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...

Absolute Hodge classes first appear in Deligne’s proof of the Weil conjectures for K3 surfaces in [14] and are explicitly introduced in [16]. The notion of absolute Hodge classes in the singular cohomology of a smooth projective variety stands between that of Hodge classes and classes of algebraic cycles. While it is not known whether absolute Hodge classes are algebraic, their definition is bo...

The matroid is called loopless if the empty subset of E is closed, and is called a combinatorial geometry if in addition all single element subsets of E are closed. A closed subset of E is called a flat of M, and every subset of E has a well-defined rank and corank in the poset of all flats of M. The notion of matroid played a fundamental role in graph theory, coding theory, combinatorial optim...

Let X be an R-fold, and let π : E −→ X be a real vector bundle, of rank r, equipped with a positive definite symmetric bilinear form. If e1, . . . , er ∈ π −1(X) are orthonormal, then e1 ∧ · · · ∧ er is a non-trivial vector in ∧r E. Proposition: If f1, . . . , fr is any other orthonormal basis for π −1(X), then e1 ∧ · · · ∧ er = ±f1 ∧ · · · ∧ fr. Proof. Note that fi = g · ei for g ∈ O(r), so de...

In the framework of an extended BRST formalism, it is shown that the four (3 + 1)-dimensional (4D) free Abelian 2-form (notoph) gauge theory presents an example of a tractable field theoretical model for the Hodge theory.

Here we survey questions and results on the Hodge theory of hyperkähler quotients, motivated by certain S-duality considerations in string theory. The problems include L2 harmonic forms, Betti numbers and mixed Hodge structures on the moduli spaces of Yang-Mills instantons on ALE gravitational instantons, magnetic monopoles on R3 and Higgs bundles on a Riemann surface. Several of these spaces a...

Introduction §1. Statement of the Main Results §2. Technical Roots: the Work of Mumford and Zhang §3. Conceptual Roots: the Search for a Global Hodge Theory §3.1. From Absolute Differentiation to Comparison Isomorphisms §3.2. A Function-Theoretic Comparison Isomorphism §3.3. The Meaning of Nonlinearity §3.4. Hodge Theory at Finite Resolution §3.5. Relationship to Ordinary Frobenius Liftings and...

We describe an alternate construction of some of the basic rings introduced by Fontaine in p-adic Hodge theory. In our construction, the central role is played by the ring of p-typical Witt vectors over a p-adic valuation ring, rather than theWitt vectors over a ring of positive characteristic. This suggests the possibility of forming a meaningful global analogue of p-adic Hodge theory.

In this note we clarify some subtle points on the limit mixed Hodge structures and on the spectrum. These are more or less well-known to the specialists, but do not seem to be stated explicitly in the literature. However, as they do not seem to be obvious to the beginners, we consider them to be worth writing down explicitly. The general constructions are exemplified by considering the isolated...

It is well-known (see eg [22]) that the topology of a compact Kähler manifold X is strongly restricted by Hodge theory. In fact, Hodge theory provides two sets of data on the cohomology of a compact Kähler manifold. The first data are the Hodge decompositions on the cohomology spaces H(X,C) (see (1.1) where V = H(X,Q)); they depend only on the complex structure. The second data, known as the Le...