[Paper Review] The Albanese functor commutes with the Hodge realization

Vadim Vologodsky|arXiv (Cornell University)|Sep 17, 2008

Algebraic structures and combinatorial models21 references4 citations

TL;DR

This paper proves that the Albanese functor and its associated embedding into Voevodsky's category of effective motives commute with the Hodge realization functor. This result provides a new proof of the rational form of Deligne’s conjecture on 1-motives by establishing a compatibility between motivic and Hodge-theoretic realizations via the derived category of 1-motives up to isogeny.

ABSTRACT

Abstract. We prove that the embedding of the derived category of 1-motives up to isogeny into the triangulated category of effective Voevodsky motives, as well as its left adjoint functor LAlbQ, commute with the Hodge realization. This result yields a new proof of the rational form of Deligne’s conjecture on 1-motives. 1.

Motivation & Objective

• To establish the compatibility of the Albanese functor with the Hodge realization in the context of motives.
• To investigate the behavior of the embedding of 1-motives up to isogeny into Voevodsky’s triangulated category of effective motives.
• To provide a new proof of the rational form of Deligne’s conjecture on 1-motives using realization functors.
• To clarify the relationship between motivic and Hodge-theoretic invariants in the derived category of 1-motives.

Proposed method

• Utilizes the derived category of 1-motives up to isogeny as a central framework.
• Applies the embedding functor into Voevodsky’s category of effective Voevodsky motives.
• Constructs the left adjoint functor LAlbQ to the embedding, ensuring compatibility with realization functors.
• Employs the Hodge realization functor to relate motivic structures to Hodge structures.
• Demonstrates that the composition of the Albanese functor with Hodge realization factors through the motivic realization.
• Relies on triangulated category techniques and adjunction properties in derived categories.

Experimental results

Research questions

• RQ1 Does the Albanese functor commute with the Hodge realization functor in the derived category of 1-motives up to isogeny?
• RQ2 How does the embedding of 1-motives into Voevodsky’s category of effective motives interact with Hodge-theoretic realizations?
• RQ3 Can the rational form of Deligne’s conjecture on 1-motives be re-proven using realization functors and motivic categories?
• RQ4 What is the precise relationship between the left adjoint LAlbQ and the Hodge realization in this context?
• RQ5 Is the Hodge realization compatible with the triangulated structure of the category of effective motives?

Key findings

• The embedding of the derived category of 1-motives up to isogeny into Voevodsky’s category of effective motives commutes with the Hodge realization functor.
• The left adjoint functor LAlbQ to this embedding also commutes with the Hodge realization.
• The compatibility of these functors with Hodge realization provides a new proof of the rational form of Deligne’s conjecture on 1-motives.
• The result establishes a deep compatibility between motivic and Hodge-theoretic invariants in the derived setting.
• The proof relies on structural properties of triangulated categories and adjunctions in the context of motives.
• The findings confirm that Hodge realization respects the motivic structure of 1-motives at the level of isogeny categories.