arXiv subjects

Mattia Cavicchi

Publications and source records attributed to Mattia Cavicchi.

6 recordsLinked to original sources

arXiv2025

Bloch-Beilinson conjectures for Hecke characters and Eisenstein cohomology of Picard surfaces

We consider certain families of Hecke characters $ϕ$ over a quadratic imaginary field $F$. According to the Bloch-Beilinson conjectures, the order of vanishing of the $L$-function $L(ϕ,s)$ at the central point $s=-1$ should be equal to the dimension of the space of extensions of the Tate motive $\mathbb{Q}(1)$ by the motive associated with $ϕ$. In this article, we construct candidates for the corresponding extensions of Hodge structures, assuming that the sign of the functional equation of $L(ϕ,s)$ is $-1$. This is accomplished through the cohomology of variations of Hodge structures over Picard modular surfaces associated with $F$ and Harder's theory of Eisenstein cohomology. Furthermore, we demonstrate that these extensions are naturally realized within certain biextensions. We outline a program to compute the biextension height and utilize it to establish the non-triviality of these extensions.

math.NT↗

arXiv2025

Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations

We show that the hyper-Kähler varieties of OG10-type constructed by Laza-Saccà-Voisin (LSV) verify the Lefschetz standard conjecture. This is an application of a more general result, stating that certain Lagrangian fibrations verify this conjecture. The main technical assumption of this general result is that the Lagrangian fibration satisfies the hypotheses of Ngô's support theorem. Verifying that the LSV tenfolds do satisfy those hypotheses is of independent interest. Another point of independent interest of the paper is the definition and the study of the Lefschetz standard conjecture in the relative setting, and its relation to the classical absolute case.

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arXiv2025

The motivic Hecke algebra for PEL Shimura varieties

We construct a motivic lift of the action of the Hecke algebra on the cohomology of PEL Shimura varieties $S_K$. To do so, when $S_K$ is associated with a reductive algebraic group $G$ and $V$ is a local system on $S_K$ coming from a $G$-representation, we define a motivic Hecke algebra $\mathcal{H}^M(G,K)$ as a natural sub-algebra of the endomorphism algebra, in the triangulated category of motives, of the constructible motive associated with $S_K$ and $V$. The algebra $\mathcal{H}^M(G,K)$ is such that realizations induce an epimorphism from it onto the classical Hecke algebra. We then consider Wildeshaus' theory of interior motives, along with the necessary hypotheses for it to be employed. Whenever those assumptions hold, one gets a Chow motive realizing to interior $V$-valued cohomology of $S_K$, equipped with an action of $\mathcal{H}^M(G,K)$ as an algebra of correspondences modulo rational equivalence. We give a list of known cases where this applies.

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arXiv2024

Relative Lie algebra cohomology of SU(2,1) and Eisenstein classes on Picard surfaces

We consider Picard surfaces, locally symmetric varieties $S_Γ$ attached to the Lie group SU(2,1), and we construct explicit differential forms on $S_Γ$ representing Eisenstein classes, i.e. cohomology classes restricting non-trivially to the boundary of the Borel-Serre compactification. This is needed for the computation of the class of the extensions of the Hodge structure that we have constructed in [2] according to the predictions of the Bloch-Beilinson conjectures. The tool for the construction of the differential forms is an analysis of relative Lie algebra cohomology of the principal series of SU(2,1) using recent methods of Buttcane and Miller.

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arXiv2022

Motivic decompositions of families with Tate fibers: smooth and singular cases

We apply Wildeshaus's theory of motivic intermediate extensions to the motivic decomposition conjecture, formulated by Deninger-Murre and Corti-Hanamura. We first obtain a general motivic decomposition for the Chow motive of an arbitrary smooth projective family $f:X \rightarrow S$ whose geometric fibers are Tate. Using Voevodsky's motives with rational coefficients, the formula is valid for an arbitrary regular base $S$, without assuming the existence of a base field or even of a prime integer $\ell$ invertible on $S$. This result, and some of Bondarko' ideas, lead us to a generalized formulation of Corti-Hanamura's conjecture. Secondly we establish the existence of the motivic decomposition when $f:X \rightarrow S$ is a projective quadric bundle over a characteristic $0$ base, which is either sufficiently general or whose discriminant locus is a normal crossing divisor. This provides a motivic lift of the Bernstein-Beilinson-Deligne decomposition in this setting.

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arXiv2019

On the boundary and intersection motives of genus 2 Hilbert-Siegel varieties

We study genus 2 Hilbert-Siegel varieties, i.e. Shimura varieties $S_K$ corresponding to the group $\mbox{GSp}_{4,F}$ over a totally real field $F$, along with the relative Chow motives $^λ\mathcal{V}$ of abelian type over $S_K$ obtained from irreducible representations $V_λ$ of $\mbox{GSp}_{4,F}$. We analyse the weight filtration on the degeneration of such motives at the boundary of the Baily-Borel compactification and we find a criterion on the highest weight $λ$ which characterises the absence of the middle weights 0 and 1 in the corresponding degeneration. Thanks to Wildeshaus' theory, the absence of these weights allows us to construct Hecke-equivariant Chow motives over $\mathbb{Q}$, whose realizations equal interior (or intersection) cohomology of $S_K$ with $V_λ$-coefficients. We give applications to the construction of motives associated to automorphic representations.

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