arXiv subjects

Johann Bouali

Publications and source records attributed to Johann Bouali.

9 recordsLinked to original sources

arXiv2026

De Rham logarithmic classes and Tate conjecture

We introduce the definition of De Rham logarithmic classes. We show that the De Rham class of an algebraic cycle of a smooth algebraic variety over a field of characteristic zero is logarithmic and conversely that a logarithmic class of bidegree $(d,d)$ is the De Rham class of an algebraic cycle (of codimension $d$). We also give for smooth algebraic varieties over a $p$-adic field an analytic version of this result. We deduce from the analytic case the Tate conjecture for smooth projective varieties over fields of finite type over $\mathbb Q$.

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arXiv2025

Hodge conjecture for projective hypersurface

We show that a Hodge class of a complex smooth projective hypersurface is an analytic logarithmic De Rham class. On the other hand we show that for a complex smooth projective variety an analytic logarithmic De Rham class of of type $(d,d)$ is the class of codimension $d$ algebraic cycle. We deduce the Hodge conjecture for smooth projective hypersurfaces.

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arXiv2024

Degeneration of families of projective hypersurfaces and Hodge conjecture

We prove by induction on dimension the Hodge conjecture for smooth complex projective varieties. Let $X$ be a smooth complex projective variety. Then $X$ is birational to a possibly singular projective hypersurface, hence to a smooth projective variety $E_0$ which is a component of a normal crossing divisor $E=\cup_{i=0}^rE_i\subset Y$ which is the singular fiber of a pencil $f:Y\to\mathbb A^1$ of smooth projective hypersurfaces. Using the smooth hypersurface case by a previous result of the autor, the nearby cycle functor on mixed Hodge module with rational de Rham factor, and the induction hypothesis, we prove that an Hodge class of $E_0$ is absolute Hodge, more precisely the locus of Hodge classes inside the algebraic vector bundle given the De Rham cohomology the rational deformation of $E_0$ is defined over $\mathbb Q$. By another previous result of the autor, we get the Hodge conjecture for $E_0$. By the induction hypothesis we also have the Hodge conjecture for $X$.

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arXiv2023

Complex vs etale Abel Jacobi map and algebraicity of the zero locus of etale normal functions

We prove, using $p$-adic Hodge theory for open algebraic varieties, that for a smooth projective variety over a subfield $k\subset\mathbb C$ which is of finite type over $\mathbb Q$, the complex abel jacobi map vanishes if the etale abel jacobi map vanishes. This implies that for a smooth projective morphism $f:X\to S$ of smooth complex algebraic varieties over $k\subset\mathbb C$ which is of finite type over $\mathbb Q$ and $Z\in\mathcal Z^d(X,n)^{f,\partial=0}$ an algebraic cycle flat over $S$ whose cohomology class vanishes on fibers, the zero locus of the etale normal function associated to $Z$ is contained in the zero locus of the complex normal function associated to $Z$. From the work of Saito or Charles, we deduce that the zero locus of the complex normal function associated to $Z$ is defined over the algebraic closure $\bar k$ of $k$ if the zero locus of the etale normal function associated to $Z$ is not empty. We also prove an algebraicity result for the zero locus of an etale normal function associated to an algebraic cycle over a field of finite type over $\mathbb Q$. By the way, for a smooth morphism $f:X\to S$ of smooth algebraic varieties over a field of finite type over $\mathbb Q$, we embed the locus of Hodge-Tate classes of $f$ inside the locus of Hodge classes of $f$.

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arXiv2022

The De Rham, complex Hodge and $p$-adic Hodge realization functors on the derived category of relative motives over a field of characteristic zero

We introduce the categories of geometric mixed Hodge modules on algebraic varieties over a subfield $k\subset\mathbb C$, and for a prime number $p$, the categories of geometric $p$-adic mixed Hodge modules on algebraic varieties over a subfield $k\subset K\subset\mathbb C_p$ of a p-adic field $K$. We then give an Hodge realization functor on the de derived category of relative motives over $k\subset\mathbb C$, and for a prime number $p$ a $p$-adic Hodge realization functor on the derived category of relative motives over $k\subset K\subset\mathbb C_p$.

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arXiv2022

The Hodge realization functor on the derived category of relative motives

We give, for a complex algebraic variety $S$, a Hodge realization functor $\mathcal F_S^{Hdg}$ from the derived category of constructible motives $DA_c(S)$ to the derived category $D(MHM(S))$ of algebraic mixed Hodge modules over $S$. Moreover, for $f:T\to S$ a morphism of complex quasi-projective algebraic varieties, $\mathcal F_{-}^{Hdg}$ commutes with the four operation $f^*$,$f_*$,$f_!$,$f^!$ on $DA_c(-)$ and $D(MHM(-))$, making the Hodge realization functor a morphism of 2-functor wich for a given $S$ sends $DA_c(S)$ to $D(MHM(S)$, moreover $\mathcal F_S^{Hdg}$ commutes with tensor product. We also give an algebraic and analytic Gauss-Manin realization functor from which we obtain a base change theorem for algebraic De Rham cohomology and for all smooth morphisms a realtive version of the comparaison theorem of Grothendieck between the algebaric De Rham cohomology and the analytic De Rham cohomology.

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arXiv2017

On the realization functor of the derived category of mixed motives

We give an alternative construction of the Betti realization functor on the derived category of motives of complex algebraic varieties via the category of CW complexes instead of the category of complex analytic spaces. In particular we show that the functor we define via the category of CW complexes coincide with Ayoub's one. We deduce from this construction that Ayoub's realization functor on geometric motives factors trough Nori motives and that the image of this functor on the morphisms between the motive of a point and a shift of a Tate twist of the motive with compact support of a complex algebraic variety coincide with the classical cycle class map on higher Chow groups.

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arXiv2016

Some results on the higher Abel jacobi map for open varieties

In this article, we study the infinitemisal invariant of the relative higher Abel Jacobi map of a smooth open morphism. We give a generalization of a theorem of Voisin to open varieties and higher Chow groups and as a corollary a non vanishing criterion for the higher Abel Jacobi map of a general open smooth hypersurface section of high degree of a smooth projective variety Y. On the other side, using Nori connectness theorem, the image of the primitive part of the higher Abel Jacobi map of a general open smooth hypersurface section of high degree of a smooth projective variety Y is generated by the image of the restriction of a primitive cycle on the corresponding affine subset of Y

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arXiv2013

Motives of quadric bundles

This article is about motives of quadric bundles. In the case of odd dimensional fibers and where the basis is of dimension two we give an explicit relative and absolute Chow-Künneth decomposition. This shows that the motive of the quadric bundle is isomorphic to the direct sum of the motive of the base and the Prym motive of a double cover of the discriminant. In particular this is a refinement with $\mathbb Q$ coefficients of a result of Beauville concerning the cohomology and the Chow groups of an odd dimensional quadric bundle over $\mathbb P^2$. This Chow-Künneth decomposition satisfies Murre's conjectures II and III. This article is a generalization of an article of Nagel and Saito on conic bundles \cite{NS}.

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