[Paper Review] Boundary motive, relative motives and extensions of motives

Jörg Wildeshaus|arXiv (Cornell University)|Apr 9, 2010

Algebraic Geometry and Number Theory18 references3 citations

TL;DR

This paper establishes the boundary motive as a central tool for constructing and resolving extensions of Chow motives via localization, particularly in the context of weight structures and relative motives. It proves a canonical correspondence between weight filtrations on the boundary motive and factorizations of the motive morphism $ M(X) \to M^c(X) $, enabling functorial constructions of interior motives and Hecke-equivariant structures on cohomology of modular curves.

ABSTRACT

We explain the role of the boundary motive in the construction of certain Chow motives, and of extensions of Chow motives. Our two main examples concern proper, singular surfaces and fibre products of a universal elliptic curve.

Motivation & Objective

• To clarify the role of the boundary motive in constructing and resolving extensions of Chow motives using localization.
• To establish a canonical correspondence between weight filtrations on the boundary motive and factorizations of the morphism $ M(X) \to M^c(X) $.
• To provide a functorial construction of the interior motive via idempotent endomorphisms and weight-avoiding conditions.
• To realize the Hecke action on cohomology of modular curves without compactifications, using boundary motive structures.
• To generalize the theory of motives for singular surfaces and fiber products of universal elliptic curves via weight structures.

Proposed method

• Uses the canonical exact triangle $ \partial M(X) \to M(X) \to M^c(X) \to \partial M(X)[1] $ in $ DM^{\mathrm{eff}}_{\mathrm{gm}}(k) $ to relate boundary motives to motives with compact support.
• Applies weight structures on triangulated motives to define weight filtrations on $ \partial M(X) $, especially for smooth compactifications.
• Constructs a bijective correspondence between isomorphism classes of weight filtrations on $ \partial M(X) $ and effective Chow motives $ M_0 $ through which $ M(X) \to M^c(X) $ factors.
• Introduces rigidification via motives avoiding weights $-1$ and $0$, enabling functoriality of the interior motive construction.
• Utilizes idempotent endomorphisms of the boundary triangle to decompose motives and define $ e $-parts of the interior motive.
• Constructs Hecke-equivariant structures on cohomology via morphisms induced by isogenies between universal elliptic curves and their products.

Experimental results

Research questions

• RQ1 How does the boundary motive control the construction of extensions of Chow motives via localization?
• RQ2 What is the precise relationship between weight filtrations on the boundary motive and factorizations of the morphism $ M(X) \to M^c(X) $?
• RQ3 Can the interior motive be canonically and functorially reconstructed from the boundary motive using weight-avoiding conditions?
• RQ4 How can the Hecke action on cohomology of modular curves be realized without compactifications, using boundary motives?
• RQ5 What is the role of relative motives and weight structures in resolving extensions of motives?

Key findings

• There is a canonical bijective correspondence between isomorphism classes of weight filtrations on $ \partial M(X) $ and effective Chow motives $ M_0 $ through which $ M(X) \to M^c(X) $ factors.
• For smooth compactifications, any smooth compactification induces a weight filtration on the Hodge realization of $ \partial M(X) $, generalizing to motivic weight structures.
• When a direct factor $ \partial M(X)^e $ avoids weights $-1$ and $0$, the associated $ M_0 $ is canonically and functorially defined as the $ e $-part of the interior motive.
• The Hecke algebra acts on the cohomology $ {}^r_n\mathcal{W} $ via morphisms induced by isogenies between base-changed universal elliptic curves, realized through the boundary motive triangle.
• The construction of $ \varphi(g_1, g_2) $ extends uniquely to the entire diagram of exact triangles, yielding a Hecke-equivariant structure on $ {}^r_n\mathcal{W} $.
• The boundary motive provides a motivic framework for resolving extensions, with applications to intersection motives of surfaces and products of elliptic curves.