Hodge-theoretic Invariants for Algebraic Cycles
نویسندگان
چکیده
In this paper we use Hodge theory to define a filtration on the Chow groups of a smooth, projective algebraic variety. Assuming the generalized Hodge conjecture and a conjecture of Bloch-Beilinson, we show that this filtration terminates at the codimension of the algebraic cycle class, thus providing a complete set of period-type invariants for a rational equivalence class of algebraic cycles. Outline (1) Introduction (2) Spreads; explanation of the idea (3) Construction of the filtration on CH(X)Q (4) Interpretations and proofs (5) Remarks and examples (6) Appendix: Reformation of the construction
منابع مشابه
Algebraic cycles, Hodge theory, and Arithmetic
The antique origins of Algebraic Geometry lie in the study of solution sets of polynomial equations, in which complex, symplectic, and arithmetic geometry are bound tightly together. Many of the most spectacular recent developments in the subject have occurred through the consideration of these aspects in tandem: for example, the duality between symplectic and complex geometry that is mirror sy...
متن کاملar X iv : 0 70 5 . 46 61 v 1 [ m at h . A G ] 3 1 M ay 2 00 7 ALGEBRAIC CYCLES AND MUMFORD - GRIFFITHS INVARIANTS
Let X be a projective algebraic manifold and let CH(X) be the Chow group of algebraic cycles of codimension r on X, modulo rational equivalence. Working with a candidate Bloch-Beilinson filtration {F }ν≥0 on CH(X)⊗Q due to the second author, we construct a space of arithmetic Hodge theoretic invariants ∇J(X) and corresponding map φ X : Gr F CH(X)⊗Q → ∇J(X), and determine conditions on X for whi...
متن کاملMotives and Algebraic De Rham Cohomology
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...
متن کاملHodge Genera of Algebraic Varieties , I
The aim of this paper is to study the behavior of Hodge-theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We obtain formulae that relate (parametrized families of) global invariants of a complex algebraic variety X to such invariants of singularities of proper algebraic maps defined on X . Such formulae se...
متن کاملCharacteristics of Algebraic Varieties Sylvain
The aim of this note is to study the behavior of intersection homology Euler characteristic under morphisms of algebraic varieties. The main result is a direct application of the BBDG decomposition theorem. Similar formulae for Hodge-theoretic invariants of algebraic varieties were announced by the first and third authors in [4, 11].
متن کامل