arXiv subjects

Florian Ivorra

Publications and source records attributed to Florian Ivorra.

7 recordsLinked to original sources

arXiv2024

A Proof of the Integral Identity via Braden's Theorem

The purpose of this paper is to provide a very short proof of a generalized categorified version, within the motivic stable homotopy category of Morel and Voevodsky, of the integral identity for virtual motives conjectured by Kontsevich and Soibelman. Our proof is an application of an important result in geometric representation theory due to Braden and known as the hyperbolic localization/restriction theorem. Though originally proved in the context of etale sheaves (or sheaves on the associated complex analytic space in the case of complex algebraic varieties) Braden's theorem turns out to hold also in the context of motivic sheaves, at least in the special case of vector bundles with a linear G_m-action.

math.AG↗

arXiv2023

The four operations on perverse motives

Let $k$ be a field of characteristic zero with a fixed embedding $σ:k\hookrightarrow \mathbb{C}$ into the field of complex numbers. Given a $k$-variety $X$, we use the triangulated category of étale motives with rational coefficients on $X$ to construct an abelian category $\mathscr{M}(X)$ of perverse mixed motives. We show that over $\mathrm{Spec}(k)$ the category obtained is canonically equivalent to the usual category of Nori motives and that the derived categories $\mathrm{D}^{\mathrm{b}}(\mathscr{M}(X))$ are equipped with the four operations of Grothendieck (for morphisms of quasi-projective $k$-varieties) as well as nearby and vanishing cycles functors and a formalism of weights. In particular, as an application, we show that many classical constructions done with perverse sheaves, such as intersection cohomology groups or Leray spectral sequences, are motivic and therefore compatible with Hodge theory. This recovers and strengthens work by Zucker, Saito, Arapura and de Cataldo-Migliorini and provide an arithmetic proof of the pureness of intersection cohomology with coefficients in a geometric variation of Hodge structures.

math.AG↗

arXiv2018

Mixed Hodge structures with modulus

We define a notion of mixed Hodge structure with modulus that generalizes the classical notion of mixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts introduced by Kato and Russell in their description of Albanese varieties with modulus. With modulus triples of any dimension we attach mixed Hodge structures with modulus. We combine this construction with an equivalence between the category of level one mixed Hodge structures with modulus and the category of Laumon $1$-motives to generalize Kato-Russell's Albanese varieties with modulus to $1$-motives.

math.AG↗

arXiv2017

Nori motives of curves with modulus and Laumon 1-motives

Let $k$ be a number field. We describe the category of Laumon 1-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 effective divisors and a notion of $H^1_\dR$ for such "curves with modulus". This result extends and relies on the theorem of J. Ayoub and L. Barbieri-Viale that describes Deligne's category of 1-isomotives in terms of Nori's Abelian category of motives.

math.AG↗

arXiv2015

K-groups of reciprocity functors

In this work we introduce reciprocity functors, construct the associated K-group of a family of reciprocity functors, which itself is a reciprocity functor, and compute it in several different cases. It may be seen as a first attempt to get close to the notion of reciprocity sheaves imagined by B. Kahn. Commutative algebraic groups, homotopy invariant Nisnevich sheaves with transfers, cycle modules or Kähler differentials are examples of reciprocity functors. As commutative algebraic groups do, reciprocity functors are equipped with symbols and satisfy a reciprocity law for curves.

math.AG↗

arXiv2009

Cycle modules and the intersection A-infinity algebra

Given a cycle module M with a ring structure we show that the cycle complex with coefficients in M of a smooth scheme of finite type over a field has a A-infinity algebra structure. In the case of Milnor K-theory this gives a homotopy model for the classical intersection theory of algebraic cycles.

math.AG↗

arXiv2007

Levine's motivic comparison theorem revisited

For a field of characteristic zero, M. Levine has proved that his category of triangulated motives is equivalent to the one constructed by V. Voevodsky. In this paper we show that the strategy of Levine's proof can be applied on every perfect field to the categories of triangulated motives with rational coefficients.

math.AG↗