We discuss and extend some of the results obtained in "Arakelov inequalities and the uniformization of certain rigid Shimura varieties" (math.AG/0503339), restricting ourselves to the two dimensional case, i.e. to surfaces Y mapping generically finite to the moduli stack of Abelian varieties. In particular we show that Y is a Hilber modular surfaces if and only if the dergee of the Hodge bundle satisfies the Arakelov equality.
In the revised version, we corrected some minor mistakes, pointed out by the referee, and we tried to improve the presentation of the text.
We study the Torelli locus T_g in the moduli space A_g of abelian varieties. We consider special subvarieties (Shimura subvarieties) contained in the Torelli locus. We review the construction of some non-trivial examples, and we discuss some conjectures, techniques and recent progress.
We use the language and tools available in model theory to redefine and clarify the rather involved notion of a {\em special subvariety} known from the theory of Shimura varieties (mixed and pure).
We consider smooth codimension two subcanonical subvarieties in $\mathbb{P}^n$ with $n \geq 5$, lying on a hypersurface of degree $s$ having a linear subspace of multiplicity $(s-2)$. We prove that such varieties are complete intersections. We also give a little improvement to some earlier results on the non existence of rank two vector bundles on $\mathbb{P}^4$ with small Chern classes.
We consider families of cyclic covers of the projective line, where we fix the covering group and the local monodromies and we vary the branch points. We prove that there are precisely twenty such families that give rise to a special subvariety in the moduli space of abelian varieties. Our proof uses techniques in mixed characteristics due to Dwork and Ogus.
We introduce the notion of dR-absolutely special subvarieties in motivic variations of Hodge structure as special subvarieties cut out by (de Rham-)absolute Hodge cycles and conjecture that all special subvarieties are dR-absolutely special. This is implied by Deligne's conjecture that all Hodge cycles are absolute Hodge cycles, but is a much weaker conjecture. We prove our conjecture for subvarieties satisfying a simple monodromy condition introduced by Klingler-Otwinowska-Urbanik. We study applications to typical respectively atypical intersections and $\bar{\mathbb{Q}}$-bialgebraic subvarie…
We study Shimura (special) subvarieties in the moduli space $A_{p,D}$ of complex abelian varieties of dimension $p$ and polarization type $D$. These subvarieties arise from families of covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to $\mathbb{P}^1$. We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, obtain 210 Shimura subvarieties contained in the Prym locus.
Let $S$ be a Shimura variety with reflex field $E$. We prove that the action of $\operatorname{Gal}(\overline{\mathbb{Q}}/E)$ on $S$ maps special points to special points and special subvarieties to special subvarieties. Furthermore, the Galois conjugates of a special point all have the same complexity (as defined in the theory of unlikely intersections). These results follow from Milne and Shih's construction of canonical models of Shimura varieties, based on a conjecture of Langlands which was proved by Borovoi and Milne.
A strongly special subvariety of a Shimura variety $S$ is (essentially) a subvariety associated to a semi-simple sub-Shimura datum. We prove that the set of probability measures canonically associated to to strongly special subvarieties is compact. More precisely: If $μ_n$ is a sequence of such probability measures associated to strongly special subvarieties $Z_n$ of $S$ there exists a subsequence $μ_{n_k}$ which converge to a measure $μ_Z$ canonically associated to a strongly special subvariety $Z$ and for all $k>>0$ $Z_{n_k}$ is contained in $Z$. We give some application to the Andre-Oort co…
This survey article discusses some results on the structure of families f:V-->U of n-dimensional manifolds over quasi-projective curves U, with semistable reduction over a compactification Y of U. We improve the Arakelov inequality for the direct images of powers of the dualizing sheaf. For families of Abelian varieties we recall the characterization of Shimura curves by Arakelov equalities. For families of curves we recall the characterization of Teichmueller curves in terms of the existence of certain sub variation of Hodge structures. We sketch the proof that the moduli scheme of curves of …
Let $Z$ be an algebraic subvariety of a Shimura variety. We extend results of the first author to prove an effective upper bound for the degree of a non-facteur maximal special subvariety of $Z$.
We consider the problem to determine which blow-ups along subvarieties in products of two projective spaces are log Fano. By describing the nef cones of such blow-ups with special centers, we give a partial classification result. For each example, explicit boundary divisors are also given.
Let X be a Mumford-Tate variety, i.e., a quotient of a Mumford-Tate domain D by a discrete subgroup. Mumford-Tate varieties are generalizations of Shimura varieties. We define the notion of a special subvariety Y in X (of Shimura type), and formulate necessary criteria for Y to be special. Our method consists in looking at finitely many compactified special curves C_i in Y, and testing whether the inclusion of the union of all C_i in Y satisfies certain properties. One of them is the so-called relative proportionality condition. In this paper, we give a new formulation of this numerical criter…
We introduce $\ell$-Galois special subvarieties as an $\ell$-adic analog of the Hodge-theoretic notion of a special subvariety. The Mumford-Tate conjecture predicts that both notions are equivalent. We study some properties of these subvarieties and prove this equivalence for subvarieties satisfying a simple monodromy condition. As applications, we show that the $\ell$-Galois exceptional locus is a countable union of algebraic subvarieties and, if the derived group of the generic Mumford-Tate group of a family is simple, its part of positive period dimension coincides with the Hodge locus of p…
We explore an analogue of the André-Oort conjecture for subvarieties of Drinfeld modular varieties. The conjecture states that a subvariety $X$ of a Drinfeld modular variety contains a Zariski-dense set of complex multiplication (CM) points if and only if $X$ is a "special" subvariety (i.e. $X$ is defined by requiring additional endomorphisms). We prove this conjecture in two cases. Firstly when $X$ contains a Zariski-dense set of CM points with a certain behaviour above a fixed prime (which is the case if these CM points lie in one Hecke orbit), and secondly when $X$ is a curve containing inf…
A conjecture by Yves Andre and Frans Oort says that closed subvarieties of Shimura varieties that contain a Zariski dense subset of special points are subvarieties of Hodge type. We prove this in the case where the subvariety is a curve that contains infinitely many special points that lie in one Hecke orbit. Via work of Juergen Wolfart, Paula Cohen and Gisbert Wuestholz, this result has an application to algebraicity of values of hypergeometric functions.
Let $Y$ be a subvariety contained in a smooth Mumford compactification of an orthogonal Shimura variety $M \subset A_g$, where $A_g$ is the moduli space of principally polarized abelian varieties of dimension $g$ with some level structure, such that $Y$ intersects the boundary of $A_g$ transversally. Then we give necessary and sufficient conditions of André-Oort type for $Y$ itself being the compactification of a special subvariety $Y^0 \subset M$
We prove a non-archimedean analogue of the fact that a closed subvariety of a semi-abelian variety is hyperbolic modulo its special locus, and thereby generalize a result of Cherry.
We make precise the structure of the first two reduction morphisms associated with codimension two nonsingular subvarieties of quadrics $\Q{n}$, $n\geq 5$. We give a coarse classification of the same class of subvarieties when they are assumed to be not of log-general-type. Keywords: Adjunction Theory, classification, codimension two, conic bundles, low codimension, non log-general-type, quadric, reduction, special variety
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 subvarietie…
We introduce a method to construct special holomorphic tensors on orthogonal modular varieties from scalar-valued modular forms, and give applications to the Lang conjecture on the birational type of subvarieties of orthogonal modular varieties.
In this paper we show how some known weak forms of the Zilber--Pink conjecture can be strengthened by combining them with the Mordell--Lang conjecture or its variants. We illustrate this idea by proving some theorems on atypical intersections in the semiabelian and modular settings. Given a "finitely generated" set $Γ$ with a certain structure, we consider $Γ$-special subvarieties -- weakly special subvarieties containing a point of $Γ$ -- and show that every variety $V$ contains only finitely many maximal $Γ$-atypical subvarieties, i.e. atypical intersections of $V$ with $Γ$-special varieties…
In this paper we study various aspects of the Ekedahl-Serre problem. We formulate questions of Ekedahl-Serre type and Coleman-Oort type for general weakly special subvarieties in the Siegel moduli space, propose a conjecture relating these two questions, and provide examples supporting these questions. The main new result is an upper bound of genera for curves over number fields whose Jacobians are isogeneous to products of elliptic curves satisfying the Sato-Tate equidistribution, and we also refine previous results showing that certain weakly special subvarieties only meet the open Torelli l…
In the moduli space of degree d polynomials, the special subvarieties are those cut out by critical orbit relations, and then the special points are the post-critically finite polynomials. It was conjectured that in the moduli space of degree d polynomials, the subvarieties containing a Zariski-dense set of special points are exactly these special subvarieties. In this article, we prove the first non-trivial case for this conjecture: the case of cubic polynomials.
We prove several results on symplectic varieties with a Hamiltonian action of a reductive group having invariant Lagrangian subvarieties. Our main result states that the images of the moment maps of a Hamiltonian variety and of the cotangent bundle over an invariant Lagrangian subvariety coincide. This implies that the complexity and rank of the Lagrangian subvariety are equal to the half of the corank and to the defect of the Hamiltonian variety, respectively. This result generalizes a theorem of Panyushev on the complexity and rank of a conormal bundle. A simple elementary proof of this theo…
Let $\mathbb{V}$ be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety $S$. In this paper, we show that the union of the non-factor special subvarieties for $(S, \mathbb{V})$, which are of Shimura type with dominant period maps, is a finite union of special subvarieties of $S$. This generalizes previous results of Clozel and Ullmo arXiv:math/0404131, Ullmo \cite{Ullmo07} on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part …
We prove that under some conditions on the monodromy, families of abelian covers of the projective line do not give rise to (higher dimensional) Shimura subvarieties in $A_g$. This is achieved by a reduction to $p$ argument. We also use another method based on monodromy computations to show that two dimensional subvarieties in the above locus are not special. In particular it is shown that such families have usually large monodromy groups. Together with our earlier results, the above mentioned results contribute to classifying the special families in the moduli space of abelian varieties and p…
This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of $X_{a, b} = (\mathbf{H}^2)^a \times (\mathbf{H}^3)^b$. A special case describes all Shimura subvarieties of type $\mathrm{A}_1$ Shimura varieties. We produce, for any $n\geq 1$, examples of manifolds/Shimura varieties with precisely $n$ commensurability classes of totally geodesic submanifolds/Shimura subvarieties. This is in stark contrast with the previously studied cases of arithmetic hyperbolic $3$-manifolds and quaternionic Shimura surfaces, wher…
We study the problem of classifying the irreducible projective varieties $X$ of dimension $n\ge 2$ in $\Bbb P^N$ which contain an algebraic family $\Cal F$ of dimension $h+1$ ($h<n$) of subvarieties $Y$ of dimension $n-h$, each one contained in a $\Bbb P^{N-h-1}$. We prove that one of the following happens: (i) there exists an integer $r$, $r<N-n$ such that $X$ is contained in a variety $V_r$ of dimension at most $N-r$ containing a family of dimension $h+1$ of subvarieties of dimension $N-h-r$, each one contained in a linear space of dimension $N-h-1$; (ii) The degree of $Y$ is bounded by a fu…
We consider Calabi-Yau $n$-folds $X$ arising from certain hyperplane arrangements. Using Fu-Vial's theory of distinguished cycles for varieties with motive of abelian type, we show that the subring of the Chow ring of $X$ generated by divisors, Chern classes and intersections of subvarieties of positive codimension injects into cohomology. We also prove Voisin's conjecture for $X$, and Voevodsky's smash-nilpotence conjecture for odd-dimensional $X$.
In this note we study linear systems on complete toric varieties $X$ with an invariant point, whose orbit under the action of the automorphism group of $X$ contains the dense torus $T$ of $X$. We give a characterization of such varieties in terms of its defining fan and introduce a new definition of expected dimension of linear systems which consider the contribution given by certain toric subvarieties. Finally, we study degenerations of linear systems on these toric varieties induced by toric degenerations.
Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional properties. The most remarkable case is the Legendrian subvarieties of projective space and prior to the author's research only few smooth examples of these were known. The first series of results of this thesis is related to the automorphism group of any Legendrian subvariety in any projective contact manifold. The connected component of this group (under suitable m…
We construct a family of examples of Legendrian subvarieties in some projective spaces. Although most of them are singular, a new example of smooth Legendrian variety in dimension 8 is in this family. The 8-fold has interesting properties: it is a compactification of the special linear group, a Fano manifold of index 5 and Picard number 1.
Recently continuous rational maps between real algebraic varieties have attracted the attention of several researchers. In this paper we continue the investigation of approximation properties of continuous rational maps with values in spheres. We propose a conjecture concerning such maps and show that it follows from certain classical conjectures involving transformation of compact smooth submanifolds of nonsingular real algebraic varieties onto subvarieties. Furthermore, we prove our conjecture in a special case and obtain several related results.
We introduce the notion of Mukai regularity (M-regularity) for coherent sheaves on abelian varieties. The definition is based on the Fourier-Mukai transform, and in a special case depending on the choice of a polarization it parallels and strenghtens the usual Castelnuovo-Mumford regularity. Mukai regularity has a large number of applications, ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties, to higher dimensional type statements and to a study of special classes of vector bundles. Some of these applications are explained here, w…
The André-Pink-Zannier conjecture concerns the intersection of subvarieties and the generalized Hecke orbit of a given point in mixed Shimura varieties. It is part of the Zilber-Pink conjecture. In this paper we focus on the universal family of principally polarized abelian varieties. We explain the moduli interpretation of the conjecture in this case and prove several different cases for this conjecture: its overlap with the André-Oort conjecture; when the subvariety is contained in an abelian scheme over a curve and the point is a torsion point on its fiber; when the subvariety is a curve.
We describe the application of the results of Kudla-Millson on the modularity of generating series for cohomology classes of special cycles to the case of lattice polarized K3 surfaces. In this case, the special cycles can be interpreted as higher Noether-Lefschetz loci. These generating series can be paired with the cohomology classes of complete subvarieties of the moduli space to give classical Siegel modular forms with higher Noether-Lefschetz numbers as Fourier coefficients. Examples of such complete families associated to quadratic spaces over totally real number fields are constructed. …
We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective n-dimensional varieties, n\geq 4, onto smooth projective varieties, arising from classical projective geometry and defined over sufficiently small fields, not necessarily algebraically closed. The examples considered come from particular special homaloidal and subhomaloidal linear systems, which usually are degenerations of general phenomena classically investigated by Bordiga, Severi, Todd, Room, Fano, Semple and Tyrrell and more recently by Ein and Shepherd-Barron. Th…
Let $X\subset Y(1)^n$ be a subvariety defined over a number field $\mathbb F$ and let $(P_1,\ldots,P_n)\in X$ be a special point not contained in a positive-dimensional special subvariety of $X$. We show that the if a coordinate $P_i$ corresponds to an order not contained in a single exceptional Siegel-Tatuzawa imaginary quadratic field $K_*$ then the associated discriminant $|Δ(P_i)|$ is bounded by an effective constant depending only on $\operatorname{deg} X$ and $[{\mathbb F}:{\mathbb Q}]$. We derive analogous effective results for the positive-dimensional maximal special subvarieties. From…
When $p$ is inert in the quadratic imaginary field $E$ and $m<n$, unitary Shimura varieties of signature $(n,m)$ and a hyperspecial level subgroup at $p$, carry a natural foliation of height 1 and rank $m^2$ in the tangent bundle of their special fiber $S$. We study this foliation and show that it acquires singularities at deep Ekedahl-Oort strata, but these singularities are resolved if we pass to a natural smooth moduli problem $S^\sharp$, a successive blow-up of $S$. Over the ($μ$-)ordinary locus we relate the foliation to Moonen's generalized Serre-Tate coordinates. We study the quotient o…
We take a sum $C_1+r C_2,\ r\in\mathbb Q$ of a line $C_1$ and a complete intersection curve $C_2$ of type $(3,3)$ inside a smooth surface of degree $8$ and with $C_1\cap C_2=\emptyset$. We gather evidences to the fact that for all except a finite number of $r$, the Noether-Lefschetz loci attached to the cohomology classes of $C_1+ r C_2$ are distinct $31$ codimensional subvarieties intersecting each other in a $32$ codimensional subvariety of the ambient space. The maximum codimension for components of the Noether-Lefschetz locus in this case is $35$, and hence, we provide a conjectural descri…
We prove a $p$-adic analogue of the André-Oort conjecture for subvarieties of the universal abelian varieties containing a dense set of special points. Let $g$ and $n$ be integers with $n \geq 3$ and $p$ a prime number not dividing $n$. Let $R$ be a finite extension of $W[{\mathbb F}_p^{\mathrm alg}]$, the ring of Witt vectors of the algebraic closure of the field of $p$ elements. The moduli space $\cA = \cA_{g,1,n}$ of $g$-dimensional principally polarized abelian varieties with full level $n$-structure as well as the universal abelian variety $π:\cX \to \cA$ over $\cA$ may be defined over $R…
