The supersingular locus of the Shimura variety of GU(1,<Emphasis Type="Italic">n</Emphasis>−1) II

We complete the study of the supersingular locus M^ss\mathcal{M}^{\mathrm{ss}} in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n−1) over ℚ in the case that p is inert. This was started by the first author in Can. J. Math. 62, 668–720 (2010) where complete results were obtained for n=2,3. The supersingular locus M^ss\mathcal{M}^{\mathrm{ss}} is uniformized by a formal scheme N\mathcal{N} which is a moduli space of so-called unitary p-divisible groups. It depends on the choice of a unitary isocrystal N. We define a stratification of N\mathcal{N} indexed by vertices of the Bruhat-Tits building attached to the reductive group of automorphisms of N. We show that the combinatorial behavior of this stratification is given by the simplicial structure of the building. The closures of the strata (and in particular the irreducible components of N_red\mathcal{N}_{\mathrm{red}}) are identified with (generalized) Deligne-Lusztig varieties. We show that the Bruhat-Tits stratification is a refinement of the Ekedahl-Oort stratification and also relate the Ekedahl-Oort strata to Deligne-Lusztig varieties. We deduce that M^ss\mathcal{M}^{\mathrm{ss}} is locally a complete intersection, that its irreducible components and each Ekedahl-Oort stratum in every irreducible component is isomorphic to a Deligne-Lusztig variety, and give formulas for the number of irreducible components of every Ekedahl-Oort stratum of M^ss\mathcal{M}^{\mathrm{ss}}.     

Cohomology of local systems coming from<Emphasis Type="Italic">p</Emphasis>-adic hyperplane arrangements

For ap-adic hyperplane arrangement in a vector spaceV, we consider a local system of De Shalit on the Bruhat-Tits building ofPGL(V). We express this local system in terms of Orlik-Solomon algebras, and calculate its cohomology in the case where the arrangement is finite.     

Unramified Brauer group of the moduli spaces of PGL r (ℂ)-bundles over curves

Let X be an irreducible smooth complex projective curve of genus g, with g ≥ 2. Let N be a connected component of the moduli space of semistable principal PGL _r (?)-bundles over X; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of N is trivial.     

The zeta functions of complexes from PGL(3): A representation-theoretic approach

The zeta function attached to a finite complex X _Γ arising from the Bruhat-Tits building for PGL₃(F) was studied in [KL], where a closed form expression was obtained by a combinatorial argument. This identity can be rephrased using operators on vertices, edges, and directed chambers of X _Γ. In this paper we re-establish the zeta identity from a different aspect by analyzing the eigenvalues of these operators using representation theory. As a byproduct, we obtain equivalent criteria for a Ramanujan complex in terms of the eigenvalues of the operators on vertices, edges, and directed chambers, respectively.     

On base manifolds of Lagrangian fibrations

We consider base spaces of Lagrangian fibrations from singular symplectic varieties.After defining cohomologically irreducible symplectic varieties,we construct an example of Lagrangian fibration whose base space is isomorphic to a quotient of the projective space.We also prove that the base space of Lagrangian fibration from a cohomologically symplectic variety is isomorphic to the projective space provided that the base space is smooth.     

Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties.In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group G^Mis equal to G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension ? 2. A Galois cover f :X ? Y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pal], it is determined by its building data, i.e. by the branch divisors and by some line bundles on Y, satisfying appropriate compatibility conditions. Natural deformations of an abelian cover are also introduced in [Pal]. In this paper we prove two results about abelian covers:first, that if the building data are sufficiently ample, then the natural deformations surject on the Kuranishi family of X; second, that if the building data are sufficiently ample and generic, then Aut(X)= G.     

Compactification of the Bruhat-Tits building of PGL by seminorms

We construct a compactification of the Bruhat-Tits building X associated to the group PGL(V) which can be identified with the space of homothety classes of seminorms on V endowed with the topology of pointwise convergence. Then we define a continuous map from the projective space to which extends the reduction map from Drinfelds p-adic symmetric domain to the building X.Mathematics Subject Classification (2000): 20E42, 20G25in final form: 4 October 2003     

Geometric and Combinatorial Realizations of Crystal Graphs

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For type A_n⁽¹⁾, we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids. Presented by P. Littleman Mathematics Subject Classifications (2000) Primary 16G10, 17B37. Alistair Savage: This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and was partially conducted by the author for the Clay Mathematics Institute.     

Algebraic degeneracy of non-Archimedean analytic maps

We prove non-Archimedean analogs of results of Noguchi and Winkelmann showing algebraic degeneracy of rigid analytic maps to projective varieties omitting an effective divisor with sufficiently many irreducible components relative to the rank of the group they generate in the Néron-Severi group of the variety.     

The multiplicative group action on singular varieties and Chow varieties

We answer two questions of Carrell on a singular complex projective variety admitting the multiplicative group action, one positively and the other negatively. The results are applied to Chow varieties and we obtain Chow groups of 0-cycles and Lawson homology groups of 1-cycles for Chow varieties. A brief survey on the structure of Chow varieties is included for comparison and completeness. Moreover, we give counterexamples to Shafarevich's problem on the rationality of the irreducible components of Chow varieties.     

Wild automorphisms of varieties with Kodaira dimension 0

An automorphism σ of a projective variety X is said to be wild if σ(Y) ≠ Y for every non-empty subvariety Y \subsetneq X{Y \subsetneq X} . In [1] Z. Reichstein, D. Rogalski, and J.J. Zhang conjectured that if X is an irreducible projective variety admitting a wild automorphism then X is an abelian variety, and proved this conjecture for dim(X) ≤ 2. As a step toward answering this conjecture in higher dimensions we prove a structure theorem for projective varieties of Kodaira dimension 0 admitting wild automorphisms. This essentially reduces the Kodaira dimension 0 case to a study of Calabi-Yau varieties, which we also investigate. In support of this conjecture, we show that there are no wild automorphisms of certain Calabi-Yau varieties.     

Quasi-projective reduction of toric varieties

We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties. For a given toric variety X we first construct a toric quasi–projective reduction. Then we show that X has a quasi–projective reduction if and only if its toric quasi–projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a quasi–projective toric variety admits a categorical quotient in the category of quasi–projective varieties. Received October 29, 1998; in final form December 28, 1998     

Fundamental groups of algebraic fiber spaces

Let be a dominant morphism, where E and B are smooth irreducible complex quasi-projective varieties. Suppose that the general fiber of $f$ is connected. We present an algebro-geometric condition under which the boundary homomorphism is well-defined, and makes the sequence exact. As an application, we calculate the fundamental group of the complement to the dual hypersurface of a smooth projective curve. Received: October 3, 2001     

Stringy E-functions of varieties with A-D-E singularities

The stringy E-function for normal irreducible complex varieties with at worst log terminal singularities was introduced by Batyrev. It is defined by data from a log resolution. If the variety is projective and Gorenstein and the stringy E-function is a polynomial, Batyrev also defined the stringy Hodge numbers as a generalization of the Hodge numbers of nonsingular projective varieties, and conjectured that they are nonnegative. We compute explicit formulae for the contribution of an A-D-E singularity to the stringy E-function in arbitrary dimension. With these results we can say when the stringy E-function of a variety with such singularities is a polynomial and in that case we prove that the stringy Hodge numbers are nonnegative. Research Assistant of the Fund for Scientific Research - Flanders (Belgium) (F.W.O.),     

Order of elements in the groups related to the general linear group

For a natural number n and a prime power q the general, special, projective general and projective special linear groups are denoted by GL_n(q), SL_n(q), PGL_n(q) and PSL_n(q), respectively. Using conjugacy classes of elements in GL_n(q) in terms of irreducible polynomials over the finite field GF(q) we demonstrate how the set of order elements in GL_n(q) can be obtained. This will help to find the order of elements in the groups SL_n(q), PGL_n(q) and PSL_n(q). We also show an upper bound for the order of elements in SL_n(q).     

Cohomology of discrete groups in harmonic cochains on buildings

Modules of harmonic cochains on the Bruhat-Tits building of the projective general linear group over ap-adic field were defined by one of the authors, and were shown to represent the cohomology of Drinfel’d’sp-adic symmetric domain. Here we define certain non-trivial natural extensions of these modules and study their properties. In particular, for a quotient of Drinfel’d’s space by a discrete cocompact group, we are able to define maps between consecutive graded pieces of its de Rham cohomology, which we show to be (essentially) isomorphisms. We believe that these maps are graded versions of the Hyodo-Kato monodromy operatorN.