According to general relativity, the curvature of spacetime in the frame of the spinning object is exactly zero. Curvature is a tensorial quantity and doesn't depend on the observer.
Now you might be wondering how you can get apparent centrifugal forces if there isn't any curvature, if everything in relativity is supposed to follow from local curvature. This is a very common question that troubles a lot of philosophers, and it's not really done justice in the standard textbooks or by pop-science.
The solution is that the fundamental object in relativity really isn't the spacetime curvature, it's the metric, which determines the distances between nearby points. From the metric, you can construct the curvature, but you can also construct a notion of straight lines. These are the curves between two points that are locally as short as possible. An inertial observer is defined to be one which travels along such a straight line. If you don't follow such a line, you will observe fictitious forces, because objects that do freely fall along these lines will appear to be accelerating relative to you, and you'd attribute that to a fictitious force like the centrifugal force.
In other words, there's no real philosophical puzzle here. Everything is determined by local information just as it should be.