[Paper Review] Deligne-Beilinson Cohomology of Affine Groups

Richard Hain|arXiv (Cornell University)|Jul 11, 2015

Algebraic Geometry and Number Theory15 references3 citations

TL;DR

This paper develops Deligne–Beilinson cohomology for affine group schemes endowed with a mixed Hodge structure, establishing an isomorphism between the Deligne–Beilinson cohomology of the relative completion of a fundamental group and the Ext groups in the category of admissible variations of mixed Hodge structures on a quasi-projective variety. The key contribution is a natural isomorphism compatible with products, extending earlier results in the unipotent case and providing a cohomological framework for periods of iterated Eisenstein integrals.

ABSTRACT

The goal of this paper is to develop the theory of Deligne-Beilinson cohomology of affine groups with a mixed Hodge structure. The motivation comes from Hodge theory and the study of motives, where such groups appear. Several of Francis Brown's period computations (arXiv:1407.5167) are interpreted as elements of the DB cohomology of the relative unipotent completion of $SL_2(Z)$ and their cup products. The results in this paper are used in arXiv:1403.6443 where they are used to prove that Pollack's quadratic relations are motivic.

Motivation & Objective

• To extend the theory of Deligne–Beilinson cohomology to affine group schemes with mixed Hodge structures, generalizing prior results in the unipotent case.
• To interpret Brown’s computation of periods of iterated Eisenstein integrals in terms of Deligne–Beilinson cohomology.
• To establish a natural isomorphism between Deligne–Beilinson cohomology of the relative completion of a fundamental group and Ext groups in the category of admissible variations of mixed Hodge structures.
• To prove that the natural map from the cohomology of the relative completion to the cohomology of the base variety is a morphism of mixed Hodge structures.
• To extend the theory to orbifolds by showing that the cohomological results for smooth varieties lift to orbifold quotients via equivariant constructions.

Proposed method

• Define Deligne–Beilinson cohomology of an affine group G with a mixed Hodge structure as the Ext group in the category of Hodge representations: $ H_{\mathcal{D}}^{\bullet}(G,V) := \operatorname{Ext}^{\bullet}_{{\sf HRep}(G)}(\mathbb{F},V) $.
• Use the natural exact sequence involving Ext and global sections: $ 0\to\operatorname{Ext}^{1}_{\mathsf{MHS}}(\mathbb{F},H^{m-1}(G,V))\to H^{m}_{\mathcal{D}}(G,V)\to\Gamma H^{m}(G,V)\to 0 $.
• Construct the mixed Hodge structure on the coordinate ring of the relative completion $ \mathcal{G}_x $ of the fundamental group using the theory of mixed Hodge complexes.
• Establish an equivalence between the category of admissible variations of mixed Hodge structures on a quasi-projective variety $ X $ and the category of Hodge representations of $ \mathcal{G}_x $, the relative completion of $ \pi_1(X,x) $.
• Lift the comparison map $ H^\bullet(\mathcal{G}_x,V_x) \to H^\bullet(X,\mathbb{V}) $ to a morphism of Deligne–Beilinson cohomology: $ \theta: H^\bullet_{\mathcal{D}}(\mathcal{G}_x,V_x) \to H^\bullet_{\mathcal{D}}(X,\mathbb{V}) $.
• Prove that this map $ \theta $ is an isomorphism in degrees 0 and 1, and injective in degree 2, using a resolution via a simply connected cover $ W $ with free $ \Gamma $-action.

Experimental results

Research questions

• RQ1 How can Deligne–Beilinson cohomology be generalized to affine group schemes with mixed Hodge structures?
• RQ2 What is the relationship between the Deligne–Beilinson cohomology of the relative completion of a fundamental group and the cohomology of variations of mixed Hodge structures on a base variety?
• RQ3 Is the natural comparison map from the cohomology of the relative completion to the cohomology of the base variety compatible with mixed Hodge structures?
• RQ4 Can the isomorphism between Deligne–Beilinson cohomology and Ext groups in the category of admissible variations be extended to orbifolds?
• RQ5 How do the cup product structures in Deligne–Beilinson cohomology relate to Brown’s computation of periods of iterated Eisenstein integrals?

Key findings

• The Deligne–Beilinson cohomology of an affine group $ G $ with a mixed Hodge structure is isomorphic to the Ext group $ \operatorname{Ext}^{\bullet}_{{\sf HRep}(G)}(\mathbb{F},V) $, where $ \mathbb{F} $ denotes the trivial Hodge structure $ \mathbb{F}(0) $.
• There is a natural isomorphism $ H^\bullet_{\mathcal{D}}(\mathcal{G}_x,V) \cong \operatorname{Ext}^\bullet_{{\mathsf{MHS}}(X,\mathbb{H})}(\mathbb{F},{\mathbb{V}}) $, compatible with all natural products, linking the cohomology of the relative completion to variations of mixed Hodge structures.
• The natural map $ H^\bullet(\mathcal{G}_x,V_x) \to H^\bullet(X,\mathbb{V}) $ is a morphism of mixed Hodge structures, with the cohomology on the right endowed with Saito’s mixed Hodge structure.
• The comparison map $ \theta: H^\bullet_{\mathcal{D}}(\mathcal{G}_x,V_x) \to H^\bullet_{\mathcal{D}}(X,\mathbb{V}) $ is an isomorphism in degrees 0 and 1 and injective in degree 2.
• The theory extends to orbifolds: for $ X = Y//\Gamma $, the category of admissible variations of MHS on $ X $ is equivalent to the category of $ \Gamma $-invariant variations on $ Y $, and the cohomological results lift via a smooth $ \Gamma $-cover $ W $ with trivial fundamental group.
• The orbifold version of the main theorem holds: the Deligne–Beilinson cohomology of the relative completion of $ \pi_1(X,x) $ is isomorphic to the Ext groups in the category of admissible variations on $ X $, compatible with products.