I don't exactly know what your problem is, but the Jazz alternator should be producing between 13.9 and 14.7 volts when the engine is running and the battery needs some charge to run lights, fans etc.
However, the Jazz is also "smart" in that it has an additional circuit inline that will decide whether the battery needs current and if it senses that it doesn't need any, then it switches off the feed used to excite the alternator windings and the battery voltage will drop to around 12.5 volts.
I didn't really believe this but checked it by buyng a cheap USB adaptor that plugs into the cigar lighter that also shows the battery voltage on a small led.
Sure enough, every so often the battery voltage will drop from around 14 volts down to around 12.5 volts as this fuel saving device cuts in.
Turning on the headlights will immediately turn the alternator feed back on as big current flows out of the battery.
If you don't see around 14+ volts just after starting, then something is wrong .... alternator not spinning or faulty or the fuel saving load shedding device has failed in the no charge position.