Can't log in or start session

I turn on the computer, it boots ok. I can decrypt the drive ok. That gets me to my log-in screen.

That's where the trouble starts. When I enter the correct password, it almost appears to be starting, because the log-in screen disappears. But then it reappears. (When I enter a bad password, it says 'Invalid password', so we know that's not the issue.)

If I boot into the Recovery Menu and query system status, I see 'LVM state. Physical volumes: not ok (BAD)'

dpkg and fsck do nothing, but that could be to do with the drive being an encrypted one