Satake Compactification Research Articles

Satake-Furstenberg compactifications and gradient map

Let $G$ be a real semisimple Lie group with finite center and let $\mathfrak g=\mathfrak k \oplus \mathfrak p$ be a Cartan decomposition of its Lie algebra. Let $K$ be a maximal compact subgroup of $G$ with Lie algebra $\mathfrak k$ and let $\tau$ be an irreducible representation of $G$ on a complex vector space $V$. Let $h$ be a Hermitian scalar product on $V$ such that $\tau(G)$ is compatible with respect to $\mathrm{U}(V,h)^{\mathbb C}$. We denote by $\mu_{\mathfrak p}:\mathbb P(V) \longrightarrow \mathfrak p$ the $G$-gradient map and by $\mathcal O$ the unique closed orbit of $G$ in $\mathbb P(V)$, which is a $K$-orbit, contained in the unique closed orbit of the Zariski closure of $\tau(G)$ in $\mathrm{U}(V,h)^{\mathbb C}$. We prove that up to equivalence the set of irreducible representations of parabolic subgroups of $G$ induced by $\tau$ are completely determined by the facial structure of the polar orbitope $\mathcal E=\mathrm{conv}(\mu_{\mathfrak p} (\mathcal O))$. Moreover, any parabolic subgroup of $G$ admits a unique closed orbit which is well-adapted to $\mathcal O$ and $\mu_{\mathfrak p}$ respectively. These results are new also in the complex reductive case. The connection between $\mathcal E$ and $\tau$ provides a geometrical description of the Satake compactifications without root data. In this context the properties of the Bourguignon-Li-Yau map are also investigated. Given a measure $\gamma$ on $\mathcal O$, we construct a map $\Psi_\gamma$ from the Satake compactification of $G/K$ associated to $\tau$ and $\mathcal E$. If $\gamma$ is a $K$-invariant measure then $\Psi_\gamma$ is an homeomorphism of the Satake compactification and $\mathcal E$. Finally, we prove that for a large class of measures the map $\Psi_\gamma$ is surjective.

The space of homogeneous probability measures on overline{Gamma backslash X}^S_mathrm{max} is compact

In this paper we prove that the space of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space is compact. As an application, we explain some consequences for the distribution of weakly special subvarieties of Shimura varieties.

Metrics and compactifications of Teichmüller spaces of flat tori

Using the identification of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ with the Teichm\"uller space of flat $n$-tori of unit volume, we explore several metrics and compactifications of these spaces, drawing inspiration both from Teichm\"uller theory and symmetric spaces. We define and study analogs of the Thurston, Teichm\"uller, and Weil-Petersson metrics. We show the Teichm\"uller metric is a symmetrization of the Thurston metric, which is a polyhedral Finsler metric, and the Weil-Petersson metric is the Riemannian metric of $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ as a symmetric space. We also construct a Thurston-type compactification using measured foliations on $n$-tori, and show that the horofunction compactification with respect to the Thurston metric is isomorphic to it, as well as to a minimal Satake compactification.

Compactification of Drinfeld moduli spaces as moduli spaces of 𝐴-reciprocal maps and consequences for Drinfeld modular forms

We construct a compactification of the moduli space of Drinfeld modules of rank r r and level N N as a moduli space of A A -reciprocal maps. This is closely related to the Satake compactification but not exactly the same. The construction involves some technical assumptions on N N that are satisfied for a cofinal set of ideals N N . In the special case where A = F q [ t ] A=\mathbb {F}_q[t] and N = ( t n ) N=(t^n) , we obtain a presentation for the graded ideal of Drinfeld cusp forms of level N N and all weights and can deduce a dimension formula for the space of cusp forms of any weight. We expect similar results in general, but the proof will require more ideas.

Satake compactification of analytic Drinfeld modular varieties

We construct a normal projective rigid analytic compactification of an arbitrary Drinfeld modular variety whose boundary is stratified by modular varieties of smaller dimensions. This generalizes work of Kapranov. Using an algebraic modular compactification that generalizes Pink and Schieder's, we show that the analytic compactification is naturally isomorphic to the analytification of Pink's normal algebraic compactification. We interpret analytic Drinfeld modular forms as the global sections of natural ample invertible sheaves on the analytic compactification and deduce finiteness results for spaces of such modular forms.

Complete moduli of cubic threefolds and their intermediate Jacobians

The intermediate Jacobian map, which associates to a smooth cubic threefold its intermediate Jacobian, does not extend to the GIT compactification of the space of cubic threefolds, not even as a map to the Satake compactification of the moduli space of principally polarized abelian fivefolds. A much better "wonderful" compactification of the space of cubic threefolds was constructed by the first and fourth authors --- it has a modular interpretation, and divisorial normal crossing boundary. We prove that the intermediate Jacobian map extends to a morphism from the wonderful compactification to the second Voronoi toroidal compactification of the moduli of principally polarized abelian fivefolds --- the first and fourth author previously showed that it extends to the Satake compactification. Since the second Voronoi compactification has a modular interpretation, our extended intermediate Jacobian map encodes all of the geometric information about the degenerations of intermediate Jacobians, and allows for the study of the geometry of cubic threefolds via degeneration techniques. As one application we give a complete classification of all degenerations of intermediate Jacobians of cubic threefolds of torus rank 1 and 2.

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=Gamma 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={mathbf{SL}}_3({{mathbb {R}}}) and Gamma ={mathbf{SL}}_3({{mathbb {Z}}}).

On the variety associated to the ring of theta constants in genus 3

Due to fundamental results of Igusa and Mumford the $N=2^{g-1}(2^g+1)$ theta constants of first kind $$ \sum_{n\;{\rm integral}}\exp\pi{\rm i}\big(Z[n+a/2]+2b’(n+a/2)\big),\quad a,b\ {\rm integral}. $$ define for each genus $g$ an injective holomorphic map of the Satake compactification $X_g(4,8)= \overline{\scr{H}_g/\Gamma_g[4,8]}$ into the projective space $P^{N-1}$. Moreover, this map is biholomorphic onto the image outside the Satake boundary. It is not biholomorphic on the whole in the cases $g\ge 6$. Igusa also proved that in the cases $g\le 2$ this map is biholomorphic onto the image. In this paper we extend this result to the case $g=3$. So we show that the theta map $$ X_3(4,8)\longrightarrow\Bbb{P}^{35} $$ is biholomorphic onto the image. This is equivalent to the statement that the image is a normal subvariety of $\Bbb{P}^{35}$.

Finsler bordifications of symmetric and certain locally symmetric spaces

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces $X=G/K$ of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable $G$-invariant Finsler metric on $X$. As an application, we establish the existence of natural bordifications, as orbifolds-with-corners, of locally symmetric spaces $X/\Gamma$ for arbitrary discrete subgroups $\Gamma< G$. These bordifications result from attaching $\Gamma$-quotients of suitable domains of proper discontinuity at infinity. We further prove that such bordifications are compactifications in the case of Anosov subgroups. We show, conversely, that Anosov subgroups are characterized by the existence of such compactifications among uniformly regular subgroups. Along the way, we give a positive answer, in the torsion free case, to a question of Ha\"issinsky and Tukia on convergence groups regarding the cocompactness of their actions on the domains of discontinuity.

ON THE RESIDUE OF EISENSTEIN CLASSES OF SIEGEL VARIETIES

AbstractEisenstein classes of Siegel varieties are motivic cohomology classes defined as pull-backs by torsion sections of the polylogarithm prosheaf on the universal abelian scheme. By reduction to the Hilbert–Blumenthal case, we prove that the Betti realization of these classes on Siegel varieties of arbitrary genus have non-trivial residue on zero-dimensional strata of the Baily–Borel–Satake compactification. A direct corollary is the non-vanishing of a higher regulator map.

Goresky–Pardon lifts of Chern classes and associated Tate extensions

Let $X$ be an irreducible complex-analytic variety, ${\mathcal{S}}$ a stratification of $X$ and ${\mathcal{F}}$ a holomorphic vector bundle on the open stratum ${X\unicode[STIX]{x0030A}}$. We give geometric conditions on ${\mathcal{S}}$ and ${\mathcal{F}}$ that produce a natural lift of the Chern class $\operatorname{c}_{k}({\mathcal{F}})\in H^{2k}({X\unicode[STIX]{x0030A}};\mathbb{C})$ to $H^{2k}(X;\mathbb{C})$, which, in the algebraic setting, is of Hodge level ${\geqslant}k$. When applied to the Baily–Borel compactification $X$ of a locally symmetric variety ${X\unicode[STIX]{x0030A}}$ and an automorphic vector bundle ${\mathcal{F}}$ on ${X\unicode[STIX]{x0030A}}$, this refines a theorem of Goresky–Pardon. In passing we define a class of simplicial resolutions of the Baily–Borel compactification that can be used to define its mixed Hodge structure. We use this to show that the stable cohomology of the Satake ($=$ Baily–Borel) compactification of ${\mathcal{A}}_{g}$ contains nontrivial Tate extensions.

The intersection cohomology of the Satake compactification of $${\mathcal {A}}_g$$ for $$g \le 4$$

We completely determine the intersection cohomology of the Satake compactifications \({{\mathcal {A}}_{2}^{\mathrm{Sat}}},{{\mathcal {A}}_{3}^{\mathrm{Sat}}}\), and \({{\mathcal {A}}_{4}^{\mathrm{Sat}}}\), except for \({ IH}^{10}({{\mathcal {A}}_{4}^{\mathrm{Sat}}})\). We also determine all the ingredients appearing in the decomposition theorem applied to the map from a toroidal compactification to the Satake compactification in these genera. As a byproduct we obtain in addition several results about the intersection cohomology of the link bundles involved.

Compactification de Thurston d’espaces de réseaux marqués et de l’espace de Torelli

We define a compactification of symmetric spaces of noncompact type, seen as spaces of isometry classes of marked lattices, analogous to the Thurston compactification of the Teichmüller space, and we show that it is equivariantly isomorphic to a Satake compactification. We then use it to define a new compactification of the Torelli space of a hyperbolic surface with marked points, and we show that it is equivariantly isomorphic to the Satake compactification of the image of the period mapping. Finally, we describe the natural stratification of a subset of the boundary.

On Jacobian Kummer surfaces

We give explicit equations of smooth Jacobian Kummer surfaces of degree 8 in $\mathbb P^5$ by theta functions. As byproducts, we can write down Rosenhain's 80 hyperplanes and 32 lines on these Kummer surfaces explicitly. Moreover we study the fibration of Kummer surfaces over the Satake compactification of the Siegel modular 3-fold of level (2,4). The total space is a smooth projective 5-fold which is regarded as a higher-dimensional analogue of Shioda's elliptic modular surfaces.

Some Siegel Threefolds with a Calabi-Yau Model II

In a previous paper, we described some Siegel modular threefolds which ad- mit a Calabi-Yau model. Using a different method we give in this paper an enlarged list of such varieties. Basic for this method is a paper of van Geemen and Nygaard. They study a variety X that is the complete intersection of four quadrics in P 7 (C). This is biholomorphic equivalent to the Satake compactification of H2=Γ ' for a certain subgroup Γ ' Sp(2;Z) and it will be the starting point of our investigation. It has been pointed out that a (projective) small resolution of this variety is a rigid Calabi-Yau manifold ˜ X. Then we will consider the action of quotients of modular groups on X and study possible resolutions that admits a Calabi-Yau model in the category of complex spaces.

On a ring of modular forms related to the theta gradients map in genus 2

The level moduli space of ppavs with a level structure Ag4,8 can be mapped to the projective space by means of the gradients of odd theta functions. Although this map is generically injective for g⩾3, it is not injective in the genus 2 case. However, there exists a congruence subgroup Γ, contained in Γ2(2,4) and containing Γ2(4,8), such that the theta gradient map factors on the quotient AΓ of the Siegel upper half-plane by the group Γ and the new map is injective on AΓ; we provide a description of this group together with some of its properties. We also prove a structure theorem for the ring of modular forms A(Γ) with respect to Γ. We finally provide a set of generators for the ideal of cusp forms S(Γ) to give an algebraic description of the desingularization ProjS(Γ) of the Satake compactification ProjA(Γ).

THE CUSP STRUCTURE OF THE PARAMODULAR GROUPS FOR DEGREE TWO

We describe the one-dimensional and zero-dimensional cusps of the Satake compactification for the paramodular groups in degree two for arbitrary levels. We determine the crossings of the one-dimensional cusps. Applications to computing the dimensions of Siegel modular forms are given.

Satake-Furstenberg compactifications, the moment map and λ 1

Let $G$ be a complex semisimple Lie group, $K$ a maximal compact subgroup and $\tau$ an irreducible representation of $K$ on $V$. Denote by $M$ the unique closed orbit of $G$ in $\Bbb{P}(V)$ and by $\cal{O}$ its image via the moment map. For any measure $\gamma$ on $M$ we construct a map $\Psi_\gamma$ from the Satake compactification of $G/K$ (associated to $V$) to the Lie algebra of $K$. If $\gamma$ is the $K$-invariant measure, then $\Psi_\gamma$ is a homeomorphism of the Satake compactification onto the convex envelope of $\cal{O}$. For a large class of measures the image of $\Psi_\gamma$ is the convex envelope. As an application we get sharp upper bounds for the first eigenvalue of the Laplacian on functions for an arbitrary K\"ahler metric on a Hermitian symmetric space.

Compactification of Drinfeld modular varieties and Drinfeld modular forms of arbitrary rank

We give an abstract characterization of the Satake compactification of a general Drinfeld modular variety. We prove that it exists and is unique up to unique isomorphism, though we do not give an explicit stratification by Drinfeld modular varieties of smaller rank which is also expected. We construct a natural ample invertible sheaf on it, such that the global sections of its k-th power form the space of (algebraic) Drinfeld modular forms of weight k. We show how the Satake compactification and modular forms behave under all natural morphisms between Drinfeld modular varieties; in particular we define Hecke operators. We give explicit results in some special cases.

Some intersection numbers of divisors on toroidal compactifications of 𝒜_{ℊ}

We study the top intersection numbers of the boundary and Hodge class divisors on toroidal compactifications of the moduli space A g \mathcal {A}_g of principally polarized abelian varieties and compute those numbers that live away from the stratum which lies over the closure of A g − 3 \mathcal {A}_{g-3} in the Satake compactification.