## Frequent Links

# 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

- <math>\Gamma(E_0) \stackrel{P_1}{\longrightarrow} \Gamma(E_1) \stackrel{P_2}{\longrightarrow} \ldots \stackrel{P_k}{\longrightarrow} \Gamma(E_k)</math>

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

- <math>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</math>

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.

## See also

**Lua error in package.lua at line 80: module 'Module:Buffer' not found.**