Stable Reductive Varieties I: Affine Varieties

0. Introduction 1 1. Main definitions and results 3 2. General criteria 6 2.1. Seminormality and connectedness of isotropy groups 6 2.2. Finiteness of number of orbits and group–like condition 9 3. Orbits in stable reductive varieties 11 3.1. Isotropy groups 11 3.2. Algebras of regular functions 14 4. Reductive varieties 18 4.1. Classification 18 4.2. Associated stable toric varieties 20 5. Stable reductive varieties 24 6. Self–adjoint stable reductive semigroups 26 7. Families 32 7.1. General remarks 32 7.2. Moduli of embedded stable reductive varieties 35 7.3. Local isotriviality 38 7.4. One–parameter degenerations 39 7.5. The Vinberg family 42 7.6. Local structure of families of reductive varieties 45 References 45