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Elliptic complex

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated April 30, 2026

In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features common to the de Rham complex and the Dolbeault complex which are essential for performing Hodge theory. They also arise in connection with the Atiyah-Singer index theorem and Atiyah-Bott fixed point theorem.

01Definition

If E0, E1, ..., Ek are vector bundles on a smooth manifold M (usually taken to be compact), then a differential complex is a sequence

\Gamma (E_{0}){\stackrel {P_{1}}{\longrightarrow }}\Gamma (E_{1}){\stackrel {P_{2}}{\longrightarrow }}\ldots {\stackrel {P_{k}}{\longrightarrow }}\Gamma (E_{k})

of differential operators between the sheaves of sections of the Ei such that Pi+1 \circ Pi=0. A differential complex with first order operators is elliptic if the sequence of symbols

0\rightarrow \pi ^{*}E_{0}{\stackrel {\sigma (P_{1})}{\longrightarrow }}\pi ^{*}E_{1}{\stackrel {\sigma (P_{2})}{\longrightarrow }}\ldots {\stackrel {\sigma (P_{k})}{\longrightarrow }}\pi ^{*}E_{k}\rightarrow 0

is exact outside of the zero section. Here π is the projection of the cotangent bundle T*M to M, and π* is the pullback of a vector bundle.

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This article is adapted from the Wikipedia article “Elliptic complex”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Chain complex

In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel of the next. Associated to a chain complex is its homology, which is (loosely speaking) a measure of the failure of a chain complex to be exact.