Algebraic Symplectic Analogues of Additive Quotients

Motivated by the study of hyperkähler structures in moduli problems and hyperkähler implosion, we initiate the study of non-reductive hyperkähler and algebraic symplectic quotients with an eye towards those naturally tied to projective geometry, like cotangent bundles of blow-ups of linear arrangements of projective space. In the absence of a Kempf-Ness theorem for non-reductive quotients, we focus on constructing algebraic symplectic analogues of additive quotients of affine spaces, and obtain hyperkähler structures on large open subsets of these analogues by comparison with reductive analogues. We show that the additive analogue naturally arises as the central fibre in a one-parameter family of isotrivial but non-symplectomorphic varieties coming from the variation of the level set of the moment map. Interesting phenomena only possible in the non-reductive theory, like non-finite generation of rings, already arise in easy examples, but do not substantially complicate the geometry.