As others have mentioned, the gate voltage can NOT vary beyond the source voltage by more than +-20V in this case, which is by the way a common range.
As such to turn it on you need to apply a voltage greater than the Vgs threshold voltage but not more than than 20V more.
However, stating that is 420V is not quite accurate. When off, the voltage on the source of the high side gates will be near ground, since, presumably, the other end of the load is being pulled low by one of the other channels.
As such, initially you can not exceed 20V relative to ground. Lets say you apply 10V absolute. As the device turns on, you want the source voltage will rapidly rise to 400V, which means the gate voltage needs to rise at the same rate to 410V.
Similarly, when turning the thing off you need to initially drop it to 400V, or more accurately no more than 20V less than the 400V, then have it follow the source voltage as it falls when the load or lower transistor pulls the source down.
Just to add another wrinkle... when initially powered up with all devices deactivated, or when they are all turned off, the sources will be floating at some voltage dictated by the capacitances of the MOSFETS.
I'm sure you are already asking yourself... "How the expletive do I do that!" It is a bit tricky but fortunately there are numerous drivers specifically designed for the purpose that basically provide a gate voltage that is a voltage ADDED to the current source voltage.