Homogeneous varieties under split solvable algebraic groups

We present a modern proof of a theorem of Rosenlicht, asserting that every variety as in the title is isomorphic to a product of affine lines and punctured affine lines.     

Derivation Algebras of Toric Varieties

We study the Lie algebra of derivations of the coordinate ring of affine toric varieties defined by simplicial affine semigroups and prove the following results:Such toric varieties are uniquely determined by their Lie algebra if they are supposed to be Cohen–Macaulay of dimension 2 or Gorenstein of dimension =1.In the Cohen–Macaulay case, every automorphism of the Lie algebra is induced from a unique automorphism of the variety.Every derivation of the Lie algebra is inner.     

Almost all minimal idempotent varieties are congruence modular

We show that, up to term equivalence, the only minimal idempotent varieties that are not congruence modular are the variety of sets and the variety of semilattices. From this it follows that a minimal idempotent variety that is not congruence distributive is term equivalent to the variety of sets, the variety of semilattices, or a variety of affine modules over a simple ring. Received March 29, 1999; accepted in final form February 8, 2000.     

Affine complete varieties are congruence distributive

An algebra is affine complete iff its polynomial operations are the same as all the operations over its universe that are compatible with all its congruences. A variety is affine complete iff all its algebras are. We prove that every affine complete variety is congruence distributive, and give a useful characterization of all arithmetical, affine complete varieties of countable type. We show that affine complete varieties with finite residual bound have enough injectives. We also construct an example of an affine complete variety without finite residual bound.? We prove several results concerning residually finite varieties whose finite algebras are congruence distributive, while leaving open the question whether every such variety must be congruence distributive. Received February 28, 1997; accepted in final form December 9, 1997.     

Canonical decomposition of projective and affine killing vectors on the tangent bundle

For an affine connection on the tangent bundle T(M) obtained by lifting an affine connection on M, the structure of vector fields on T(M) which generate local one-parameter groups of projective and affine collineations is described. On the T(M) of a complete irreducible Riemann manifold, every projective collineation is affine. On the T(M) of a projectively Euclidean space, every affine collineation preserves the fibration of T(M), and on the T(M) of a projectively non-Duclidean space which is maximally homogeneous (in the sense of affine collineations) there exist affine collineations permuting the fibers of T(M).Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 247–258, February, 1976.     

Wavelet frames for (not necessarily reducing) affine subspaces II: The structure of affine subspaces

This is a continuation of the investigation into the theory of wavelet frames for general affine subspaces. The main focus of this paper is on the structural properties of affine subspaces. We show that every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces, while every reducing subspace (with respect to the dilation and translation operators) is the orthogonal direct sum of two purely non-reducing ones. This result is obtained through considering the basic question as to when the orthogonal complement of an affine subspace in another one is still affine. Motivated by the fundamental question as to whether every affine subspace is singly-generated, and by a recent result that every singly generated purely non-reducing subspace admits a singly generated wavelet frame, we prove that every affine subspace can be decomposed into the direct sum of a singly generated affine subspace and some space of “small size”. As a consequence we establish a connection between the above mentioned two questions.     

Affine Diffusions with Non-Canonical State Space

Multidimensional affine diffusions have been studied in detail for the case of a canonical state space. We present results for general state spaces and provide a complete characterization of all possible affine diffusions with polyhedral and quadratic state space. We give necessary and sufficient conditions on the behavior of drift and diffusion on the boundary of the state space in order to obtain invariance and to prove strong existence and uniqueness.     

Vector Hulls of Affine Spaces and Affine Bundles

Every affine space A can be canonically immersed as a hyperplane into a vector space  , which is called the vector hull of A. This immersion satisfies a universal property for affine functions defined on A. In the same way, every affine map between affine spaces has a linear prolongation to their vector hulls. Though not much known, this construction is greatly clarifying, both for affine geometry and for its applications. The goal of this paper is to perform a thorough study of the vector hull functor and to describe its counterpart in the framework of affine bundles. With this respect, it is shown that the vector hull of some interesting affine bundles, and more specifically some jet bundles, can be identified with certain vector bundles.      

Uniform Zariski’s theorem on fundamental groups

The Zariski theorem says that for every hypersurface in a complex projective (resp. affine) space and for every generic plane in the projective (resp. affine) space the natural embedding generates an isomorphism of the fundamental groups of the complements to the hypersurface in the plane and in the space. If a family of hypersurfaces depends algebraically on parameters then it is not true in general that there exists a plane such that the natural embedding generates an isomorphism of the fundamental groups of the complements to each hypersurface from this family in the plane and in the space. But we show that in the affine case such a plane exists after a polynomial coordinate substitution. The research was partially supported by an NSA grant.     

On a construction of affine Grassmannians and spine spaces

In this paper we introduce the notion of spine space, generalizing the notion of affine grassmannians. We describe the set of strong subspaces of a spine space (Thm. 2.6, 2.7), construct the horizon of a spine space, and study the automorphisms of a spine space. Received 20 October 1999; revised 15 March 2000.     

Spherical functions on Euclidean space

We study spherical functions on Euclidean spaces from the viewpoint of Riemannian symmetric spaces. Here the Euclidean space En=G/K where G is the semidirect product Rn⋅K of the translation group with a closed subgroup K of the orthogonal group O(n). We give exact parameterizations of the space of (G,K)—spherical functions by a certain affine algebraic variety, and of the positive definite ones by a real form of that variety. We give exact formulae for the spherical functions in the case where K is transitive on the unit sphere in En.     

Free

It has been conjectured that every free algebraic action of the additive group of complex numbers on complex affine three space is conjugate to a global translation. The main result lends support to this conjecture by showing that the morphism to the variety defined by the ring of invariants of the associated action on the coordinate ring is smooth. As a consequence, the graph morphism is an open immersion, and simple proofs of certain cases of the conjecture are obtained.

     

Equivariant deformations of the affine multicone over a flag variety

We prove that the invariant Hilbert scheme parameterising the equivariant deformations of the affine multicone over a flag variety is, under certain hypotheses, an affine space. More specifically, we obtain that the isomorphism classes of equivariant deformations of such a multicone are in correspondence with the orbits of a well-determined wonderful variety.     

Blocking-sets in infinite projective and affine spaces

In every projective or affine space of infinite order there exist blockingsets. Moreover, we prove that every bijective map which preserves blockingsets of a fixed level 1 is a collineation.     

On exact actions of unipotent groups

Any affine variety with a d-exact action of a unipotent group can be embedded in an affine space preserving d-exactness. Furthermore, we can find such an ambient space which has some other good properties. The key idea of the proof is describing the property “d-exact” by means of inequalities.     

Ornament Groups on a Minkowski Plane

We are engaged in classifying up to isomorphism of discrete subgroups of an affine transformation group on a plane (a two-dimensional space) preserving the Minkowski metric. It is proved that, for subgroups that do not coincide with Euclidean ones, the orbit of almost every point is everywhere dense.     

Toric degenerations of spherical varieties

We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program.     

Toric degenerations of spherical varieties

We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program.     

Quasicrystallographic groups on Minkowski spaces

We generalize the quasicrystallographic groups in the sense of Novikov and Veselov from Euclidean spaces to pseudo-Euclidean and affine spaces. We prove that the quasicrystallographic groups on Minkowski spaces whose rotation groups satisfy an additional assumption are projections of crystallographic groups on pseudo-Euclidean spaces. An example shows that the assumption cannot be dropped. We prove that each quasicrystallographic group is a projection of a crystallographic group on an affine space.     

On locally trivial G a -actions

If the additive group of complex numbers acts algebraically on a normal affine variety, then the associated ring of invariants need not be finitely generated, but is an ideal transform of some normal affine algebra (Theorem 1). We investigate such normal affine algebras in the case of a locally trivial action on a factorial variety. If the variety is a complex affine space and the ring of invariants is isomorphic to a polynomial ring, then the action is conjugate to a translation (Theorem 3). Equivalently, ifC ⁿ , is the total space for a principalG _a -bundle over some open subset ofC ^n–1 then the bundle is trivial. On the other hand, there is a locally trivialG _a -action on a normal affine variety with nonfinitely generated ring of invariants (Theorem 2).Supported in part by NSA Grant No. MDA904-96-1-0069