Although it is safe to declare a static ObjectMapper in terms of thread safety, you should be aware that constructing static Object variables in Java is considered bad practice. For more details, see Why are static variables considered evil? (and if you'd like, my answer)
In short, statics should be avoided because the make it difficult to write concise unit tests. For example, with a static final ObjectMapper, you can't swap out the JSON serialization for dummy code or a no-op.
In addition, a static final prevents you from ever reconfiguring ObjectMapper at runtime. You might not envision a reason for that now, but if you lock yourself into a static final pattern, nothing short of tearing down the classloader will let you re-initialize it.
In the case of ObjectMapper its fine, but in general it is bad practice and there is no advantage over using a singleton pattern or inversion-of-control to manage your long-lived objects.