# On the complex structure of symplectic quotients

## TL;DR

This paper investigates the local invariance of complex structures on symplectic quotients of Hamiltonian Kähler manifolds as parameters vary, using symplectic implosion and GIT quotient techniques.

## Contribution

It demonstrates the local invariance of complex structures on symplectic quotients under parameter variation, employing two distinct geometric approaches.

## Key findings

- Complex structure on symplectic quotients is locally invariant with parameter changes.
- Uses symplectic implosion to analyze complex structure behavior.
- Investigates variation of GIT quotients to support invariance results.

## Abstract

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.