Symmetric Varieties Research Articles

Mixed Hodge structures and Siegel operators

Abstract In this paper, we study mixed Hodge structures on the cohomology of locally symmetric varieties and give an application to modular forms. After proving vanishing of some Hodge numbers, we focus on the weight filtration on the last Hodge subspace of the middle degree cohomology. We prove that the weight filtration coincides with the corank filtration on the space of modular forms of canonical weight defined by the Siegel operators, and calculate the graded quotients. As an application, we deduce surjectivity of the total Siegel operators in many cases, and identify an obstruction space in the remaining case.

Multiplicative Higgs bundles and involutions

In this paper we generalize the theory of multiplicative G-Higgs bundles over a curve to pairs (G,θ), where G is a reductive algebraic group and θ is an involution of G. This generalization involves the notion of a multiplicative Higgs bundle taking values in a symmetric variety associated to θ, or in an equivariant embedding of it. We also study how these objects appear as fixed points of involutions of the moduli space of multiplicative G-Higgs bundles, induced by the involution θ.

Equivariant Grothendieck ring of a complete symmetric variety of minimal rank

Equivariant Grothendieck ring of a complete symmetric variety of minimal rank

Integral points on algebraic subvarieties of period domains: from number fields to finitely generated fields

We show that for a variety which admits a quasi-finite period map, finiteness (resp. non-Zariski-density) of S-integral points implies finiteness (resp. non-Zariski-density) of points over all mathbb {Z}-finitely generated integral domains of characteristic zero. Our proofs rely on foundational results in Hodge theory due to Deligne, Griffiths, and Schmid, and Bakker-Brunebarbe-Tsimerman. We give straightforward applications to arithmetic locally symmetric varieties, the moduli space of smooth hypersurfaces in projective space, and the moduli of smooth divisors in an abelian variety.

On the Eisenstein functoriality in cohomology for maximal parabolic subgroups

In his paper, ’On torsion in the cohomology of locally symmetric varieties’, Peter Scholze has introduced a new, purely topological method to construct the cohomology classes on arithmetic quotients of symmetric spaces of reductive groups over $$\mathbb {Q}$$ originating from the cohomology of the similar quotients of Levi subgroups of maximal parabolic subgroups. We extend this construction beyond the cases he considers, and, in the complex case, to the cohomology of local systems.

The perfect cone compactification of quotients of type IV domains

The perfect cone compactification is a toroidal compactification which can be defined for locally symmetric varieties. Let overline{D_{L}/widetilde{O}^{+}(L)}^{p} be the perfect cone compactification of the quotient of the type IV domain D_{L} associated to an even lattice L. In our main theorem we show that the pair { (overline{D_{L}/widetilde{O}^{+}(L)}^{p}, Delta /2) } has klt singularities, where Delta is the closure of the branch divisor of { D_{L}/widetilde{O}^{+}(L) }. In particular this applies to the perfect cone compactification of the moduli space of 2d-polarised K3 surfaces with ADE singularities when d is square-free.

The essential dimension of congruence covers

AbstractConsider the algebraic function $\Phi _{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $\Phi _{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$-divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$-dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$-dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $M$ is proper. As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety.

The action of a mirabolic subgroup on a symmetric variety

Let F be a local field of characteristic zero. Let E be a quadratic field extension of F. We show that any PE-invariant linear functional on a GLn(E)-distinguished irreducible smooth admissible representation π of GL2n(F) is also GLn(E)-invariant, where PE is the standard mirabolic subgroup of GLn(E). Then we give a simple proof to show that the exterior square L-function has a pole at s=0 is a necessary condition such that a unitary and unramified principal series representation π is distinguished by GLn(E) when E/F is unramified.

GIT versus Baily-Borel compactification for K3's which are double covers of [formula omitted

In previous work, we have introduced a program aimed at studying the birational geometry of locally symmetric varieties of Type IV associated to moduli of certain projective varieties of K3 type. In particular, a concrete goal of our program is to understand the relationship between GIT and Baily-Borel compactifications for quartic K3 surfaces, K3's which are double covers of a smooth quadric surface, and double EPW sextics. In our first paper [36], based on arithmetic considerations, we have given conjectural decompositions into simple birational transformations of the period maps from the GIT moduli spaces mentioned above to the corresponding Baily-Borel compactifications. In our second paper [35] we studied the case of quartic K3's; we have given geometric meaning to this decomposition and we have partially verified our conjectures. Here, we give a full proof of the conjectures in [36] for the moduli space of K3's which are double covers of a smooth quadric surface. The main new tool here is VGIT for (2,4) complete intersection curves.

Kähler–Einstein Metrics on Smooth Fano Symmetric Varieties with Picard Number One

Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces, and they are interesting examples of spherical varieties. We prove that all smooth Fano symmetric varieties with Picard number one admit Kähler–Einstein metrics by using a combinatorial criterion for K-stability of Fano spherical varieties obtained by Delcroix. For this purpose, we present their algebraic moment polytopes and compute the barycenter of each moment polytope with respect to the Duistermaat–Heckman measure.

Moduli spaces of symmetric cubic fourfolds and locally symmetric varieties

In this paper we realize the moduli spaces of cubic fourfolds with specified automorphism groups as arithmetic quotients of complex hyperbolic balls or type IV symmetric domains, and study their compactifications. Our results mainly depend on the well-known works about moduli space of cubic fourfolds, including the global Torelli theorem proved by Voisin ([Voi86]) and the characterization of the image of the period map, which is given by Laza ([Laz09, Laz10]) and Looijenga ([Loo09]) independently. The key input for our study of compactifications is the functoriality of Looijenga compactifications, which we formulate in the appendix (section A). The appendix can also be applied to study the moduli spaces of singular K3 surfaces and cubic fourfolds, which will appear in a subsequent paper.

Type D quiver representation varieties, double Grassmannians, and symmetric varieties

We unify aspects of the equivariant geometry of type D quiver representation varieties, double Grassmannians, and symmetric varieties GL(a+b)/GL(a)×GL(b); in particular we translate results about singularities of orbit closures, combinatorics of orbit closure containment, and torus equivariant K-theory between these three families. These results are all obtained from our generalization of a construction of Zelevinsky for type A quivers to the type D setting. More precisely, we give explicit embeddings with nice properties of homogeneous fiber bundles over type D quiver representation varieties into these symmetric varieties.

The Bruhat order on Hermitian symmetric varieties and on abelian nilradicals

Let G be a simple algebraic group and P a parabolic subgroup of G with abelian unipotent radical P^u , and let B be a Borel subgroup of G contained in P . Let \mathfrak {p^u} be the Lie algebra of P^u and L a Levi factor of P . Then L is a Hermitian symmetric subgroup of G and B acts with finitely many orbits both on \mathfrak {p^u} and on G/L . In this paper we study the Bruhat order of the B -orbits in \mathfrak {p^u} and in G/L , proving respectively a conjecture of Panyushev and a conjecture of Richardson and Ryan.

Shimura varieties at level and Galois representations

We show that the compactly supported cohomology of certain $\text{U}(n,n)$- or $\text{Sp}(2n)$-Shimura varieties with $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$-level vanishes above the middle degree. The only assumption is that we work over a CM field $F$ in which the prime $p$ splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for $\text{GL}_{n}/F$. More precisely, we use the vanishing result for Shimura varieties to eliminate the nilpotent ideal in the construction of these Galois representations. This strengthens recent results of Scholze [On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), 945–1066; MR 3418533] and Newton–Thorne [Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum Math. Sigma 4 (2016), e21; MR 3528275].

Notes on diagonals of the product and symmetric variety of a surface

Let X be a smooth quasi-projective algebraic surface and let Δn be the big diagonal in the product variety Xn. We study cohomological properties of the ideal sheaves IΔnk and their invariants (IΔnk)Sn by the symmetric group, seen as ideal sheaves over the symmetric variety SnX. In particular we obtain resolutions of the sheaves of invariants (IΔn)Sn for n=3,4 in terms of invariants of sheaves over Xn whose cohomology is easy to calculate. Moreover, we relate, via the Bridgeland-King-Reid equivalence, powers of determinant line bundles over the Hilbert scheme to powers of ideals of the big diagonal Δn. We deduce applications to the cohomology of double powers of determinant line bundles over the Hilbert scheme with 3 and 4 points and we give universal formulas for their Euler-Poincaré characteristic. Finally, we obtain upper bounds for the regularity of the sheaves IΔnk over Xn with respect to very ample line bundles of the form L⊠⋯⊠L and of their sheaves of invariants (IΔnk)Sn on the symmetric variety SnX with respect to very ample line bundles of the form DL.

A strong hyperbolicity property of locally symmetric varieties

We show that all subvarieties of a quotient of a bounded symmetric domain by a sufficiently small arithmetic discrete group of automorphisms are of general type. This result corresponds through the Green-Griffiths-Lang's conjecture to a well-known result of Nadel.

Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety

Let $X$ be a holomorphic symplectic variety with a torus $\mathsf{T}$ action and a finite fixed point set of cardinality $k$. We assume that elliptic stable envelope exists for $X$. Let $A_{I,J}= \operatorname{Stab}(J)|_{I}$ be the $k\times k$ matrix of restrictions of the elliptic stable envelopes of $X$ to the fixed points. The entries of this matrix are theta-functions of two groups of variables: the K\"ahler parameters and equivariant parameters of $X$. We say that two such varieties $X$ and $X'$ are related by the 3d mirror symmetry if the fixed point sets of $X$ and $X'$ have the same cardinality and can be identified so that the restriction matrix of $X$ becomes equal to the restriction matrix of $X'$ after transposition and interchanging the equivariant and K\"ahler parameters of $X$, respectively, with the K\"ahler and equivariant parameters of $X'$. The first examples of pairs of 3d symmetric varieties were constructed in [Rim\'anyi R., Smirnov A., Varchenko A., Zhou Z., arXiv:1902.03677], where the cotangent bundle $T^*\operatorname{Gr}(k,n)$ to a Grassmannian is proved to be a 3d mirror to a Nakajima quiver variety of $A_{n-1}$-type. In this paper we prove that the cotangent bundle of the full flag variety is 3d mirror self-symmetric. That statement in particular leads to nontrivial theta-function identities.

On the deformation rigidity of smooth projective symmetric varieties with Picard number one

Symmetric varieties are normal equivariant open embeddings of symmetric homogeneous spaces and they are interesting examples of spherical varieties. The principal goal of this article is to study the rigidity under Kähler deformations of smooth projective symmetric varieties with Picard number one.

Birational geometry of the moduli space of quartic surfaces

By work of Looijenga and others, one understands the relationship between Geometric Invariant Theory (GIT) and Baily–Borel compactifications for the moduli spaces of degree-$2$ $K3$ surfaces, cubic fourfolds, and a few other related examples. The similar-looking cases of degree-$4$ $K3$ surfaces and double Eisenbud–Popescu–Walter (EPW) sextics turn out to be much more complicated for arithmetic reasons. In this paper, we refine work of Looijenga in order to handle these cases. Specifically, in analogy with the so-called Hassett–Keel program for the moduli space of curves, we study the variation of log canonical models for locally symmetric varieties of Type IV associated to $D$-lattices. In particular, for the $19$-dimensional case, we conjecturally obtain a continuous one-parameter interpolation between the GIT and Baily–Borel compactifications for the moduli of degree-$4$ $K3$ surfaces. The analogous $18$-dimensional case, which corresponds to hyperelliptic degree-$4$ $K3$ surfaces, can be verified by means of Variation of Geometric Invariant Theory (VGIT) quotients.

Stanley symmetric functions for signed involutions

An involution in a Coxeter group has an associated set of involution words, a variation on reduced words. These words are saturated chains in a partial order first considered by Richardson and Springer in their study of symmetric varieties. In the symmetric group, involution words can be enumerated in terms of tableaux using appropriate analogues of the symmetric functions introduced by Stanley to accomplish the same task for reduced words. We adapt this approach to the group of signed permutations. We show that involution words for the longest element in the Coxeter group Cn are in bijection with reduced words for the longest element in An=Sn+1, which are known to be in bijection with standard tableaux of shape (n,n−1,…,2,1).