A polarizable variation of Hodge structure over a smooth complex quasi projective variety $S$ is said to be defined over a number field $L$ if $S$ and the algebraic connection associated to the variation are both defined over $L$. Conjecturally any special subvariety (also called "an irreducible component of the Hodge locus) for such variations is defined over $\overline{\mathbb{Q}}$, and its Galois conjugates are also special subvarieties. We prove this conjecture for special subvarieties satisfying a simple monodromy condition. As a corollary we reduce the conjecture that special subvarieties for variation of Hodge structures defined over a number field are defined over $\overline{\mathbb{Q}}$ to the case of special points.
We prove the equidistribution of the Hodge locus for certain non-isotrivial, polarized variations of Hodge structure of weight $2$ with $h^{2,0}=1$ over complex, quasi-projective curves. Given some norm condition, we also give an asymptotic on the growth of the Hodge locus. In particular, this implies the equidistribution of elliptic fibrations in quasi-polarized, non-isotrivial families of $K3$ surfaces.
We present a conjecture on the geometry of the Hodge locus of a (graded polarizable, admissible) variation of mixed Hodge structure over a complex smooth quasi-projective base, generalizing to this context the Zilber-Pink Conjecture for mixed Shimura varieties (in particular the André-Oort conjecture).
This paper contains two results on Hodge loci in the moduli space of curves. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibres and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fibers is contained in a proper Hodge locus. The second result deals with divisors in the moduli space of curves. It is proved that the image of a divisor in the moduli of principally polariz…
We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex quasi-projective variety $S$, are topologically tame. As an easy corollary of these results and of Peterzil-Starchenko's o-minimal GAGA theorem we obtain that the Hodge locus of $(S, \mathbb{V})$ is a countable union of algebraic subvarieties of $S$ (a result originally due to Cattani-Deligne-Kaplan).
Let $\mathbb V$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$. In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, \emph{the pull-push form}. In particular, it is always analytically dense when the pull-push form does not vanish. When the weight is $2$, the Hodge numbers are $(q,p,q)$ and the dimension of $S$ is least $rq$, we prove that the typical locus where the Picard rank is at least $r$ is equidistribut…
We define two tau functions, $τ$ and $\hatτ$ , on moduli spaces of spectral covers of $GL(n)$ Hitchin's systems. Analyzing the properties of $τ$, we express the divisor class of the universal Hitchin's discriminant in terms of standard generators of the rational Picard group of the moduli spaces of spectral covers with variable base. The function $\hatτ$ is used to compute the divisor of canonical 1-forms with multiple zeros.
In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures. $2)$ We prove that the period map associated to any pure polarized variation of integral Hodge structures $\mathbb{V}$ on a smooth complex quasi-projective variety $S$ is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the…
We describe the Hodge theory of brilliant families of K3 surfaces. Their characteristic feature is a close link between the Hodge structures of any two fibres over points in the Noether-Lefschetz locus. Twistor deformations, the analytic Tate-Safarevic group, and one-dimensional Shimura special cycles are covered by the theory. In this setting, the Brauer group is viewed as the Noether-Lefschetz locus of the Brauer family or as the specialization of the Noether-Lefschetz loci in a family of approaching twistor spaces. Passing from one algebraic twistor fibre to another, which by construction i…
In previous work, we constructed for a smooth complex variety $X$ and for a linear algebraic group $G$ a mixed Hodge structure on the complete local ring $\widehat{\mathcal{O}}_ρ$ to the moduli space of representations of the fundamental group $π_1(X,x)$ into $G$ at a representation $ρ$ underlying a variation of mixed Hodge structure. We now show that the jump ideals $J_k^i \subset \widehat{\mathcal{O}}_ρ$, defining the locus of representations such the the dimension of the cohomology of $X$ in degree $i$ of the associated local system is greater than $k$, are sub-mixed Hodge structures; this …
The generalized roof F(1,2,n): Hodge structures and derived categories
Sapienza - Università di Roma（罗马大学）；Institute of Mathematics of the Polish Academy of Sciences（波兰科学院数学研究所）；Alma Mater Studiorum Università di Bologna（博洛尼亚大学）
The motivation for this paper is the study of arithmetic properties of Shimura varieties, in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure. This is closely related to the structure of Rapoport-Zink spaces and of affine Deligne-Lusztig varieties. We prove a Hodge-Newton decomposition for affine Deligne-Lusztig varieties and for the special fibres of Rapoport-Zink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (Hodge-Newton decomposability) which can b…
Let $G$ be a reductive group scheme over the $p$-adic integers, and let $μ$ be a minuscule cocharacter for $G$. In the Hodge-type case, we construct a functor from nilpotent $(G,μ)$-displays over $p$-nilpotent rings $R$ to formal $p$-divisible groups over $R$ equipped with crystalline Tate tensors. When $R/pR$ has a $p$-basis étale locally, we show that this defines an equivalence between the two categories. The definition of the functor relies on the construction of a $G$-crystal associated with any adjoint nilpotent $(G,μ)$-display, which extends the construction of the Dieudonné crystal ass…
The notion of a d-critical locus is an ingredient in the definition of motivic Donaldson-Thomas invariants by [BJM19]. There is a canonical d-critical locus structure on the Hilbert scheme of dimension zero subschemes on local toric Calabi-Yau 3-folds. This is obtained by truncating the $-1$-shifted symplectic structure on the derived moduli stack [BBBBJ15]. In this paper we show the canonical d-critical locus structure has critical charts consistent with the description of Hilbert scheme as a degeneracy locus [BBS13]. In particular, the canonical d-critical locus structure is isomorphic to th…
