Complexity of the isomorphism relation for metric structures

For a given sentence s in the infinitary logic, one can consider the Borel space M of all countable models satisfying s, and the isomorphism equivalence relation E on M. Hjorth and Kechris found a natural characterization in model-theoretic terms of the situations when E is smooth or essentially countable. We generalize their results to the context of Polish metric structures, and we give a short proof of a theorem of Kechris to the effect that orbit equivalence relations induced by continuous actions of locally compact groups are essentially countable. This is joint work in progress with Todor Tsankov.