tudio Monitor flashing off and on after macOS 27.0.1 update

It is difficult to troubleshoot without necessary information. Your tagline lists an AppleTV but no Mac. What model of Mac do you have? What display are you talking about? Apple makes does not make a "Studio Monitor" but they multiple products called Studio Display:

Identify your Apple display model - Apple Support

On the display side, assuming you have some model of Studio Display, hard reset it. Because the Studio Display runs its own internal iOS-based chip, it doesn't reboot when you restart your Mac.

• Action: Unplug the Studio Monitor's power cable directly from the wall, wait at least 60 seconds to fully drain its capacitors, and plug it back in

On the Mac side, have you rebooted your Mac? If not, that's the first step. If that does not help, then see if the problem is present when you boot in safe mode. That process performs some system cleanup and clears caches, and also disables 3rd party extensions that may cause problems.

Start up your Mac in safe mode - Apple Support

If that does not correct the issue, test in a new user account. A newly-created user account will have all settings at default, and if the issue does not occur in that new account then it provides guidance on what may need to be changed to correct the issue.

Add a user or group on Mac - Apple Support

Once the testing is complete, the new account can easily be deleted.

Delete a user or group on Mac – Apple Support

Those steps will help to isolate the problem.

It might help to download and run EtreCheck, written by a long-time forum member. The free version is fine. Post the full report using the Additional Text button. No personal information is revealed.