Large affine spaces of symplectic forms
arXiv:2607.14648
The paper classifies affine spaces of symplectic forms on a 2n‑dimensional vector space that attain the maximal possible dimension n(n‑1), reducing the problem to the classification of non‑isotropic quadratic and Hermitian forms over the base field.
Abstract
Let F be a field and V be a 2n-dimensional vector space over F. In a previous article, we have proved that if F has more than 2n-2 elements then the greatest possible dimension for an affine space of symplectic forms on V is n(n-1). Here, under the same cardinality assumption we study the spaces that have the critical dimension n(n-1). In particular, if the characteristic of F is not 2 the classification of these spaces up to congruence is reduced to: (1) the classification of nonisotropic quadratic forms over F, up to equivalence and multiplication with a nonzero scalar; (2) the classification of nonisotropic Hermitian forms over all quadratic extensions of F, up to equivalence and multiplication by . In particular, for quadratically closed fields it is shown that there is exactly one solution up to congruence.
89 pages (including a table of contents)