7 - A simple construction of regulator indecomposable higher Chow cycles in elliptic surfaces

Summary

Introduction

Let CH^j(X, i) be Bloch's higher Chow groups of a projective smooth variety X over ℂ. A higher Chow cycle z ∈ CH^j(X, i) is called indecomposable if it does not belong to the image of the map of the product

Of particular interest to us is CH²(X,1). For A = ℚ or ℝ, we say z ∈ CH²(X,1) A-regulator indecomposable if the regulator class reg(z) ∈ H_D³(X,A(2)) in the Deligne-Beilinson cohomology group with coefficients in A does not belong to the image of H_D¹ (X,ℤ(1))⨂H_D² (X,A(1)) ≅ = ℂ^x⊗CH¹(X)⊗A. In other words, z is A-regulator indecomposable if and only if

Obviously ℝ-reg. indecomp. ⇒ ℚ-reg. indecomp. ⇒ indecomposable.

Quite a lot of examples of ℚ or ℝ-regulator indecomposable cycles are obtained by many people ([1], [3], [4], [5], [6] and more).

In this note we construct R-regulator indecomposable cycles for X an elliptic surface which satisfies certain conditions. The main theorem is the following.

Theorem 1.1Let S be a smooth irreducible curve over ℂ. Let

be an elliptic fibration over S with a section s. This means that g and h are projective smooth morphisms of relative dimension 2 and 1 respectively, and the general fiber of f is an elliptic curve. For a point t ∈ S we denote X_t =g⁻¹(t) or C_t= h⁻¹ (t) the fibers over t. Assume that the following conditions hold.

1. (1) Let η be the generic point of S. Then there is a split multiplicative fiber Dη = f⁻¹(P) ⊂ Xη of Kodaira type I_n, n ≥ 1 (see [8] VII, §5 for the terminology of “split multiplicative fiber”).

2. (2) Let D ⊂ X be the closure of Dη. Then there is a closed point 0 ∈ S(C) such that the specialization is multiplicative of type Im with m > n.

Then the composition

is non-zero for a general t ∈ S(ℂ). Here NF(X_t)⊂NS(X_t) denotes the subgroup generated by components of singular fibers and the section s(C_t).