In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences.
They also arise in connection with the Atiyah-Singer index theorem and Atiyah-Bott fixed point theorem.
If E0, E1, ..., Ek are vector bundles on a smooth manifold M (usually taken to be compact), then a differential complex is a sequence of differential operators between the sheaves of sections of the Ei such that Pi+1 ∘ Pi=0.
A differential complex with first order operators is elliptic if the sequence of symbols is exact outside of the zero section.
This differential geometry-related article is a stub.