Among the invariants that have a double life in geometry and in conformal field theory, there are the Euler characteristic and its refinements to complex elliptic genera. We argue that superconformal field theory motivates further refinements, resulting in a choice of several new invariants, the so-called Hodge-elliptic genera. At least for K3 surfaces and K3 theories, higher algebra and conformal field theory select the same refinement as the most natural one, allowing insights into generic features of K3 theories.