TOday I received the following email:

Dear Manager,
We are a Network Service Company which is the domain name registration center in Shanghai, China. On August,19th,2010, We received HUATAI Company's application that they are registering the name "mymusictheory" as their Internet Trademark and "mymusictheory.cn","mymusictheory.com.cn" ,"mymusictheory.asia"domain names etc.,It is China and ASIA domain names.But after auditing we found the brand name been used by your company. As the domain name registrar in China, it is our duty to notice you, so I am sending you this Email to check.According to the principle in China,your company is the owner of the trademark,In our auditing time we can keep the domain names safe for you firstly, but our audit period is limited, if you object the third party application these domain names and need to protect the brand in china and Asia by yourself, please let the responsible officer contact us as soon as possible. Thank you!
Kind regards

It's all a bit puzzling. Any ideas what I'm supposed to do or what my rights are? Do I have some automatic rights as the owner of the .com address?

It's a scam.

They expect you to rush for your credit card and pay their inflated prices for a hostname with a TLD you don't need and would probably never bother with.

Aha! That sounds about right! Thanks Chris! x

The fact that it involves China and "protecting" you isn't a good sign. Just mark it as spam and be on your way.

Originally Posted by GCM

The fact that it involves China and "protecting" you isn't a good sign. Just mark it as spam and be on your way.

Very well said. They can refer you to their a bit strange government once you have any problems with them. So please do not risk.