients of the homology of abelian covers of complex algebraic varieties with a canonical and functorial mixed Hodge…

Let $f:X-->Y$ be a map of algebraic varieties. Barthel, Brasselet, Fieseler, Gabber and Kaup have shown that there exists a homomorphism of intersection homology groups $f^*:IH^*(Y)-->IH^*(X)$ compatible with the induced homomorphism on…

This paper has two aims. The former is to give an introduction to our earlier work on the Hodge theory of algebraic maps and more generally to some of the main themes of the theory of perverse sheaves and to some of its geometric…

We prove the decomposition theorem for Hodge modules with integral structure along proper K\"ahler morphisms, partially generalizing M. Saito's theorem for projective morphisms. Our proof relies on compactifications of period maps of…

We show that the description of Deligne--Beilinson cohomology is improved by using log Hodge theory. We consider the log relative version of it, and also present a fundamental conjecture in log Hodge theory.

Suppose that Y is a cyclic cover of projective space branched over a hyperplane arrangement D, and that U is the complement of the ramification locus in Y. The first theorem implies that the Beilinson-Hodge conjecture holds for U if certain…

We introduce higher analytic geometry, a novel framework extending Lurie's derived complex analytic spaces. This theory generalizes classical complex analytic geometry, enabling the study of derived K\"ahler spaces with non-trivial higher…

Over the past few years, it is gradually understood that de Rham Cohomology Theory is closely related to Saint-Venant's compatibility condition in the Elasticity Theory. In this article, we will discuss the Hodge Theory and de Rham…

I will discuss recent progress by many people in the program of extending natural topological invariants from manifolds to singular spaces. Intersection homology theory and mixed Hodge theory are model examples of such invariants. The past…

In this short note we show that the homotopy category of smooth compactifications of smooth algebraic varieties is equivalent to the homotopy category of smooth varieties over a field of characteristic zero. As an application we show that…

A new proof of the decomposition theorem is established using a relation with a version of the local purity theorem of Deligne and Gabber adapted to complex algebraic varieties.

We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of C_p. It takes values in a mixed-characteristic analogue of Dieudonne modules, which was previously defined by Fargues as a version of…

We use the Decomposition Theorem to derive several generalizations of the Clemens-Schmid sequence, relating asymptotic Hodge theory of a degeneration to the mixed Hodge theory of its singular fiber(s).

We introduce a Hopf algebroid associated to a proper Lie group action on a smooth manifold. We prove that the cyclic cohomology of this Hopf algebroid is equal to the de Rham cohomology of invariant differential forms. When the action is…

In this article, we will explore the fundamental concepts, including various basic concepts on $d$-complex manifolds, along with several differential operators and examine the relationships between them. A $d$-K\"ahler manifold is a…

In this paper, we construct a differential graded Lie algebra whose Maurer-Cartan elements are given by crossed homomorphisms on Leibniz algebras. This allows us to define cohomology for a crossed homomorphism. Finally, we study linear…

In this paper, we establish an innovative framework in logarithmic Hodge theory for toroidal varieties, introducing weighted toroidal structures and developing a systematic obstruction theory for Hodge classes. Building upon recent advances…

We establish a positive characteristic analogue of intersection cohomology theory for variations of Hodge structure. It includes: a) the de Rham-Higgs comparison theorem for the intersection de Rham complex; b) the $E_1$-degeneration…