Well, for one thing it really depends on the chroming quality.
Chromium has a very high corrosive effect on steel. Really good chrome plating has a series of different metal plating layers with chrome on top. For the best weather resistance it's usually a copper layer, then a nickel layer, finally chrome - sometimes more layers beneath. These under layer's plating thickness must be enough that there's no micro perforations, if the upper layers can make electrical contact through to beneath, there will be galvanic corrosion.
"Show Chrome" may mean super excellent multi layer plating for durability, or mean a really poor process that is shiny as long as it stays dry.
That said chrome is very resistant to corrosion.
The problem with real world use on steel is that chrome plating is great as long as it isn't disturbed. Any dent or scratch that goes through to the steel is really bad news. We've all seen chrome flaking off, that's because the steel is corroding underneath the plating. The galvanic potential between steel and chrome is very high, so it just spreads underneath and that creates a loose layer of plating over the steel.
Aluminum is even worse, the galvanic potential for Cr-Al is close to the maximum on the chart... so chrome plating engine cases is not a good plan for any road bike, for trailer queen show bikes OK.
I don't know if there's any great coating that will protect chrome on a swingarm. It will get dings from road crap flying off the rear tire or flipped up by the front tire. Wax or whatever will coat the surface but once the plating is broken, corrosion beneath it is inevitable.
So just put it on and enjoy the shininess as long as it looks good. It's not much use sitting in a box in your shed, after all.