Abelian simply transitive affine groups of symplectic type

Abstract

We construct a model space C(\gsp(\bR^{2n})) for the variety of Abelian simply transitive groups of affine transformations of type {\rm Sp}(\bR^{2n}). The model is stratified and its principal stratum is a Zariski-open subbundle of a natural vector bundle over the Grassmannian of Lagrangian subspaces in \bR^{2n}. \noindent Next we show that every flat special K\"ahler manifold may be constructed locally from a holomorphic function whose third derivatives satisfy some algebraic constraint. In particular global models for flat special K\"ahler manifolds with constant cubic form correspond to a subvariety of C(\gsp(\bR^{2n})).Comment: corrected typos, updated reference