Characterizations of symplectic polar spaces

A polar space ℐ is called symplectic if it admits a projective embedding ϵ: ℐ → PG(V) such that the image ϵ( ℐ) of ℐ by ϵ is defined by an alternating form of V. In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especially when ℐ admits different (non-isomorphic) embeddings, as it is the case when ℐ is defined over a field of characteristic 2.