How to drive a MOSFET from AC wih max efficiency?
The following is somewhat conceptual with details "yet to be dealt with".

The AVERAGE gate current is much lower than the peak.
For a given Capacitance C charged to V volts the energy required is 0.5 x C x V^2. This must be charged and discharged F_PWM times per second so
Power = f/2 x C x V^2.
For C = 1 nF, C = 10V, f = 20 kHz
Power = 20,000/2 x 1E-9 x 10^2  = 1 mW mean
At 10V that's 0.1 mA mean with a CC charger or 0.2 mA mean if you "waste" half the charge energy due to the voltage ramp.

If you fed this from 300 VDC the power is VI = 300 x 0.2 mA = 60 mW.
ie if you use a zener clamped reservoir capacitor and feed it via a resistor from rectified mains peak at 300 VDC and allow suitably more than 0.2 mA 

Here D1 C1 provide an ~= 300 VDC source.
R1 R2 nomunailly float C2 at half V_C! when not clamped to some other poit, but their main function is to form an effective CC (constant current) feed to C2 and to allow C2 D2 to be driven to whatever voltage is required by the driver. Z2 clamps C2 to whatever voltage is desired. 







simulate this circuit &ndash; Schematic created using CircuitLab



Somebody queried your 20 kHz PWM frequency. 
My mean current calculations assume either that this is the "PWM frame rate" with the FET being driven on or off once per frame (= effectively phase control) OR that the frame rate is much lower and that the PWM level can change at most once every 1/20 kHz = 50 uS.
IF your PWM completes a complete 256 level PWM signal in 50 uS with multiple on/off transitions in this period then a much higher gate drive current is needed and this scheme is not suitable.

In practice you can probably use a much lower frame rate