From the perspective of the Bit Torrent swarm, your virtual server will appear as a client and the swarm will learn its IP address. Virtual servers rented from IaaS providers usually have static IP addresses, so the provider knows which IP address belongs to which one of their customers.
When someone wants to press legal actions for copyright violation, they will notice that the IP belongs to a block of IP addresses registred by the IaaS provider, and press charges on them first. Their complaint will include the IP address of your server. The provider will then look up who pays for the server with that IP address and forward the complaint to them.
When the provider does not cooperate, they could be made liable for the copyright violation themselves. So you likely can not rely on your IaaS provider to protect your anonymity in case of legal problems.