Elliptic complex

Elliptic complex

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.

Definition

If "E"0, "E"1, ..., "E""k" 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 "E""i" such that "P""i"+1 o "P""i"=0. A differential complex is elliptic if the sequence of symbols

:0 ightarrow 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 ightarrow 0

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

ee also

*Chain complex


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