Complete algebraic vector fields on affine surfaces

Let \AAutH (X) be the subgroup of the group \AutH (X) of holomorphic automorphisms of a normal affine algebraic surface $X$ generated by elements of flows associated with complete algebraic vector fields. Our main result is a classification of all normal affine algebraic surfaces $X$ quasi-homogeneous under \AAutH (X) in terms of the dual graphs of the boundaries \bX \setminus X of their SNC-completions \bX.Comment: 44 page

Toric varieties and spherical embeddings over an arbitrary field

We are interested in two classes of varieties with group action, namely toric varieties and spherical embeddings. They are classified by combinatorial objects, called fans in the toric setting, and colored fans in the spherical setting. We characterize those combinatorial objects corresponding to varieties defined over an arbitrary field $k$. Then we provide some situations where toric varieties over $k$ are classified by Galois-stable fans, and spherical embeddings over $k$ by Galois-stable colored fans. Moreover, we construct an example of a smooth toric variety under a 3-dimensional nonsplit torus over $k$ whose fan is Galois-stable but which admits no $k$-form. In the spherical setting, we offer an example of a spherical homogeneous space $X_0$ over \mr of rank 2 under the action of SU(2,1) and a smooth embedding of $X_0$ whose fan is Galois-stable but which admits no \mr-form

An existence theorem for G-structure preserving affine immersions

We prove an existence result for local and global G-structure preserving affine immersions between affine manifolds. Several examples are discussed in the context of Riemannian and semi-Riemannian geometry, including the case of isometric immersions into Lie groups endowed with a left-invariant metric, and the case of isometric immersions into products of space forms.Comment: 31 page