Algebraic symplectic analogues of additive quotients

TL;DR

This paper develops algebraic symplectic analogues of additive quotients in non-reductive settings, revealing new geometric phenomena and constructing hyperkahler structures on these quotients.

Contribution

It introduces a framework for non-reductive hyperkahler and algebraic symplectic quotients, especially related to projective geometry, and constructs their hyperkahler structures.

Findings

Algebraic symplectic analogues of additive quotients are constructed.

Hyperkahler structures are obtained on large open subsets of these analogues.

Non-reductive phenomena like non-finite ring generation are observed in simple examples.

Abstract

Motivated by the study of hyperkahler structures in moduli problems and hyperkahler implosion, we initiate the study of non-reductive hyperkahler 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 hyperkahler 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.