There is an AlwaysOn Primary replica configured with sync commit automatic failover secondary replica. If anyone of the databases of availability group is down/corrupted, will the failover process be initiated?
Could anybody please help me with this?
Broadly speaking, yes. There's a lot more to consider here with AlwaysOn and clustering though. It isn't for the faint of heart. Additionally, Down and Corrupted are two different things.
If there is corruption on the primary, specifically page corruption, then automatic page repair will attempt to restore the page from the secondary replica and visa versa. You can learn more about that here, in the Microsoft documentation. It's important to focus in on File header pages, the database boot page, and GAP, SGAM, PFS pages (allocation pages) can not be automatically repaired. If you have database corruption, you have a big problem. Call Microsoft immediately... and perhaps follow this guide for what to do when you have corruption.
Since you are using synchronous commit, which is required for automatic fail-over, it will automatically fail over if there is a a secondary replica that is in a healthy synchronization state. This prevents data loss, which is what happens in a manual fail over in asynchronous commit configuration, and manual fail-overs in synchronous set ups where the databases aren't synchronized.
An automatic failover occurs in response to a failure that causes a synchronized secondary replica to transition to the primary role (with guaranteed data protection). When the former primary replica becomes available, it transitions to the secondary role. Automatic failover requires that both the primary replica and the target secondary replica are running under synchronous-commit mode with the failover mode set to "Automatic". In addition, the secondary replica must already be synchronized, have WSFC quorum, and meet the conditions specified by the flexible failover policyof the availability group.