mate desktop does not start on my ubuntu 21.10 system (lightdm doesn't start)

I just found out, what the reason was looking into the log using:

sudo tail -n 100 /var/log/lightdm/lightdm.log | less

There were two problems, first, the config contained xlocal (probably something is wrong in the installation routine, causing this phenomenon) and the second problem was that the config of the "arctica-greeter" was missing. I fixed the first problem by searching all config files stated by lightdm --show-config (in the source section of the output) for xlocal and then fixed it in /usr/share/lightdm/lightdm.conf.d/90-nvidia.conf. For the second problem, I just installed arctica-greeter with apt-get install arctica-greeter. Now it works :-) Maybe that helps if someone has a similar problem.

[+0.01s] WARNING: Seat type 'xlocal' is deprecated, use 'type=local' instead
[+0.01s] DEBUG: Seat seat0: Loading properties from config section Seat:*
[+0.01s] DEBUG: Seat seat0 has property CanMultiSession=no
[+0.01s] DEBUG: Seat seat0: Starting
[+0.01s] DEBUG: Seat seat0: Creating greeter session
[+0.01s] DEBUG: Seat seat0: Failed to find session configuration arctica-greeter
[+0.01s] DEBUG: Seat seat0: Failed to create greeter session
[+0.01s] DEBUG: Failed to start seat: seat0
[+0.00s] DEBUG: Logging to /var/log/lightdm/lightdm.log
[+0.00s] DEBUG: Starting Light Display Manager 1.30.0, UID=0 PID=1396