Minimum connections needed between stm32 MCU and the st-link programmer?
Minimum SWD connections:
- SWCLK
- SWIO
- GND  

Optional:
- Vcc for non-3.3V targets.
- NRST if SWD pins are re-used.

Debug:
- SWO, provides CoreSight SWO debug output.

Of course, the minimum of your chip itself are:
- All Vdd pins, including Vdda, and Vbat if available.
- All Vss pins, including Vssa.
- BOOT0 pin to known state.
- RESET pin to high.