Symplectic implosion and non-reductive quotients

There is a close relationship between Mumford’s geometric invariant theory (GIT) in (complex) algebraic geometry and the process of reduction in symplectic geometry. GIT was developed to construct quotients of algebraic varieties by reductive group actions and thus to construct and study moduli spaces [28, 29]. When a moduli space (or a compactification of a moduli space) over C can be constructed as a GIT quotient of a complex projective variety by the action of a complex reductive group G, then it can be identified with a symplectic reduction by a maximal compact subgroup K of G and techniques from symplectic geometry can be used to study its topology (for example [2, 16, 17, 20, 21, 22, 23]). Many moduli spaces arise as quotients of algebraic group actions, but the groups concerned are not necessarily reductive, so that classical GIT does not apply and different methods need to be used to construct the quotients (cf. e.g. [19, 25]). Nonetheless, in suitable situations GIT can be generalised to allow us to construct GIT-like quotients (and compactified quotients) for these actions [7, 8, 24]. This paper describes some ways in which such non-reductive compactified quotients can be studied using symplectic techniques closely related to the ‘symplectic implosion’ construction of Guillemin, Jeffrey and Sjamaar [15].