The dimension of some affine Deligne-Lusztig varieties

We prove Rapoport's dimension conjecture for affine Deligne-Lusztig varieties for GLh and superbasic b. From this case the general dimension formula for affine Deligne-Lusztig varieties for special maximal compact subgroups of split groups follows, as was shown in a recent paper by Görtz, Haines, Kottwitz, and Reuman.     

Cancellation over affine varieties

It is proved that if X is a smooth affine curve over a field F of characteristic , then the group SK₁(X)/ SK₁(X) is isomorphic to a subgroup of the étale cohomology group H _et ³ (X, _e ² ) and if F is algebraically closed, then SK₁(X) is a uniquely divisible group. The following cancellation theorem is obtained from results about SK₁ for curves: If X is a normal affine variety of dimension n over a field F, and if char F > n and C.d._e(F)1 for any prime >/n then any stably trivial vector bundle of rank n over X is trivial.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 187–195, 1982.     

Combinatorics in affine flag varieties

The Littelmann path model gives a realization of the crystals of integrable representations of symmetrizable Kac–Moody Lie algebras. Recent work of Gaussent and Littelmann [S. Gaussent, P. Littelmann, LS galleries, the path model, and MV cycles, Duke Math. J. 127 (1) (2005) 35–88] and others [A. Braverman, D. Gaitsgory, Crystals via the affine Grassmannian, Duke Math. J. 107 (3) (2001) 561–575; S. Gaussent, G. Rousseau, Kac–Moody groups, hovels and Littelmann's paths, preprint, arXiv: math.GR/0703639, 2007] has demonstrated a connection between this model and the geometry of the loop Grassmanian. The alcove walk model is a version of the path model which is intimately connected to the combinatorics of the affine Hecke algebra. In this paper we define a refined alcove walk model which encodes the points of the affine flag variety. We show that this combinatorial indexing naturally indexes the cells in generalized Mirkovi?–Vilonen intersections.     

Eigenvalue completions by affine varieties

In this paper we provide new necessary and sufficient conditions for a general class of eigenvalue completion problems.

     

Classification of smooth affine spherical varieties

Let G be a complex reductive group. A normal G-variety X is called spherical if a Borel subgroup of G has a dense orbit in X. Of particular interest are spherical varieties which are smooth and affine since they form local models for multiplicity free Hamiltonian K-manifolds, K a maximal compact subgroup of G. In this paper, we classify all smooth affine spherical varieties up to coverings, central tori, and -fibrations.     

Smooth affine varieties and complete intersections

This paper consists of two independent parts. First I give a Chern class condition that is sufficient for a smooth surface in affinen-space to be a set-theoretic complete intersection. In the second part I show the existence of a smooth affine fourfold over C which is not a complete intersection in anyA ⁿ although its canonical bundle is trivial.     

Complete intersection points on affine varieties

This paper addresses the following problem: given a commutative ring A, what is the structure of the set of “CI points,” i.e., those maximal ideals generated by dim(A) elements? When A is finitely generated over an algebraically closed field, we conjecture that this set is a countable union of closed subsets of Max(A). When A is regular of dimension ? or 3, we verify this conjecture, as well as an analogous set-theoretic conjecture.     

Vector bundles on real affine varieties

Studiamo i fibrati fortemente algebrici sulle superficie reali aperte e i moduli proiettivi su certi anelli di funzioni razionali reali in tre variabili.     

Parameters of the codes over affine varieties

We derive a lower bound of the generalized Hamming weights of the codes over affine varieties, which are defined by appropriate sequences of rational polynomials over varieties.     

Euler characteristic of certain affine flag varieties

The purpose of this note is to prove, as Lusztig stated, that the Euler characteristic of the variety of Iwahori subalgebras containing a certain nil-elliptic elementn _t istc^l wherel is the rank of the associated finite type Lie algebra. The author's research is supported in part by a National Science Foundation postdoctoral fellowship.     

Fixed point varieties on affine flag manifolds

We study the space of Iwahori subalgebras containing a given element of a semisimple Lie algebra over C((ɛ)). We also define and study a map from nilpotent orbits in a semisimple Lie algebra over C to conjugacy classes in the Weyl group. Both authors were supported in part by the National Science Foundation.     

A generalised Skolem-Mahler-Lech theorem for affine varieties

We give a corrected and strengthened statement and proof ofthe ‘p-adic analytic arc lemma’ in a paper of theauthor (J. London Math. Soc. (2) 73 (2006) 367–379). Weshow that the analytic arc is guaranteed to exist for p 5 andgive a counterexample showing that this sometimes cannot bedone when p = 2. Footnotes 2000 Mathematics Subject Classification 11D45 (primary), 14R10,11D88 (secondary). Received September 16, 2007; revised January 22, 2008; published online March 30, 2008.     

Dimensions of Cantor and post varieties

A dimension of a finitely based variety V of algebras is the greatest length of a basis (that is, an independent generating set) for the SC-theory SC(V) with the strong Mal'tsev conditions satisfied in V. A dimension is said to be infinite if the lengths of bases in SC(V) are unbounded. We prove that the dimension of a Cantor variety C_m,n in the general form, i.e., with n>m≥1, is infinite. The algorithm of constructing a basis of any given length in SC(C_m,n) is presented. By contrast, any Post variety P_n generated by a primal algebra of order n>1 is shown to have a finite dimension not exceeding the number of maximal subalgebras in the iterative Post algebra over the set {0,1,…,n−1}. Specifically, the dimension of the variety of Boolean algebras is at most four. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 359–369, May–June, 1996.     

Smooth and palindromic Schubert varieties in affine Grassmannians

We completely determine the smooth and palindromic Schubert varieties in affine Grassmannians, in all Lie types. We show that an affine Schubert variety is smooth if and only if it is a closed parabolic orbit. In particular, there are only finitely many smooth affine Schubert varieties in a given Lie type. An affine Schubert variety is palindromic if and only if it is a closed parabolic orbit, a chain, one of an infinite family of “spiral” varieties in type A, or a certain 9-dimensional singular variety in type B ₃. In particular, except in type A there are only finitely many palindromic affine Schubert varieties in a fixed Lie type. Moreover, in types D and E an affine Schubert variety is smooth if and only if it is palindromic; in all other types there are singular palindromics. The proofs are for the most part combinatorial. The main tool is a variant of Mozes’ numbers game, which we use to analyze the Bruhat order on the coroot lattice. In the proof of the smoothness theorem we also use Chevalley’s cup product formula.     

Twisted loop groups and their affine flag varieties

We develop a theory of affine flag varieties and of their Schubert varieties for reductive groups over a Laurent power series local field k((t)) with k a perfect field. This can be viewed as a generalization of the theory of affine flag varieties for loop groups to a “twisted case”; a consequence of our results is that our construction also includes the flag varieties for Kac–Moody Lie algebras of affine type. We also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k((t)) and some applications to local models of Shimura varieties.     

Torsion 0-cycles with modulus on affine varieties

In this note, we show that given a smooth affine variety X over an algebraically closed field k and an effective (possibly non-reduced) Cartier divisor on it, the Chow group of zero cycles with modulus ${\mathrm{CH}}_0(X|D)$ is torsion free, except possibly for p-torsion if the characteristic of k is $p>0$. This generalizes to the relative setting classical results of Rojtman (for X smooth) and of Levine (for X singular).