LCD Glass driving waveform
That's a static drive display, I believe. 

You must keep the (average) DC voltage across the display (common to any segment) to less than some small amount like 50mV (read the specifications). 

What you generally want to do is drive the common with a square wave at, say, 60Hz, and drive each segment with a square wave that is either in phase or out of phase with the common. You can do that with a microcontroller, with an FPGA or with discrete logic. Ideally a microcontroller with an LCD peripheral on board, but it's not necessary if you have enough I/O pins. 

It might be possible to tie the common to Vcc/2 and either drive the segment with a square wave or make it high-Z, but the capacitive coupling could make the segment partly on in the second case, and (in)accuracy of the divider might result in some DC voltage across the segment. The latter is probably okay if you use 1% resistors and the same supply as the driver.