Makita makes a 12 volt chain saw. In the British market they called it a trimmer/clipper or something like that to avoid being subject to regulations governing "chain saws." Most electric "cordless" chainsaws use 18 volt battery packs. So unlike the Makita, they can't be easily connected to common 13.8 volt automotive-type charging battery maintenance circuits. 18VDC chain saws WILL run on those lower voltage systems but cut at a much slower rate and you'd need to pull their internal battery to avoid that 18VDC verses 13.8 volt conflict. Electric motor designs typically cool by brisk air flow. But if little DC motors designed to run at higher speed are heavily loaded and run slower overheating damage may occur. So both performance and durability issues argue against lower than design-voltage operation.
Some cordless chain saws have VERY short blades. User satisfaction is probably enhanced by discouraging them from undertaking sawing tasks which would quickly exhaust their saw's limited stored power reserves.
Driving 120 volt AC chain saws with inverters is a workable solution. But that introduces AT LEAST one more efficiency dropping step, often two efficiency dropping steps as I'll explain.
Typical low-priced inverters operate well below 90% efficiency and some dip below 80%, converting the rest to waste heat. The most efficient can convert about 94% of input power to output over the most-efficient part of their efficiency curve. Engine electrical systems are typically alternator charged and buffered by a battery sufficiently substantial to spin the engine's starting motor. That starting reserve is enough to sustain most inverter loads during short periods when drawing current exceeds alternator charging capacity.
Consider an extreme case, the tiny Kubota B5100D diesel's 35 watt alternator. Clearly the duty cycle that you could drive a common 560 watt 120 volt electric chain saw with an 85% efficient inverter is very limited. Yes, it will work, but you would need to let the little charging system catch up between saw-running periods. After all, that hypothetical inverter load is (85%/1) x 560 watts = 659 watts. So during sawing, 624 watts (659 - 35 = 624) of that power must come from stored power that little alternator output earlier into the battery at 35 watt charging rate. Making this chain even worse is battery charge/dischage cycle efficiency which is only about 80%. That second inefficiency step increases charge recovery time by another 125%. Taking both inverter inefficiency and battery charging/discharging inefficiency into account, that 560 watt saw load requires 824 watts from the alternator to sustain it. 35 watts/824 watts = a fair 4% duty cycle estimate describing usable sawing time per hour. 96% of the time the little electrical system would be recovering from those high-demand 4% use periods. If you're mowing and once in a while need to cut a branch or 4 inch tree, that duty cycle time is enough to sustain operations. But that low duty cycle time is a "deal breaker" that prevents scheduling a day of sawing.
A large machine with a large alternator probably has such a high duty cycle percentage that you can ignore it. But they can't run 8 hours on a gallon of diesel fuel as my B5100D can. Trade-offs everywhere.
The highest amperage draw electric chain saw of which I know is the 13 amp Milwaukee. It would need a heavy-duty charging system to run it. That little diesel's front or rear PTO could easily drive such a large alternator and a larger inverter making that quiet but powerful Milwaukee chain saw mobile over a large acreage. I can report from my experience with system this that total noise level is less than an electric lawn mower. At diesel's current $4.40/gallon, first estimated fueling-cost rate at high duty cycle use rates is be about $.55/hour. Or you can get out your ear plugs and fire up a howling smoker chain saw.
John