mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, in particular in

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

s and

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, an elliptic complex generalizes the notion of an

elliptic operator In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which im ...

to sequences. Elliptic complexes isolate those features common to the de Rham complex and the Dolbeault complex which are essential for performing

Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...

. 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 In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...

''M'' (usually taken to be compact), then a differential complex is a sequence :

\Gamma(E_0) \stackrel \Gamma(E_1) \stackrel \ldots \stackrel \Gamma(E_k)

of differential operators between the sheaves of sections of the ''E''_''i'' such that ''P''_''i''+1

\circ

''P''_''i''=0. A differential complex with first order operators is elliptic if the sequence of

symbols A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise different concep ...

0 \rightarrow \pi^*E_0 \stackrel \pi^*E_1 \stackrel \ldots \stackrel \pi^*E_k \rightarrow 0

is exact outside of the zero section. Here π is the projection of the

cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...

''T*M'' to ''M'', and π* is the pullback of a vector bundle.

References

Differential geometry Elliptic partial differential equations {{differential-geometry-stub

Definition

See also

References