We consider the family of polynomials in $\mathbf{C}[x,y,z]$ of the form $x^{2}y-z^2-xq(x,z)$. Two such polynomials $P_1$ and $P_2$ are equivalent if there is an automorphism ${\varphi }^{*}$ of $\mathbf{C}[x,y,z]$ such that ${\varphi }^{*}\left(P_1\right)=P_2$. We give a complete classification of the equivalence classes of these polynomials in the algebraic and analytic category. As a consequence, we find the following results. There are explicit examples of inequivalent polynomials $P_1$ and $P_2$ such that the zero set of $P_1+c$ is isomorphic to the zero set of $P_2+c$ for all $c\in \mathbf{C}$. There exist polynomials which are algebraically...

De Concini and Procesi have defined the wonderful compactification $\overline{X}$ of a symmetric space $X=G/G^{\sigma }$ where $G$ is a complex semisimple adjoint group and $G^{\sigma }$ the subgroup of fixed points of $G$ by an involution $\sigma$. It is a closed subvariety of a Grassmannian of the Lie algebra $𝔤$ of $G$. In this paper we prove that, when the rank of $X$ is equal to the rank of $G$, the variety is defined by linear equations. The set of equations expresses the fact that the invariant alternate trilinear form $w$ on $𝔤$ vanishes on the $(-1)$-eigenspace...

Let $X$ be an affine conical factorial variety over an algebraically closed field of characteristic zero. We consider equidimensional and stable algebraic actions of an algebraic torus on $X$ compatible with the conical structure. We show that such actions are cofree and the nullcones of $X$ associated with them are complete intersections.

Given a complex manifold M equipped with an action of a group G, and a holomorphic principal H–bundle EH on M, we introduce the notion of a connection on EH along the action of G, which is called a G–connection. We show some relationship between the condition that EH admits a G–equivariant structure and the condition that EH admits a (flat) G–connection. The cases of bundles on homogeneous spaces and smooth toric varieties are discussed.

Soit $S$ un schéma arithmétique de dimension $1$, c’est-à-dire le spectre de l’anneau des entiers d’un corps de nombres ou une courbe algébrique, lisse, irréductible, définie sur un corps fini ou algébriquement clos. Nous associons à un $S$-espace homogène (à gauche) $X$ d’un groupe réductif $G$ dont l’isotropie est aussi un groupe réductif $H$ une classe caractéristique qui, dans le cas où $H$ est semi-simple, vit dans un $H^3$ de $S$ à valeurs dans le noyau du revêtement universel d’une $S$-forme de $H$. Cette classe...

There is a well-known procedure -induction- for extending an action of a subgroup H of a Lie group G on a topological space X to an action of G on an associated space. Induction can also extend a smooth action of a subgroup H of a Lie group G on a manifold M to a smooth action of G on an associated manifold. In this paper elementary methods are used to show that induction also works in the category of (nonsingular) real algebraic varieties and regular or entire maps if G is a compact abelian Lie...