Let $\mathfrak{g}$ be a complex reductive Lie algebra with Cartan algebra $\mathfrak{t}$. Hotta and Kashiwara defined a holonomic $\mathcal{D}$-module $\mathcal{M}$, on $\mathfrak{g}\times \mathfrak{t}$, called the Harish-Chandra module. We relate $gr\mathcal{M}$, an associated graded module with respect to a canonical Hodge filtration on $\mathcal{M}$, to the isospectral commuting variety, a subvariety of $\mathfrak{g}\times \mathfrak{g}\times \mathfrak{t}\times \mathfrak{t}$ which is a ramified cover of the variety of pairs of commuting elements of $\mathfrak{g}$. Our main result establishes an isomorphism of $gr\mathcal{M}$ with the structure sheaf of ${\mathfrak{X}}_{norm}$, the normalization of the isospectral commuting variety. We deduce, using Saito’s theory of Hodge $\mathcal{D}$-modules, that the scheme ${\mathfrak{X}}_{norm}$ is Cohen–Macaulay and Gorenstein. This confirms a conjecture of M. Haiman.

Associated with any principal nilpotent pair in $\mathfrak{g}$ there is a finite subscheme of ${\mathfrak{X}}_{norm}$. The corresponding coordinate ring is a bigraded finite-dimensional Gorenstein algebra that affords the regular representation of the Weyl group. The socle of that algebra is a $1$-dimensional space generated by a remarkable $W$-harmonic polynomial on $\mathfrak{t}\times \mathfrak{t}$. In the special case where $\mathfrak{g}=\mathfrak{g}{\mathfrak{l}}_n$ the above algebras are closely related to the $n!$-theorem of Haiman, and our $W$-harmonic polynomial reduces to the Garsia–Haiman polynomial. Furthermore, in the $\mathfrak{g}{\mathfrak{l}}_n$-case, the sheaf $gr\mathcal{M}$ gives rise to a vector bundle on the Hilbert scheme of $n$ points in ${\mathbb{C}}^2$ that turns out to be isomorphic to the Procesi bundle. Our results were used by I. Gordon to obtain a new proof of positivity of the Kostka–Macdonald polynomials established earlier by Haiman.

Soit $X$ une variété propre et lisse sur un corps de nombres $k$. Des conjectures sur l’image du groupe de Chow des zéro-cycles de $X$ dans le produit des mêmes groupes sur tous les complétés de $k$ ont été proposées par Colliot-Thélène, Kato et Saito. Nous démontrons ces conjectures pour l’espace total de fibrations en variétés rationnellement connexes vérifiant l’approximation faible, au-dessus de courbes dont le groupe de Tate–Shafarevich est fini, sous une hypothèse d’abélianité sur les fibres singulières.

Let $X$ be a smooth and proper variety over a number field $k$. Conjectures on the image of the Chow group of zero-cycles of $X$ in the product of the corresponding groups over all completions of $k$ were put forward by Colliot-Thélène, Kato and Saito. We prove these conjectures for the total space of fibrations, over curves with finite Tate–Shafarevich group, into rationally connected varieties which satisfy weak approximation, under an abelianness assumption on the singular fibers.

André used Hodge-theoretic methods to show that in a smooth proper family $\mathcal{X}\to B$ of varieties over an algebraically closed field $k$ of characteristic zero, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if $k$ is countable. We give a completely different approach to André’s theorem, which also proves the following refinement: in a family of varieties with good reduction at $p$, the locus on the base where the Picard number jumps is $p$-adically nowhere dense. Our proof uses the “$p$-adic Lefschetz $(1,1)$-theorem” of Berthelot and Ogus, combined with an analysis of $p$-adic power series. We prove analogous statements for cycles of higher codimension, assuming a $p$-adic analogue of the variational Hodge conjecture, and prove that this analogue implies the usual variational Hodge conjecture. Applications are given to abelian schemes and to proper families of projective varieties.

We prove that any reduced Noetherian quasi-excellent scheme of characteristic zero admits a strong desingularization which is functorial with respect to all regular morphisms. As a main application, we deduce that any reduced formal variety of characteristic zero admits a strong functorial desingularization. Also, we show that as an easy formal consequence of our main result one obtains strong functorial desingularization for many other spaces of characteristic zero including quasi-excellent stacks, formal schemes, and complex or nonarchimedean analytic spaces. Moreover, these functors easily generalize to noncompact settings by use of generalized convergent blow-up sequences with regular centers.