Kempf–Ness type theorems and Nahm equations

Abstract

We prove a version of the affine Kempf-Ness theorem for nonalgebraic symplectic structures and shifted moment maps, and use it to describe hyperkähler quotients of T * G, where G is a complex reductive group.where k := Lie(K) and ·, · is the standard inner-product on C n . Recall that G is reductive so M //G is an affine variety. Moreover, if µ −1 (0)/K is smooth, then its reduced symplectic form is a Kähler form on M //G. This theorem admits many generalizations and variants; for instance, there are versions for projective manifolds [33, §2] Another important version -which is closer to the spirit of this paper -is when M is affine as above but we shift the moment map (1.1). More precisely, if χ : K → S 1 is a character, then ξ := i dχ ∈ k * is central (i.e. fixed by the coadjoint action), so we can consider the symplectic reduction µ −1 (ξ)/K. Then, King [22, §6] (see also [20]) showed that µ −1 (ξ)/K is homeomorphic to the twisted GIT quotient M // χ G, i.e. the GIT quotient of M by G with respect to the trivial line bundle M × C with the G-action g · (p, z) = (g · p, χ(g)z). In other words,such that u(g · p) = χ(g) n u(p) for all p ∈ M and g ∈ G. Recall that the quasi-projective variety M// χ G is a categorical quotient