It is definitely possible, some kW is more then enough to make a very spectacular gas discharge, but you really need to know what you are doing (High Voltage Engineering and Safety aspects).
If this is your first power poject, look for something else, unless you have somebody near you that can guide you during the design and construction process. You may pay an error with severe injury or even with your life.
Regarding rectification, it will not "step down" the voltage. If you use a bridge rectifier you will get a no load voltage very likely around 330V (without galvanic isolation). This 330V will very likely be fed to a full bridge circuit (so that you still can use MOSFET instead of IGBT). As arcs show negative impedance/resistance, you need a current limiting scheme (similar as used in a "ballast" for fluorescent lighting).