Why is not the laptop not starting after a shutdown?

@zongo_boy all of this "not booting up", "charging light is not show" and "power is not going to the motherboard" means you are having issue with the AC power. Check your power supply. Make sure it has the proper voltages (20V). Check to make sure connectors are not bend etc. After that, check the charging port. Clean it of any debris etc, use a few puffs of air to clean it out. Check the DC-in jack of your laptop. Make sure there is no corrosion, debris etc. that prevents it from working. Next you would have to open up your laptop and check for power on the inside of the DC-In jack as well as going to your motherboard.

Download the hardware maintenance manual for your Lenovo from here to work on your laptop. In case all of your test show that your computer gets power to the motherboard, you will have to check the motherboard itself. Looks like your motherboard is a Compal NM-C431. Here is the schematic for the NM C431 YogaC740 14IML 15IML Page 3 gives you the information on what rails to check and what voltages should be present.