Stringy Hodge numbers are introduced by Batyrev for a mathematical formulation of mirror symmetry. However, since the stringy Hodge numbers of an algebraic variety are defined by choosing a resolution of singularities, the well-definedness is not clear from the definition. Batyrev proved the well-definedness by using the theory of motivic integration developed by Kontsevich, Denef-Loeser. The a...

Recall first that a weight k Hodge structure (L,L) has coniveau c ≤ k2 if the Hodge decomposition of LC takes the form LC = Lk−c,c ⊕ Lk−c−1,c+1 ⊕ . . .⊕ Lc,k−c with Lk−c,c 6= 0. If X is a smooth complex projective variety and Y ⊂ X is a closed algebraic subset of codimension c, then Ker (H(X,Q) → H(X \ Y,Q)) is a sub-Hodge structure of coniveau ≥ c of H(X,Q) (cf. [32, Theorem 7]). The generaliz...

In despair, as Deligne (2000) put it, of proving the Hodge and Tate conjectures, we can try to find substitutes. For abelian varieties in characteristic zero, Deligne (1982) constructed a theory of Hodge classes having many of the properties that the algebraic classes would have if the Hodge conjecture were known. In this article I investigate whether there exists a theory of " rational Tate cl...

In this paper, we define a certain Hodge-theoretic structure for an arbitrary variety X over the complex number field by using the theory of mixed Hodge module due to Morihiko Saito. We call it an arithmetic Hodge structure of X. It is shown that extension groups of arithmetic Hodge structure do not vanish even for degree ≥ 2. Moreover, we define higher Abel-Jacobi maps from Bloch’s higher Chow...

0. Introduction 229 Par t I. Summary of main results 231 1. The geometric situation giving rise to variation of Hodge structure. . . . 231 2. Data given by the variation of Hodge structure 232 3. Theorems about monodromy of homology 235 4. Theorems about Picard-Fuchs equations (Gauss-Manin connex ion) . . . . 237 5. Global theorems about holomorphic and locally constant cohomology classes 242 6...

The rational cohomology groups of complex algebraic varieties possess Hodge structures which can be used to refine their usual topological nature. Besides being useful parameters for geometric classification, these structures reflect in essential ways the conjectural category of motives, and hence, are of great interest from the viewpoint of arithmetic. A key link here is provided by the Hodge ...

R. Thomas (with a remark of B. Totaro) proved that the Hodge conjecture is essentially equivalent to the existence of a hyperplane section, called a generalized Thomas hyperplane section, such that the restriction to it of a given primitive Hodge class does not vanish. We study the relations between the vanishing cycles in the cohomology of a general fiber, and show that each relation between t...