Well, baitbegger, there are some turners who would tell you to toss it into the woodstove or fireplace, and find another piece of wood that isn't rotten in the center. Ruth Niles sells a t-shirt with a John Jordan saying, "Life's too short to turn crappy wood."
Then there are others, who don't have a continuous stream of fresh, green turnable wood, who will do what they can to save the wood they have, because it's in limited supply.
I've been in both camps, personally. But if you really need to turn that particular piece of cherry with a rotten core, and need to stabilize the core, there are a few things that you might try. If the blank isn't too big, CA glue is one choice. I say not too big, because if the blank and the rot is a large chunk, you could go through A LOT of CA glue, adding up to a lot of money.
Another option is to use a 50/50 mix of white glue and water mix; soak the blank before turning for a few days, then take it out and let it dry completely. The rotten part should stiffen up from the glue/water mix and allow you to turn it.
Another option, depending on where the rot is, woudl be some of the commercially available punky wood fixes, like PEG and Pentacryl. I've never used either of these, but from what I recall in reading past posts, the use of these chemicals may affect the type of finish that you may use at the end; i.e., some finishes may not take.
Hopefully, others may chime in with their personal experience/opinion.