A polar space S is said to be symplectic if it admits an embedding e in a
projective geometry PG(V) such that the e-image e(S) of S 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 S admits different (non
isomorphic) embeddings, as it is the case (precisely) when S is defined over a
field of characteristic 2.Comment: 20 pages/extensively revise
