A natural refinement of the Euler characteristic

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The Euler characteristic of a compact complex manifold M is a classical cohomological invariant. Depending on the viewpoint, it is most natural to interpret it as an index of an elliptic differential operator on M, or as a supersymmetric index in superconformal field theories “on M”. Refining the Euler characteristic but keeping with both index theoretic interpretations, one arrives at the notion of complex elliptic genera. We argue that superconformal field theory motivates further refinements of these elliptic genera which result in a choice of several new invariants, all of which have lost their interpretation in terms of index theory. However, at least if M is a K3 surface, then superconformal field theory and higher algebra select the same new invariant as a natural refinement of the complex elliptic genus.