On blended extensions in filtered tannakian categories and mixed motives with maximal unipotent radicals

Payman Eskandari
Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), K-Theory and Homology (math.KT), Number Theory (math.NT), Representation Theory (math.RT)
2023-07-27 16:00:00
Let $(T, W_\cdot)$ be a filtered tannakian category over a field of characteristic zero, e.g. the category of rational mixed Hodge structures or any reasonable tannakian category of mixed motives over a field of characteristic zero. Given an object $A$ that is a direct sum of pure objects of $T$, the first part of this paper concerns two classification problems, namely of (1) equivalence classes of pairs $(X,\phi)$ of objects $X$ of $T$ and isomorphisms $\phi: Gr^WX\rightarrow A$, and (2) isomorphism classes of objects $X$ of $T$ such that $Gr^WX$ is isomorphic to $A$ (with no specific choice of isomorphism made). Building on Grothendieck's concept of a blended extension (extension panach\'ee), we introduce a novel inductive approach (induction on the "level") to study the structure of the classifying set in each problem. The main theorems in this part are two results on the structure of these classifying sets. As a guiding problem, we also consider a classification problem related to blended extensions (the classification of the "middle object" up to isomorphism). In the second part of the paper, we apply these ideas to study mixed motives with maximal unipotent radicals. Building on some ideas of an earlier joint work with K. Murty, we define a notion of graded-independence and give a simple necessary and sufficient condition for a graded-independent motive to have a maximal unipotent radical. Combining this with the earlier results of the paper, we obtain a result on the structure of the set of isomorphism classes of motives with associated graded isomorphic to a given graded-independent semisimple motive $A$ and maximal unipotent radicals. The special case of the main theorem here for 3 weights was proved earlier with Murty under some extra hypotheses. As an example at the end, we apply our classification result to 4-dimensional mixed Tate motives over $Q$ with 4 weights.
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