[Paper Review] Higher Abel-Jacobi Maps

Jishnu Biswas, Gautham Dayal+2 more|ArXiv.org|Oct 3, 2000

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper proposes an alternative to Green's Higher Abel-Jacobi map using absolute Deligne-Beilinson cohomology and Morihiko Saito's theory of Hodge modules to address Bloch's conjecture on the filtration of Chow groups. It constructs a cycle class map into a cohomology theory that unifies singular cohomology and Abel-Jacobi maps, showing that Bloch's conjecture follows from standard conjectures on motives when the variety is defined over a number field.

ABSTRACT

This paper forms the major portion of a talk given at the International Colloquium on Arithmetic, Algebra and Geometry at TIFR, Mumbai in Jan 2000. We look at the problem of detecting cycles with trivial Abel-Jacobi invariant. M. Green proposed a Hodge-theoretic method to which C. Voisin found a counter-example. We present an easier example. We also propose another possible invariant to detect these classes using Hodge Theory. Similar methods have been proposed earlier by M. Asakura and M. Saito.

Motivation & Objective

• To address Bloch's conjecture on the filtration of Chow groups for algebraic cycles on smooth projective varieties over subfields of ℂ.
• To overcome the limitations of Green's Higher Abel-Jacobi map, which fails to detect certain non-torsion elements in F²CH²(S) as shown by a counterexample.
• To construct a cohomological framework—based on Deligne-Beilinson cohomology and Hodge modules—that realizes the higher extension groups Ext^k(ℚ(−p), H^{2p−k}(V)) conjectured by Bloch and Beilinson.
• To establish a cycle class map into absolute Deligne-Beilinson cohomology that induces the correct filtration on Chow groups, compatible with Hodge-theoretic expectations.
• To show that Bloch’s conjecture on F²CHᵖ(V) = 0 for varieties over number fields follows from standard conjectures on motives, assuming the proposed cohomological framework is valid.

Proposed method

• Uses the framework of Hodge modules on varieties defined over ℚ̄ ⊂ ℂ to construct a mixed Hodge structure on relative de Rham cohomology R^i_dR(𝒱/S) for a family 𝒱 → S over a base S.
• Applies the spectral sequence Ext^b_MHM(S)(ℚ(c), R^a_dR(𝒱/S)) ⇒ Ext^{a+b}_MHM(𝒱)(ℚ(c), ℚ) to define Deligne-Beilinson cohomology via Ext groups in the category of mixed Hodge modules.
• Defines absolute Deligne-Beilinson cohomology H^n_ADb(V, ℚ(c)) as the direct limit of Ext^n_MHM(𝒱)(ℚ(c), ℚ) over all models 𝒱 → S of V over ℚ̄.
• Constructs a cycle class map cl^p_ADb: CH^p(V) → H^{2p}_ADb(V, ℚ(p)) that combines the singular cycle class map and the Abel-Jacobi map.
• Uses the spectral sequence to induce a filtration on CH^p(V) that is conjectured to match Bloch’s filtration F^kCH^p(V).
• Relies on the fact that the Ext^2 groups in MHM(S) are non-trivial when dim S ≥ 1, allowing the construction of higher Abel-Jacobi maps via higher extensions.

Experimental results

Research questions

• RQ1 Can a higher Abel-Jacobi map be constructed that realizes the higher extension groups Ext^k(ℚ(−p), H^{2p−k}(V)) in the category of mixed motives?
• RQ2 Does Green’s Higher Abel-Jacobi map correctly detect elements in F²CH²(S) for smooth projective surfaces S defined over ℂ?
• RQ3 Can Bloch’s conjecture on the vanishing of F²CH^p(V) for varieties defined over number fields be deduced from standard conjectures using a cohomological framework?
• RQ4 Is the filtration on CH^p(V) induced by absolute Deligne-Beilinson cohomology compatible with Bloch’s conjectural filtration?
• RQ5 Can the absolute Deligne-Beilinson cohomology H^n_ADb(V, ℚ(c)) be used to unify the cycle class map and the Abel-Jacobi map in a way that captures the full structure of the Chow group filtration?

Key findings

• Green’s Higher Abel-Jacobi map fails to detect non-torsion elements in F²CH²(S) for certain smooth projective surfaces S, as demonstrated by a counterexample.
• The proposed absolute Deligne-Beilinson cohomology H^{2p}_ADb(V, ℚ(p)) provides a cohomological framework that unifies the cycle class map to singular cohomology and the Abel-Jacobi map.
• The spectral sequence Ext^b_MHM(S)(ℚ(−p), R^a_dR(𝒱/S)) ⇒ Ext^{a+b}_MHM(𝒱)(ℚ(−p), ℚ) realizes the higher extension groups conjectured by Bloch and Beilinson.
• The cycle class map cl^p_ADb: CH^p(V) → H^{2p}_ADb(V, ℚ(p)) induces a filtration on CH^p(V) that is conjectured to coincide with Bloch’s filtration F^kCH^p(V).
• The conjecture that F²CH^p(V) = 0 for varieties V defined over a number field follows from standard conjectures on motives if the proposed cohomological framework is valid.
• The construction shows that elements in F²CH²(S) that are not detected by the intermediate Jacobian J²₂(S) can still be detected via the absolute Deligne-Beilinson cohomology, resolving a key limitation of classical approaches.