A remark on homogeneous affine varieties and related matters

In this note we give an example of affine quotient $G/H$ where $G$ is an affine algebraic group over an algebraically closed field of characteristic 0 and $H$ is a unipotent subgroup not contained in the unipotent radical of $G$. Some remarks about symmetric algebras of centralizers of nilpotent elements in simple Lie algebras, in particular cases, are added.Comment: 23 pages - from page 15 to 22 are presented programs for calculatio

Enumerative invariants of stongly semipositive real symplectic six-manifolds

Following the approach of Gromov and Witten, we define invariants under deformation of stongly semipositive real symplectic six-manifolds. These invariants provide lower bounds in real enumerative geometry, namely for the number of real rational $J$-holomorphic curves which realize a given homology class and pass through a given real configuration of points.Comment: 18 pages, 1 figure, main result restricted to dimension six, see Remark 5.

Invariants of real symplectic 4-manifolds out of reducible and cuspidal curves

We construct invariants under deformation of real symplectic 4-manifolds. These invariants are obtained by counting three different kinds of real rational J-holomorphic curves which realize a given homology class and pass through a given real configuration of (the appropriate number of) points. These curves are cuspidal curves, reducible curves and curves with a prescribed tangent line at some real point of the configuration. They are counted with respect to some sign defined by the parity of their number of isolated real double points and in the case of reducible curves, with respect to some mutiplicity. In the case of the complex projective plane equipped with its standard symplectic form and real structure, these invariants coincide with the ones previously constructed in math.AG/0303145. This leads to a relation between the count of real rational J-holomorphic curves done in math.AG/0303145 and the count of real rational reducible J-holomorphic curves presented here.Comment: 27 pages, 5 figure

On a variety related to the commuting variety of a reductive Lie algebra

For a reductive Lie algbera over an algbraically closed field of charasteristic zero,we consider a borel subgroup $B$ of its adjoint group, a Cartan subalgebra contained inthe Lie algebra of $B$ and the closure $X$ of its orbit under $B$ in the Grassmannian.The variety $X$ plays an important role in the study of the commuting variety. In thisnote, we prove that $X$ is Gorenstein with rational singularities