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Emmanuel Ullmo

Publications and source records attributed to Emmanuel Ullmo.

At least 19 recordsLinked to original sources
arXiv2025

Bi-$\overline{\mathbb{Q}}$-structures on Hermitian symmetric spaces and quadratic relations between CM periods

In this paper, we introduce the notion of a bi-$\overline{\mathbb{Q}}$-structure on the tangent space at a CM point on a locally Hermitian symmetric domain. We prove that this bi-$\overline{\mathbb{Q}}$-structure decomposes into the direct sum of $1$-dimensional bi-$\overline{\mathbb{Q}}$-subspaces, and make this decomposition explicit for the moduli space of abelian varieties $\mathbb{A}_g$. We propose an Analytic Subspace Conjecture, which is the analogue of the Wüstholz's Analytic Subgroup Theorem in this context. We show that this conjecture, applied to $\mathbb{A}_g$, implies that all quadratic $\overline{\mathbb{Q}}$-relations among the holomorphic periods of CM abelian varieties arise from elementary ones.

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arXiv2025

Hodge cycles and quadratic relations between holomorphic periods on CM abelian varieties

In this paper, we prove the following result advocating the importance of monomial quadratic relations between holomorphic CM periods. For any simple CM abelian variety $A$, we can construct a CM abelian variety $B$ such that all non-trivial Hodge relations between the holomorphic periods of the product $A\times B$ are generated by monomial quadratic ones which are also explicit. Moreover, $B$ splits over the Galois closure of the CM field associated with $A$.

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arXiv2024

Rich representations and superrigidity

We investigate and compare applications of the Zilber-Pink conjecture and dynamical methods to rigidity problems for arithmetic real and complex hyperbolic lattices. Along the way we obtain new general results about reconstructing a variation of Hodge structure from its typical Hodge locus that may be of independent interest. Applications to Siu's immersion problem are also discussed, the most general of which only requires the hypothesis that infinitely many closed geodesics map to proper totally geodesic subvarieties under the immersion.

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arXiv2024

Non-density of the exceptional components of the Noether-Lefschetz locus

We study when the Picard group of smooth surfaces of degree $d\geq 5$ in $\mathbb{P}^3$ acquires extra classes. In particular we show that the so called exceptional components of the Noether-Lefschetz locus are not Zariski dense. This answers a 1991 question of C. Voisin. We also obtain similar results for the Noether-Lefschetz locus for suitable $(Y,L)$, where $Y$ is a smooth projective threefold and $L$ a very ample line bundle. Both results are applications of the Zilber-Pink viewpoint recently developed by the authors for arbitrary (polarized, integral) variations of Hodge structures.

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arXiv2024

Manin-Mumford in arithmetic pencils

We obtain a refinement of Manin-Mumford (Raynaud's Theorem) for abelian schemes over some ring of integers. Torsion points are replaced by special 0-cycles, that is reductions modulo some, possibly varying, prime of Galois orbits of torsion points. There is a flat/horizontal part and a vertical one. The irreducible components of the flat part are given by the Zariski closure, over the integers, of torsion cosets of the generic fibre of the abelian scheme. The vertical components are given by translates of abelian subvarieties, which 'come from characteristic zero'.

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arXiv2023

On the distribution of the Hodge locus

Given a polarizable $\mathbb{Z}$-variation of Hodge structures $\mathbb{V}$ over a complex smooth quasi-projective base $S$, a classical result of Cattani, Deligne and Kaplan says that its Hodge locus (i.e. the locus where exceptional Hodge tensors appear) is a countable union of irreducible algebraic subvarieties of $S$, called the special subvarieties for $\mathbb{V}$. Our main result in this paper is that, if the level of $\mathbb{V}$ is at least $3$, this Hodge locus is in fact a finite union of such special subvarieties (hence is algebraic), at least if we restrict ourselves to the Hodge locus factorwise of positive period dimension. For instance the Hodge locus of positive period dimension of the universal family of degree $d$ smooth hypersurfaces in $\mathbf{P}^{n+1}_\mathbb{C}$, $n\geq 3, d\geq 5$ and $(n,d)\neq (4,5)$, is algebraic. On the other hand we prove that in level $1$ or $2$, the Hodge locus is analytically dense in $S^{an}$ as soon as it contains one typical special subvariety. These results follow from a complete elucidation of the distribution in $S$ of the special subvarieties in terms of typical/atypical intersections, with the exception of the atypical special subvarieties of zero period dimension.

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arXiv2023

Special subvarieties of non-arithmetic ball quotients and Hodge Theory

Let $Γ\subset \operatorname{PU}(1,n)$ be a lattice, and $S_Γ$ the associated ball quotient. We prove that, if $S_Γ$ contains infinitely many maximal totally geodesic subvarieties, then $Γ$ is arithmetic. We also prove an Ax-Schanuel Conjecture for $S_Γ$, similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to realise $S_Γ$ inside a period domain for polarised integral variations of Hodge structures and interpret totally geodesic subvarieties as unlikely intersections.

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arXiv2018

Convergence of measures on compactifications of locally symmetric spaces

We conjecture that the set of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space $S=Γ\backslash G/K$ is compact. More precisely, given a sequence of homogeneous probability measures on $S$, we expect that any weak limit is homogeneous with support contained in precisely one of the boundary components (including $S$ itself). We introduce several tools to study this conjecture and we prove it in a number of cases, including when $G={\rm SL}_3(\mathbb{R})$ and $Γ={\rm SL}_3(\mathbb{Z})$.

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arXiv2018

The Hyperbolic Ax-Lindemann-Weierstrass conjecture

The hyperbolic Ax-Lindemann-Weierstrass conjecture is a functional algebraic independence statement for the uniformizing map of an arithmetic variety. In this paper we provide a proof of this conjecture, generalizing previous work of Pila-Tsimerman and Peterzil-Starchenko.

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arXiv2016

Algebraic flows on Shimura varieties

In this paper we formulate some conjectures about algebraic flows on Shimura varieties. In the first part of the paper we prove the `logarithmic Ax-Lindemann theorem'. We then prove a result concerning the topological closure of the images of totally geodesic subvarieties of symmetric spaces uniformising Shimura varieties. This is a special case of our conjectures.

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arXiv2016

o-minimal flows on abelian varieties

Let $A$ be an abelian variety over ${\bf C}$ of dimension $n$ and $π\colon {\bf C}^n \rightarrow A$ be the complex uniformisation. Let $X$ be an unbounded subset of ${\bf C}^n$ definable in a suitable o-minimal structure. We give a description of the Zariski closure of $π(X)$.

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arXiv2013

Galois orbits and equidistribution: towards the Andre-Oort conjecture

In this paper we develop a strategy and some technical tools for proving the Andre-Oort conjecture. We give lower bounds for the degrees of Galois orbits of geometric components of special subvarieties of Shimura varieties, assuming the Generalised Riemann Hypothesis. We proceed to show that sequences of special subvarieties whose Galois orbits have bounded degrees are equidistributed in a suitable sense.

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