Yes, Hyper-V supports many Linux distros, architectures, and versions, and that support has been improved in WS2012 R2 Hyper-V.
It’s no secret that there were some changes to the Linux Integration Services that are built into the Linux kernel. Those changes were intended for and supported on WS2012 R2 Hyper-V (not WS2012 Hyper-V). Those two changes are:
- Dynamic Memory: Linux guest OSs can use the balloon driver to have the exact same support for Dynamic Memory as Windows (add and remove). Bear in mind the constraints of the Linux distro itself. And remember the recommendations of Linux when assigning large amounts of CPU/RAM to a machine. These are Linux recommendations/limits, not Hyper-V ones.
- Online backup: You can now perform a file system freeze in the Linux guest OS to get a file system consistent backup of a Linux guest OS without pausing the VM. Linux does not have VSS (like Windows) so we cannot get application consistency. But this is still a huge step forward. According to Microsoft, WS2012 R2 Hyper-V is now the best way to virtualize and backup Linux; you can use any backup tool that supports Hyper-V to reliably backup your Linux VMs without using some script that does a dumb file copy.
Remember that online VHDX resizing is a host function, so Linux guest OSs support this too. Don’t ask me how to resize Linux partitions or make use of the new free space 
There is also a new video driver for Linux. This gives you a better video experience as with the Windows guest OS, including better mouse support – but hey, real Linux admins don’t click!
To take advantage of these features, make sure you have an up-to-date Linux kernel in your VMs, and your running them on WS2012 R2 Hyper-V.