Content uploaded by A. Collino

Author content

All content in this area was uploaded by A. Collino on Dec 10, 2014

Content may be subject to copyright.

A preview of the PDF is not available

ArticlePDF Available

We construct some natural indecomposable elements of \(CH^g(J(C),1)\) with trivial regulator, and in particular, prove that
\(\)
is uncountable for C a generic curve or a generic hyperelliptic curve of genus \(g \geq 3\).

Content uploaded by A. Collino

Author content

All content in this area was uploaded by A. Collino on Dec 10, 2014

Content may be subject to copyright.

A preview of the PDF is not available

... This does not contradicts Voisin's conjecture [42] on the countability of CH 2 ind (X, 1) Q , because CH 1 ind (X 2 , 1) Q = 0. It is known that CH p+1 ind (X, 1) Q is uncountable for certain varieties X, see [13]. The assumption on the nonvanishing of the image of ζ 2 by (4.1.2) is satisfied for some examples, see [14], [35]. ...

... It is known that CH p+1 ind (X, 1) Q is uncountable in some cases, see e.g. [13] for the case of the Jacobian of a curve, and also [14], [35] for examples such that the hypothesis of (0.3) is satisfied. (We can show that a higher cycle on a product of elliptic curves in [22] is decomposable by restricting to the generic fiber of the first projection.) ...

We introduce a class of cycles, called nondegenerate, strictly decomposable cycles, and show that the image of each cycle in this class under the refined cycle map to an extension group in the derived category of arithmetic mixed Hodge structures does not vanish. This class contains certain cycles in the kernel of the Abel-Jacobi map. The construction gives a refinement of Nori's argument in the case of a self-product of a curve. As an application, we show that a higher cycle which is not annihilated by the reduced higher Abel-Jacobi map produces uncountably many indecomposable higher cycles on the product with a variety having a nonzero global 1-form.

A general specialization map is constructed for higher Chow groups and used to prove a "going-up" theorem for algebraic cycles and their regulators. The results are applied to study the degeneration of the modified diagonal cycle of Gross and Schoen, and of the coordinate symbol on a genus-2 curve.

Notes for a mini course at the University of Science and Technology of China in Hefei, China, June 23–July 12, 2014.

We first give an elementary new proof of the vanishing of the regulator on K-1(Z) where Z subset of P-3 be a general surface of degree d greater than or equal to 5, using a Lefschetz pencil argument. By a similar argument we then show the triviality of the regulator for K-1 of a general product of two curves.

For a polynomial f (x )i n (Zp ∩ Q)(x) of degree d ≥ 3 let L( f ⊗ Fp; T) be the L function of the exponential sum of f mod p. Let NP ( f ⊗ Fp) denote the Newton polygon of L( f ⊗ Fp; T). Let HP (Ad) denote the Hodge polygon of Ad, which is the lower convex hull in R2 of the points (n, n(n+1) 2d ) for 0 ≤ n ≤ d − 1. Let A d be the space of degree-d monic polynomials parameterized by their coefficients. Let GNP (Ad; Fp) := inf f ∈Ad (Fp) NP (f ) be the lowest Newton polygon over Fp if exists. We prove that for p large enough GNP (A d ; Fp) exists and we give an explicit formula for it. We also prove that there is a Zariski dense open subset U defined over Q in A d such that for f ∈U (Q) and for p large enough we have NP ( f ⊗ Fp) = GNP (A d ; Fp); furthermore, as p goes to infinity their limit exists and is equal to HP (A d ). Finally we prove analogous results for the space of polynomials f (x )= x d + ax with one parameter. In particular, for any nonzero a ∈ Q we show that limp→∞ NP ((x d + ax) ⊗ Fp) = HP (A d ).

We explicitly describe cycle-class maps c_H from motivic cohomology to absolute Hodge cohomology, for smooth quasi-projective and (some) proper singular varieties, and compute special cases of the latter. For smooth projective varieties, we also study Hodge-theoretically defined "higher Abel-Jacobi maps" on the kernel of c_H; this leads to new results on nontrivial indecomposable higher Chow cycles in the regulator kernel. Comment: 68 pages, 3 figures

In this note we give an example of an indecomposable higher Chow cycle on a special family of quartics in ރ. The example is obtained as an extension of a cycle in the higher Chow group CH2(K,1) of a singular Kummer surface.

We express the kernel of Griffiths' Abel-Jacobi map by using the inductive limit of Deligne cohomology in the generalized sense (i.e. the absolute Hodge cohomology of A. Beilinson). This generalizes a result of L. Barbieri-Viale and V. Srinivas in the surface case. We then show that the Abel-Jacobi map for codimension 2 cycles and the Albanese map are bijective if a general hyperplane section is a surface for which Bloch's conjecture is proved. In certain cases we verify Nori's conjecture on the Griffiths group. We also prove a weak Lefschetz-type theorem for (higher) Chow groups, generalize a formula for the Abel-Jacobi map of higher cycles due to Beilinson and Levine to the smooth non proper case, and give a sufficient condition for the nonvanishing of the transcendental part of the image by the Abel-Jacobi map of a higher cycle on an elliptic surface, together with some examples.

We construct a map between Bloch's higher Chow groups and Deligne homology for smooth, complex quasiprojective varieties on the level of complexes. For complex projective varieties this results in a formula which generalizes at the same time the classical Griffiths Abel-Jacobi map and the Borel/Beilinson/Goncharov regulator type maps.

Let X be a projective algebraic manifold, and CH k(X, 1) the higher Chow group. We introduce the subgroup of decomposable cycles and the quotient space of indecomposable cycles CH kind(X, 1; ℚ). Firstly, for X a general product of two elliptic curves we construct a nonzero indecomposable higher Chow cycle in CH 2ind(X, 1; ℚ). Then by combining ideas similar to those in that construction with previous results of the second author we show that when X is a general product of three elliptic curves then CH 3ind(X, 1; ℚ) is uncountable. As a consequence, the kernel of the regulator map modulo decomposables can be very large. This result is analogous to Mumford's famous theorem on the kernel of the Albanese map on the Chow group of zero-cycles on a surface of positive genus.

Received 9 September 1994; accepted in final form 2 May 1995 We formulate a conjecture about the Chow groups of generic Abelian varieties and prove it in a few cases.

Let X be a curve over a field k with a rational point e. We define Δ e ∈Z 2 (X 3 ) hom , a canonical cycle. Suppose that k is a number field and that X has semi-stable reduction over the integers of k with fiber components non-singular. We construct a regular model of X 3 and show that the height pairing 〈τ * (Δ e ),τ * ' (Δ e )〉 is well defined where τ and τ ' are correspondences. The paper ends with a brief discussion of heights and L-functions in the case that X is a modular curve.

C'est pour étendre le théorème de Riemann-Roch à un morphisme projectif arbitraire que Grothendieck a introduit le groupe K(X) (noté aujourd'hui K 0 ( X )), construit à l'aide des -modules localement libres sur un schéma X [ 14 ]. La somme directe et le produit tensoriel de modules font de K 0 ( X ) un anneau, et les opérations de puissances extérieures lui fournissent une structure supplémentaire, que Grothendieck appelle λ-anneau. Un λ-anneau est muni d'une filtration décroissante, la γ-filtration, et un des principaux résultats de Grothendieck est que, si X est lisse sur un corps, le groupe
est isomorphe, à la torsion près, au groupe de Chow CH ¹ ( X ) des cycles de codimension i sur X , modulo l'équivalence linéaire.