Keep the Mail window from opening when mail arrives

Yes I should have spelled it out. If one of your email accounts is yourname@gmail.com or @googlemail.com you will have this problem until Google and Apple find a solution. Or so it seems from many similar threads.

From a cursory read of the forum it seems not everyone with Google has the problem, but everyone with the problem has google.

Does this happen if Mail isn't running?

Is it google mail?

But who is the Mail provider?

That's odd.

FWIW, I have two gmail accounts that I use in Apple Mail, ,one as POP, and the other as IMAP, and I do not see the problem.