arXiv research

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…

The study finds infinitely many Shimura subvarieties in Jacobian loci for curves of genus 2, 3, and 4.

We construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type O(p,2) and U(p,1), we show that the resulting local archimedean height pairings are related to special values of derivatives of Siegel Eisentein series. A conj…

Let $G$ be a connected semisimple group over ${\Bbb Q}$. Given a maximal compact subgroup $K\subset G({\Bbb R})$ such that $X=G({\Bbb R})/K$ is a Hermitian symmetric domain, and a convenient arithmetic subgroup $Γ\subset G({\Bbb Q})$, one constructs a (connected) Shimura variety $S=S(Γ)=Γ\backslash X$. If $H\subset G$ …

We prove that there are no Shimura-Teichmüller curves generated by genus five translation surfaces, thereby completing the classification of Shimura-Teichmüller curves in general. This was conjectured by Möller in his original work introducing Shimura-Teichmüller curves. Moreover, the property of being a Shimura-Teichm…

We parametrize the commensurability classes of curves on Shimura surfaces that are totally geodesic, i.e., the commensurability classes of so-called $\mathbb{C}$-Fuchsian subgroups. In particular, if a Shimura surface contains one commensurability class of totally geodesic curves, it contains infinitely many.

Two number fields are said to be Brauer equivalent if there is an isomorphism between their Brauer groups that commutes with restriction. In this paper we prove a variety of number theoretic results about Brauer equivalent number fields (e.g., they must have the same signature). These results are then applied to the ge…

Using conjugation of Shimura varieties, we produce nonisomorphic, cocompact, torsion-free lattices in $\mathrm{PU}(n,1)$ with isomorphic profinite completions for all $n \ge 2$. This disproves a conjecture of D. Kazhdan and gives the first examples nonisomorphic lattices in a semisimple Lie group of real rank one with …

We show that asymptotically the first Betti number, or the arithmetic genus, of a Shimura curve satisfies the Gauss--Bonnet equality. We also show that the first Betti number of a congruence hyperbolic 3--orbifold asymptotically vanishes relatively to hyperbolic volume.

We classify curves in the moduli space of curves that are both Shimura- and Teichmueller curves: Except for the moduli space of genus one curves there is only a single such curve. We start with a Hodge-theoretic description of Shimura curves and of Teichmueller curves that reveals similarities and differences of the tw…

We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-Kühn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms…

We consider the cohomology group $H^1(Γ, ρ)$ of a discrete subgroup $Γ\subset G=SU(n, 1)$ and the symmetric tensor representation $ρ$ on $S^m(\mathbb C^{n+1})$. We give an elementary proof of the Eichler-Shimura isomorphism that harmonic forms $H^1(Γ\backslash G/K, ρ)$ are $(0, 1)$-forms for the automorphic holomorphic…

We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings…

New method constructs non-arithmetic hyperbolic orbifolds from complex arithmetic ball quotients.

We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds generate the cohomology groups of degree $n$ of compact congruence $p$-dimensional hyperbolic manifolds "of simple type" a…

The paper describes how Hodge loci are typically equidistributed in complex varieties.

We define geometric zeta functions for locally symmetric spaces as generalizations of the zeta functions of Ruelle and Selberg. As a special value at zero we obtain the Reidemeister torsion of the manifold. For hermitian spaces these zeta functions have as special value the quotient of the holomorphic torsion of Ray an…

Computes canonical heights for arithmetic log surfaces using Hurwitz zeta function.

The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincaré disk into bounded symmetric domains. As a first result we prove that any holomorphic curve exiting the boundary of a bounded symmetric domain $Ω$ must necessarily be asymptotically totally geodesic. A…

Paper studies Wiman-Edge pencil and Wiman curve, providing uniformizations and modular interpretations.

This paper proves all Pfaffian varieties are area-minimizing except hypersurfaces.

The paper shows how certain complex projective varieties can be broken down into simpler types.

Investigates secant dimensions and identifiability in flag varieties.

Study of braid varieties and their Legendrian isotopy.

We introduce the fibred toric varieties as equivariant $\mathbb{C}P^r$ bundles over lower dimensional toric varieties. An equivalent characterization is that the natural morphisms on them degenerate to bundle projections in the context of variation of toric varieties as GIT quotients. Our main observation is that these…

The study shows boundedness of certain fibered varieties in algebraic geometry.

Study identifies subvarieties of projective varieties mapping to models.

Paper describes holomorphic polyvector fields on toric varieties.

New proof of divisibility property for certain algebraic varieties.

The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.

Cominuscule subvarieties found in flag varieties.

We consider projective varieties with degenerate Gauss image whose focal hypersurfaces are non-reduced schemes. Examples of this situation are provided by the secant varieties of Severi and Scorza varieties. The Severi varieties are moreover characterized by a uniqueness property.

Kähler-Einstein metrics found on special types of symmetric varieties.

Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in 14-dimensional projec…

Study empty polar varieties' impact on singular function-germs.

Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomog…

Character varieties get a natural Poisson structure.

The dual variety X* for a smooth n-dimensional variety X of the projective space P^N is the set of tangent hyperplanes to X. In the general case, the variety X* is a hypersurface in the dual space (P^N)*. If dim X* < N - 1, then the variety X is called dually degenerate. The authors refine these definitions for a varie…

The paper classifies certain singular projective varieties with specific properties.

Proves minimality of tensor varieties, generalizing previous results.

Each of the four critical Severi varieties arises from a minimal holomorphic nilpotent orbit in a simple regular rank 3 hermitian Lie algebra and each such variety lies as singular locus in a cubic--the chordal variety--in the corresponding complex projective space; the cubic and projective space are identified in term…

Study projective KLT varieties with projectively flat cotangent sheaves.

The paper examines when real matrix Schubert varieties are minimal submanifolds.

New connections on symmetric spaces with invariant properties.

Solves open problems on curved projective varieties.

Stratifies representation varieties of twisted Hopf links.

Classifies holomorphic parabolic geometries on complex manifolds.

Character variety of Borromean link solved, Alexander polynomial formula found.