M721: Index Theory

Ruminations of a Graduate Class

The Algebraic Hodge Theorem and the Fundamental Theorem of Elliptic Operators

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Statement of the Fundamental Theorem of Elliptic Operators

Definition.   Let V and W be inner product spaces. Let L: V\rightarrow W be a linear map. Then a linear map L^*: W\rightarrow V is called the formal adjoint of L if \langle Lv,w\rangle=\langle v,L^* w\rangle for any v\in V and any w\in W.

Lemma.   (1) If a formal adjoint exists, it is unique.  (2) If \dim V<\infty, then L^* exists.

Example.   The map L: \Bbb{R}^\infty \rightarrow \Bbb{R} defined by summing the coodinates has no formal adjoint, where \Bbb{R}^\infty is the colimit of \Bbb{R}^n.

If V and W are Hilbert spaces, then we have

Theorem.    Any continuous linear map L:V\rightarrow W of Hilbert spaces has a formal adjoint.

Example.   Let D:C^\infty (E) \rightarrow C^\infty (F) be a differential operator. Suppose E, F and T X have smooth inner product structures, then we have the “L^2-inner products” on C^\infty (E) and C^\infty (F), given by \langle s_1,s_2\rangle :=\int_X \langle s_1, \overline{s_2}\rangle. Then D has a formal adjoint D^*:C^\infty (F) \rightarrow C^\infty (E). If we write D locally as \Sigma A_\alpha (x) D^\alpha, then D^* t=\Sigma (-1)^{|\alpha|} D^\alpha (\overline{A_\alpha}^{tr}t).

An elliptic operator D: C^\infty(E) \to C^\infty(E) is self-adjoint if D = D^*.

We can now state the Fundamental Theorem of Elliptic Operators. Later in this entry we will give some corollaries and much later in the course we will outline a proof using the method of elliptic regularity.

Fundamental Theorem of Elliptic Operators. For a self-adjoint elliptic operator D : C^\infty(E) \to C^\infty(E), there is an orthogonal decomposition C^\infty(E) = \ker D \oplus \mathrm{im } D with \ker D finite-dimensional.

It is important here that the manifold X is closed.

The algebraic Hodge theorem

Suppose now we have (co)chain complex (C^\cdot,d) over \Bbb{R} or \Bbb{C}:

\dots \rightarrow C^{p-1} \xrightarrow{d} C^p\xrightarrow{d} C^{p+1}\rightarrow \cdot\cdot\cdot

Give each C^p an inner product. Assume each d has a formal adjoint d^*. Define the Laplacian \Delta :=d d^*+d^* d:C^p\rightarrow C^p. Then we have

Lemma.   \Delta\alpha=0 iff d\alpha=0 and d^*\alpha =0.

Proof.   Suppose \Delta\alpha =0. Then we have

0=\langle (d d^*+d^* d) \alpha, \alpha\rangle=\langle dd^*\alpha, \alpha\rangle +\langle d^* d\alpha,\alpha\rangle =\langle d^*\alpha, d^*\alpha\rangle +\langle d\alpha,d\alpha\rangle,

and hence d\alpha=d^*\alpha=0.

\Box

Theorem.   Let (C^\cdot,d) be a (co)chain complex over a field k. Then there exist decompositions C^p=H^p\oplus B^p \oplus \hat{B}^p such that the (co)chain complex can be written as

.

\Box

When k=\Bbb{R} or \Bbb{C} and C^p is finite dimensional for each p, setting B^p=\mathrm{Im}\ d and \hat{B}^p=\mathrm{Im}\ d^*, the theorem above becomes a corollary of the following Algebraic Hodge Theorem:

Algebraic Hodge Theorem.   Let (C^\cdot,d) be a (co)chain complex over \Bbb{R} or \Bbb{C}. Suppose that C^p has inner product for each p and that formal adjoint d^*:C^{p+1}\rightarrow C^p exists for each p. Let \mathcal{H}^p=\ker\Delta :=dd^*+d^*d:C^p\rightarrow C^p, then

(1) TFAE: (a) \Delta\alpha=0, (b) d\alpha=0 and d^*\alpha=0, (c) (d+d^*)\alpha=0.

(2) \Delta(C^p)\subset (\mathcal{H}^p)^\perp.

(3) If C^p is finite dimensional, then C^p=\Delta(C^p)\oplus \mathcal{H}^p.

(4) If C^p=\Delta(C^p)\oplus\mathcal{H}^p for any p, then there are orthogonal decompositions

C^p=\mathcal{H}^p\oplus d(C^{p-1})\oplus d^* (C^{p+1})=\mathcal{H}^p\oplus dd^*(C^p)\oplus d^* d(C^p).

Proof.   (1) (a)\Rightarrow (b) \Rightarrow (c) \Rightarrow (a).

(2) Let \beta \in \mathcal{H}^p, then \langle \Delta\alpha, \beta\rangle=\langle\alpha,\Delta\beta\rangle=0.

(3) Show the inclusion in (2) is an equality by counting dimensions.

(4) It suffices to show the following orthogonal decomposition:

\Delta (C^p)=d(C^{p-1})\oplus d^*(C^{p+1}) = dd^* (C^p)\oplus d^* d(C^p).

However easily we have

\Delta(C^p)\subset dd^*(C^p)\oplus d^*d(C^p)\subset d(C^{p-1})\oplus d^*(C^{p+1})\subset (\mathcal{H}^p)^\perp=\mathrm{Im}\ \Delta.

 \Box

By checking the decomposition diagram above, we can obtain:

Corollary.   \phi : \mathcal{H}^p=\ker \Delta \rightarrow H^p(C^\cdot,d) is an isomorphism, where \phi(\alpha)=[\alpha].

Corollary.   H^*(C^\cdot,d)=0 iff \Delta: C^p\rightarrow C^p is an isomorphism for any p.

Wrapping up

Corollary.    Algebraic Wrapping up. For (C,d) as above:  H^*(C^\cdot,d)=0 \Rightarrow d+d^*: C^{even} =: \oplus C^{2i}\rightarrow C^{odd}=: \oplus C^{2i+1} is an isomorphism. Hence \Delta : C^p \to C^p is an isomorphism for all p.

This corollary “wraps up” a (co)chain complex into a single map.

Next, we consider wrapping up an ellliptic complex.

Definition. An elliptic complex of differential operators is a cochain complex of differential operators

0 \to C^\infty(E^0) \xrightarrow{D} C^\infty(E^1) \xrightarrow{D}   \cdots \xrightarrow{D} C^\infty(E^k) \to 0

so that for all 0\neq\xi\in T_x^*X the associated symbol complex is exact.

If we define the symbol of differential operator D of order m by \sigma_D(x,\xi):=(-i)^m\Sigma_{|\alpha|=m}A_\alpha \xi^\alpha, then we have (\sigma_D)^*=\sigma_{D^*}.

Proposition.   Let (C^\infty E^\cdot, D) be an elliptic complex of differential operators. Give E^p and TX metrics for each p. Then D+D^*:C^\infty E^{even}\rightarrow C^\infty E^{odd} is an elliptic operator.

Proof.   For any 0\neq\xi\in T_x^*X, since the complex is elliptic, we have the exact sequence

\dots\rightarrow E_x^{p-1}\rightarrow E_x^p\xrightarrow{\sigma_p(\xi)} E_x^{p+1}\rightarrow\dots

Thus, \sigma_D+(\sigma_D)^*:E_x^{even}\rightarrow E_x^{odd} is an isomorphism.

Finally note that \sigma_D+(\sigma_D)^*=\sigma_D+\sigma_{D^*}=\sigma_{D+D^*}.

\Box

Consequences of the fundamental theorem.

We deduce the following corollaries of the Fundamental Theorem, the Algebraic Hodge Theorem, and Wrapping Up.

Corollary. Let (C^\infty(E),D) be an elliptic complex of differential operators.

(1) For any p, \mathcal{H}^p:=\ker \Delta : C^\infty E^p\rightarrow C^\infty E^p is finite dimensional.

(2) For any p, C^\infty E^p=\mathrm{Im}\ \Delta \oplus\mathcal{H}^p.

(3) \mathrm{Index}\ (D+D^*: C^\infty(E^{even}) \to C^\infty(E^{odd}) ) =\Sigma (-1)^p \mathrm{dim}\ \mathcal{H}^p.

Corollary.   If D: C^\infty E \rightarrow C^\infty F is an elliptic differential operator, then we have isomorphisms \ker\ D \xrightarrow{\cong} \mathrm{cok}\ D^*, \ker\ D^*\xrightarrow{\cong} \mathrm{cok}\ D. Hence the kernel and the cokernel of an elliptic differential operator are finite dimensional.

Corollary.   If D: C^\infty E \rightarrow C^\infty E is self-adjoint, then \mathrm{Index}\ D=0.

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Written by topoclyb

October 19, 2012 at 6:29 pm

Posted in Uncategorized

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