Let $K$ be a compact group. For a symplectic quotient $M_{ \lambda}$ of a compact Hamiltonian K\"ahler $K$-manifold, we show that the induced complex structure on $M_{ \lambda}$ is locally invariant when the parameter $\lambda$ varies in $\mathrm{ Lie}(K)^*$. To prove such a result, we take two different approaches: (i) by using the complex geometry properties of the symplectic implosion construction; (ii) by investigating the variation of GIT quotients.
Categories: math.DG math.AG math.SG