1 Differential Operators
Fix a manifold , and Hermitian vector bundles with inner products.
A differential operator of order is a -linear map that is local, i.e. the value of near a point depends only on the values of near , and in a coordinate chart, it is of the form
for some .
Note that under our definition, any differential operator of order
is also a differential operator of order
Just as we can define tangent vectors as derivations, we have the following coordinate-free definition of differential operators (which we will not use), with a similar proof:
Let be linear. Then
is a differential operator of order iff for all .
is a differential operator of order iff is a differential operator of order for all .
Integration by parts implies that we have
For any differential operator , there is a formal adjoint such that for any , we have
Let be a differential operator of order . The (principal) symbol of is the family of operators for given locally by
Formally, if is the projection, then .
In a coordinate-free manner, if and with , then
We say is elliptic at if is invertible for all , and is elliptic if it is elliptic everywhere.
While the coordinate-free definition seems rather artificial, it is actually useful when we want to do computations later on.
It will be convenient to note that the adjoint of an elliptic operator is elliptic. More generally,
For any operators , we have
Hence the composition and adjoints of elliptic operators is elliptic.
Consider the exterior derivative . Using the coordinate-free definition, we compute the symbol as
whenever . So the symbol of is
Note that for
, this is not invertible. Instead, what we have is an elliptic complex.
An elliptic complex is a sequence of vector bundles with first-order differential operators
such that and for any non-zero , the sequence
is exact outside of the zero section of .
The de Rham complex and Dolbeault complex are elliptic complexes.
Ultimately, we will prove Hodge decomposition for elliptic complexes, which subsumes the Hodge decomposition of Riemannian and Kähler manifolds.
To get from an elliptic complex to an elliptic operator, we use the following linear algebraic result:
Let be finite-dimensional vector spaces, and
be an exact sequence. Let . Then is an isomorphism.
It suffices to show
is injective. Suppose
So we get
So . By exactness, for some , and then
If is an elliptic complex, define . Then