Nori Motives of Curves With Modulus and Laumon 1-motives

Abstract

Abstract. Let k be a number eld. We describe the category of Laumon -isomotives over k as the universal category in the sense of Nori associated with a quiver representation built out of smooth proper k-curves with two disjoint e ective divisors and a notion of H dR for such "curves with modulus".is result extends and relies on the theorem of J. Ayoub and L. Barbieri-Viale that describes Deligne's category of -isomotives in terms of Nori's Abelian category of motives.

Mixed Hodge Structures With Modulus

“…where G(X, Y ) add denotes the additive part of G(X, Y ). It follows H 1 add ∼ = L 1 add by [14,Lemma 24]. In particular we get L 1 ∼ = H 1 .…”

“…] has been used in [14]. When k = d and Z = ∅, Ω (d)• X|Y,∅ agrees with the complex S • Y used in [16].…”

“…In [14,Definition 25], we considered a Laumon 1-motive LM(X, Y, Z). Explicitely, this is given as follows (see [14, §5.2]).…”

Modulus triples

“…Hence this category is too big to describe MCrv. (3) In [6], Ivorra and Yamazaki use MCrv to perform a construction à la Nori, which they identify with Laumon's 1-motives when k is a number field. (4) The category MHSM carries a perfect duality, but no tensor structure.…”

Motives with modulus, III: The categories of motives

“…This suggests that one should construct an even larger category based on "modulus triples" (two Cartier divisors at infinity with opposite modulus conditions). Work in this direction has been made by Binda [5] in the context of -homotopy theory (as above), and by Ivorra-Yamazaki [14,15] in the additive context. • Of course, develop the various theories over a base.…”