# M721: Index Theory

## The Algebraic Hodge Theorem and the Fundamental Theorem of Elliptic Operators

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 $\mathrm{Index}\ D=0$.