arXiv subjects
Publications and source records attributed to Salim Tayou.
We survey recent results on a conjecture of Kudla regarding the modularity of generating series of special cycle classes in toroidal compactifications of orthogonal and unitary Shimura varieties. Along the way, we formulate several conjectures on related phenomena for special cycles in other types of Shimura varieties, as well as on more general quotients of period domains.
We partially resolve conjectures of Deligne and Simpson concerning $\mathbb{Z}$-local systems on quasi-projective varieties that underlie a polarized variation of Hodge structure. For local systems with $\mathbb{Q}$-anisotropic monodromy, we prove (1) a relative form of Deligne's finiteness theorem, for any family of quasi-projective varieties, and (2) algebraicity of the corresponding non-abelian Hodge locus.
We construct natural extensions of the Kudla--Millson generating series of cohomology classes of special cycles in compactified unitary Shimura varieties of signature $(n+1,1)$ and prove that they are holomorphic Hermitian modular forms. This proves the cohomological version of a conjecture of Kudla and Bruinier--Rosu--Zemel, in all codimensions up to the middle. We also develop the theory of Hermitian quasi-modular forms, with a particular focus on polynomial weighted theta functions, and prove that the generating series of Zariski closures of special cycles is a Hermitian quasi-modular form.
Given a smooth quasi-projective complex algebraic variety $\mathcal{S}$, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over $\mathcal{S}$ of degree $d$ in $\mathbb{P}_{\mathbb C}^{n+1}$. We prove that the finiteness is uniform in $\mathcal{S}$ and we give examples where the result is sharp. We also prove similar results for certain complete intersections in $\mathbb{P}_{\mathbb C}^{n+1}$ of higher codimension and more generally for algebraic varieties whose moduli space admits a period map that satisfies the infinitesimal Torelli theorem.
In this note, we prove the Zilber--Pink conjecture for subvarieties of mixed Shimura varieties, which are not defined over~$\overline{\mathbb Q}$ in a strong sense. We prove similar results for general variations of mixed Hodge structure of geometric origin, assuming furthermore that they are absolute.
We prove that the generating series of special divisors in toroidal compactifications of orthogonal Shimura varieties is a mixed mock modular form. More precisely, we find an explicit completion using theta series associated to rays in the cone decomposition. The proof relies on intersection theory at the boundary of the Shimura variety.
Given a Brauer class on a K3 surface defined over a number field, we prove that there exists infinitely many reductions where the Brauer class vanishes, under certain technical hypotheses, answering a question of Frei--Hassett--Várilly-Alvarado.
We prove that the positive-dimensional part of the torsion locus of the Ceresa normal function in $\mathcal{M}_g$ is not Zariski dense when $g\geq 3$. Moreover, it has only finitely many components with generic Mumford-Tate group equal to $\mathrm{GSp}_{2g}$; these components are defined over $\overline{\mathbb{Q}}$, and their union is closed under the action of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb Q)$. More generally, we study the distribution of the torsion locus of arbitrary admissible normal functions.
Let $X$ be a K3 surface over a number field. We prove that $X$ has infinitely many specializations where its Picard rank jumps, hence extending our previous work with Shankar--Shankar--Tang to the case where $X$ might have potentially bad reduction. We prove a similar result for generically ordinary non-isotrivial families of K3 surfaces over curves over $\overline{\mathbb{F}}_p$ which extends previous work of Maulik--Shankar--Tang. As a consequence, we give a new proof of the ordinary Hecke orbit conjecture for orthogonal and unitary Shimura varieties.
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 equidistributed in $S$ with respect to the volume form $c_q^r$, where $c_q$ is the $q$\textsuperscript{th} Chern form of the Hodge bundle. We obtain also several equidistribution results of the typical locus in Shimura varieties: a criterion for the density of the typical Hodge loci of a variety in $\mathcal{A}_g$, equidistribution of certain families of CM points and equidistribution of Hecke translates of curves and surfaces in $\mathcal A_g$. These results are proved in the much broader context of dynamics on homogeneous spaces of Lie groups which are of independent interest. The pull-push form appear in this greater generality and we provide several tools to determine it and we compute it in many examples.
Given a K3 surface $X$ over a number field $K$ with potentially good reduction everywhere, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{\overline{K}}$ has infinitely many rational curves or $X$ has infinitely many unirational specializations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field $K$ with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of $K$.
We prove that any non-isotrivial elliptic K3 surface over an algebraically closed field $k$ of arbitrary characteristic contains infinitely many rational curves. In the case when $\mathrm{char}(k)\neq 2,3$, we prove this result for any elliptic K3 surface. When the characteristic of $k$ is zero, this result is due to the work of Bogomolov-Tschinkel and Hassett.
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 study the image of the $\ell$-adic Galois representations associated to the four vector valued Siegel modular forms appearing in the work of Chenevier and Lannes. These representations are symplectic of dimension $4$. Following a method of Dieulefait, we determine the primes $\ell$ for which these representations are absolutely irreducible. In addition, we show that their image is "full" for all primes $\ell$ such that the associated residual representation is absolutely irreducible, except in two cases.