Chapter VI Kodaira’s Projective Embedding Theorem 217 1. Hodge Manifolds 217
4. A Parametrix for Elliptic Differential Operators
In this section we want to restrict our attention to operators which generalize the classic Laplacian operator in Euclidean space and its inverse.
These will be called elliptic operators. We start with a definition, using the same notation as in the preceding sections. Let E andF be vector bundles over a differentiable manifold X.
Definition 4.1: Let s ∈ Smblk(E, F ). Then s is said to be elliptic if and only if for any (x, ξ )∈T′(X), the linear map
s(x, ξ ): Ex−→Fx is an isomorphism.
Note that, in particular, bothEandF must have the same fibre dimension.
We shall be most interested in the case where E=F.
Definition 4.2: Let L ∈ PDiffk(E, F ). Then L is said to be elliptic (of order k) if and only if σk(L) is an elliptic symbol.
Note that ifLis an elliptic operator of orderk, thenLis also an operator of order k+1, but clearly not an elliptic operator of order k+1 since σk+1(L)=0. For convenience, we shall call any operator L∈OP−1(E, F ) a smoothing operator. We shall later see why this terminology is justified.
Definition 4.3: Let L ∈ PDiff(E, F ). Then L˜ ∈ PDiff(F, E) is called a parametrix (or pseudoinverse) for L if it has the following properties,
L◦ ˜L−IF ∈OP−1(F )
˜
L◦L−IE ∈OP−1(E),
where IF and IE denote the identity operators on F andE, respectively.
The basicexistencetheorem for elliptic operators on a compact manifold X can be formulated as follows.
Theorem 4.4: Let k be any integer and let L ∈PDiffk(E, F ) be elliptic.
Then there exists a parametrix for L.
Proof: Let s=σk(L). Thens−1 exists as a linear transformation, since s is invertible,
s−1(x, ξ ): Fx−→Ex,
and s−1 ∈ Smbl−k(F, E). Let L˜ be any pseudodifferential operator in PDiff−k(F, E) such that σ−k(L)˜ = s−1, whose existence is guaranteed by Theorem 3.16. We have then that
σ0(L◦ ˜L−IF)=σ0(L◦ ˜L)−σ0(IF),
and letting σ0(IF)=1F, the identity in Smbl0(F, F ), we obtain σ0(L◦ ˜L−IF)=σk(L) • σ−k(L)˜ −1F
=1F−1F =0.
Thus, by Theorem 3.16, we see that
L◦ ˜L−IF ∈OP−1(F, F ).
Similarly, L˜ ◦L−IE is seen to be in OP−1(E, E).
Q.E.D.
This theorem tells us that modulo smoothing operators we have an inverse for a given elliptic operator. On compact manifolds, this turns out to be only a finite dimensional obstruction, as will be deduced later from the following proposition. First we need a definition. Let X be a compact differentiable manifold and suppose thatL∈OPm(E, F ). Then we say thatL iscompact (orcompletely continuous) if for everysthe extensionLs:Ws(E)→Ws−m(F ) is a compact operator as a mapping of Banach spaces.
Proposition 4.5: Let X be a compact manifold and let S ∈ OP−1(E, E).
Then S is a compact operator of order 0.
Proof: We have for anys the following commutative diagram, Ws(E) Ws(E)
Ws+1(E),
S˜s
Ss j
where Ss is the extension of S to a mapping Ws → Ws+1, given since S∈OP−1(E, E), andS˜sis the extension ofS, as a mappingWs→Ws, given by the fact that OP−1(E, E)⊂OP0(E, E). Since j is a compact operator (by Rellich’s lemma, Proposition 1.2), then S˜s must also be compact.
Q.E.D.
In the remainder of this section we shall let EandF be fixed Hermitian vector bundles over a compact differentiable manifold X. Assume that X is equipped with a smooth positive measure µ (such as would be induced by a Riemannian metric, for example) and let W0(X, E)=W0(E), W0(F ) denote the Hilbert spaces equipped with L2-inner products
(ξ,η)E=
$
X
ξ(x),η(x)Edµ, ξ,η∈E(X, E) (σ, τ )F =
$
X
σ (x), τ (x)Fdµ, σ, τ ∈E(X, F ),
as in Sec. 1. We shall also consider the Sobolev spacesWs(E), Ws(F ), defined for all integrals, as before, and shall make use of these without further men- tion. If L∈OPm(E, F ), denote by Ls: Ws(E)→Ws−m(F ) the continuous extension ofLas a continuous mapping of Banach spaces. We want to study the homogeneous and inhomogeneous solutions of the differential equation
Lξ =σ, for ξ ∈ E(X, E), σ ∈E(X, F ), where L∈Diffm(E, F ), and L∗ is the adjoint of L defined with respect to the inner products in W0(E) and W0(F ); i.e.,
(Lξ, τ )F =(ξ, L∗τ )E, as given in Proposition 2.8. If L∈Diffm(E, F ), we set
HL= {ξ ∈E(X, E): Lξ=0}, and we let
H⊥L = {η∈W0(E): (ξ,η)E =0, ξ∈HL}
denote the orthogonal complement inW0(E)ofHL. It follows immediately that the space HL⊥ is a closed subspace of the Hilbert space W0(E). As we shall see, under the assumption that L is elliptic HL turns out to be finite dimensional [and hence a closed subspace of W0(E)]. Before we get to this, we need to recall some standard facts from functional analysis, due to F. Riesz (see Rudin [1]).
Proposition 4.6: LetB be a Banach space and letSbe a compact operator, S:B→B. Then letting T =I−S, one has:
(a) Ker T =T−1(0)is finite dimensional.
(b) T (B)is closed in B, and Coker T =B/T (B) is finite dimensional.
In our applications the Banach spaces are the Sobolev spacesWs(E)which are in fact Hilbert spaces. Proposition 4.6(a) is then particularly easy in this case, and we shall sketch the proof forB, a Hilbert space. Namely, if the unit ball in a Hilbert space his compact, then it follows that there can be only a finite number of orthonormal vectors, since the distance between any two orthonormal vectors is uniformly bounded away from zero (by the distance
√2). Thus h must be finite dimensional. Proposition 4.6(a), for instance, then follows immediately from the fact that the unit ball in the Hilbert space h=Ker T must be compact (essentially the definition of a compact operator). The proof thatT (B)is closed is more difficult and again uses the compactness ofS. SinceS∗is also compact, KerT∗is finite dimensional, and the finite dimensionality of Coker T follows. More generally, the proof of Proposition 4.6 depends on the fundamental finiteness criterion in functional analysis which asserts that a locally compact topological vector space is necessarily finite dimensional. See Riesz and Nagy [1], Rudin [1], or any other standard reference on functional analysis for a discussion of this as well as a proof of the above proposition. (A good survey of this general topic can be found in Palais [1].)
An operator T on a Banach space is called a Fredholm operator if T has finite-dimensional kernel and cokernel. Then we immediately obtain the following from Theorem 4.4 and Propositions 4.5 and 4.6.
Theorem 4.7: LetL∈PDiffm(E, F )be an elliptic pseudodifferential oper- ator. Then there exists a parametrix P for L so that L◦P and P ◦L have continuous extensions as Fredholm operators: Ws(F )→ Ws(F ) and Ws(E)→Ws(E), respectively, for each integer s.
We now have the important finiteness theorem for elliptic differential operators.
Theorem 4.8: LetL∈Diffk(E, F )be elliptic. Then, letting HLs =KerLs: Ws(E)→Ws−k(F ), one has
(a) HLs ⊂E(X, E) and hence HLs =HL, all s.
(b) dimHLs =dimHL<∞ and dimWs−k(F )/Ls(Ws(E)) <∞. Proof: First we shall show that, for any s, dim HLs <∞. Let P be a parametrix for L, and then by Theorem 4.7, it follows that
(P◦L)s: Ws(E)−→Ws(F )
has finite dimensional kernel, and obviously KerLs⊂Ker(P◦L)s, since we have the following commutative diagram of Banach spaces:
Ws(E) Ws(E)
Ws+1(E),
(P◦L)
Ls Ps−k
Hence HLs is finite dimensional for all s. By a similar argument, we see that Ls has a finite dimensional cokernel. Once we show that HLs contains only C∞ sections of E, then it will follow that HLs = HL and that all dimensions are the same and, of course, finite.
To show that HLs ⊂E(X, E) is known as the regularity of the homo- geneous solutions of an elliptic differential equation. We formulate this as a theorem stated somewhat more generally, which will then complete the proof of Theorem 4.8.
Theorem 4.9: Suppose thatL∈Diffm(E, F )is elliptic, andξ ∈Ws(E)has the property that Lsξ =σ ∈E(X, F ). Then ξ ∈E(X, E).
Proof: IfP is a parametrix forL, thenP◦L−I =S∈OP−1(E). Now Lξ ∈E(X, F ) implies that (P◦L)ξ ∈E(X, E), and hence
ξ =(P ◦L−S)ξ.
Since we assumed that ξ ∈ Ws(E) and since (P ◦L)ξ ∈ E(X, E) and Sξ ∈Ws+1(E), it follows that ξ ∈Ws+1(E). Repeating this process, we see thatξ ∈Ws+k(E)for allk >0. But by Sobolev’s lemma (Proposition 1.1) it follows thatξ ∈El(X, E), for alll >0, and henceξ ∈E(X, E)(=E∞(X, E)).
Q.E.D.
We note that S is called a smoothing operator precisely because of the role it plays in the proof of the above lemma. It smooths out the weak solution ξ ∈Ws(E).
Remark: The above theorem did not need the compactness ofX which is being assumed throughout this section for convenience. Regularity of the solution of a differential equation is clearly a local property, and the above proof can be modified to prove the above theorem for noncompact manifolds.
We have finiteness and regularity theorems for elliptic operators. The one remaining basic result is the existence theorem. First we note the following elementary but important fact, which follows immediately from the definition.
Proposition 4.10: LetL∈Diffm(E, F ). ThenL is elliptic if and only ifL∗ is elliptic.
We can now formulate the following.
Theorem 4.11: Let L ∈ Diffm(E, F ) be elliptic, and suppose that τ ∈ H⊥L∗∩E(X, F ). Then there exists a unique ξ ∈E(X, E) such that Lξ =τ and such that ξ is orthogonal to HL in W0(E).
Proof: First we shall solve the equation Lξ = τ, where ξ ∈ W0(E), and then it will follow from the regularity (Theorem 4.9) of the solution ξ thatξ isC∞ since τ is C∞, and we shall have our desired solution. This reduces the problem to functional analysis. Consider the following diagram of Banach spaces,
Wm(E) Lm //
W0(F )
W−m(E)
OO
W0(F ),
L∗m
oo OO
where we note that (Lm)∗ =(L∗)0, by the uniqueness of the adjoint, and denote same byL∗m. The vertical arrows indicate the duality relation between the Banach spaces indicated. A well-known and elementary functional analy- sis result asserts that the closure of the range is perpendicular to the kernel of the transpose. Thus Lm(Wm(E)) is dense in H⊥L∗
m. Moreover, since Lm has finite dimensional cokernel, it follows that Lm has closed range, and hence the equation Lmξ =τ has a solution ξ ∈ Wm(E). By orthogonally projecting ξ along the closed subspace Ker Lm (= HL by Theorem 4.8), we obtain a unique solution.
Q.E.D.
Let L∈ Diffm(E)=Diffm(E, E). Then we say that L is self-adjoint if L=L∗. Using the above results we deduce easily the following fundamental decomposition theorem for self-adjoint elliptic operators.
Theorem 4.12: Let L∈ Diffm(E) be self-adjoint and elliptic. Then there exist linear mappings HL and GL
HL: E(X, E)−→E(X, E) GL: E(X, E)−→E(X, E) so that
(a) HL(E(X, E))=HL(E) and dimcHL(E) <∞.
(b) L◦GL+HL=GL◦L+HL=IE, where IE=identity on E(X, E).
(c) HL and GL ∈ OP0(E), and, in particular, extend to bounded operators on W0(E)(=L2(X, E)).
(d) E(X, E) = HL(X, E) ⊕ GL ◦ L(E(X, E)) = HL(X, E) ⊕ L ◦ GL(E(X, E)), and this decomposition is orthogonal with respect to the inner product in W0(E).
Proof: LetHLbe the orthogonal projection [inW0(E)] onto the closed subspace HL(E), which we know by Theorem 4.8 is finite dimensional. As we saw in the proof of Theorem 4.11, there is a bijective continuous mapping
Lm: Wm(E)∩H⊥L−→W0(E)∩H⊥L.
By the Banach open mapping theorem, Lm has a continuous linear inverse which we denote by G0:
G0: W0(E)∩HL⊥−→Wm(E)∩HL⊥.
We extend G0 to all of W0(E)by letting G0(ξ )=0 if ξ ∈HL, and noting that Wm(E)⊂W0(E), we see that
G0: W0(E)−→W0(E).
Moreover,
Lm◦G0=IE−HL, since Lm◦G0=identity on H⊥L. Similarly,
G0◦Lm=IE−HL
for the same reason. Since G0(E(X, E)) ⊂ E(X, E), by elliptic regularity (Theorem 4.9), we see that we can restrict the linear Banach space mappings above to E(X, E). Let GL=G0|E(X,E), and it becomes clear that all of the conditions (a)–(d) are satisfied.
Q.E.D.
The above theorem was first proved by Hodge for the case where E=
∧pT∗(X)and whereL=dd∗+d∗d is the Laplacian operator, defined with respect to a Riemannian metric on X (see Hodge [1] and de Rham [1]).
Hodge called the homogeneous solutions of the equation Lϕ=0 harmonic p-forms, since the operatorLis a true generalization of the Laplacian in the plane. Following this pattern, we shall call the sections in HL, forL a self- adjoint elliptic operator,L-harmonic sections, and when there is no chance of
confusion, simply harmonic sections. For convenience we shall refer to the operator GL given by Theorem 4.12 as the Green’s operator associated to L, also classical terminology.† The harmonic forms of Hodge and their generalizations will be used in our study of Kähler manifolds and algebraic geometry. We shall refine the above theorem in the next section dealing with elliptic complexes and at the same time give some examples of its usefulness.
Suppose thatE→X is a differentiable vector bundle andL: E(X, E)→ E(X, E) is an elliptic operator. Then theindex of L is defined by
i(L)=dim Ker L−dim Ker L∗,
which is a well-defined integer (Theorem 4.8). The Atiyah-Singer index theorem asserts that i(L)is a topological invariant, depending only on (a) the Chern classes of E and (b) a cohomology class in H∗(X,C) defined by the top-order symbol of the differential operator L. Moreover, there is an explicit formula for i(L) in terms of these invariants (see Atiyah and Singer [1, 2]). We shall see a special case of this in Sec. 5 when we discuss the Hirzebruch-Riemann-Roch theorem for compact complex manifolds.
We would like to give another application of the existence of the para- metrix to prove a semicontinuity theorem for a family of elliptic operators.
Suppose that E −→ X is a differentiable vector bundle over a compact manifold X, and let {Lt} be a continuous family of elliptic operators,
(4.1) Lt: E(X, E)−→E(X, E),
where t is a parameter varying over an open setU⊂Rn. By this we mean that for a fixed t∈U, Lt is an elliptic operator and that the coefficients of Lt in a local representation for the operator should be jointly continuous in x∈X and t ∈U.
Theorem 4.13: Let{Lt}be a continuous family of elliptic differential oper- ators of order mas in (4.1). Then dim Ker Lt is an upper semicontinuous function of the parameter t; moreover ift0∈U, then forǫ >0 sufficiently small,
dim Ker Lt ≤dim Ker Lt0 for |t−t0|< ǫ.
Proof: Suppose that t0 =0, let B1 =W0(X, E) and B2 =W−m(X, E), and letP be a parametrix for the operatorL=L0. Denoting the extensions of the operators Lt and P by the same symbols, we have
Lt: B1−→B2, t ∈U P: B2−→B1.
We shall continue the proof later, but first in this context we have the following lemma concerning the single operator L=L0, whose proof uses
†Note that the Green’s operatorGLis aparametrixforL, but such thatGL◦L−I= −HL
is a smoothing operator of infinite order which is orthogonal to GL, a much stronger parametrix than that obtained from Theorem 4.4.
the existence of the parametrix P at t =0. Let Ht =Ker Lt, t ∈U, and
1, 2 denote the norms in B1 and B2.
Lemma 4.14: There exists a constant C >0 such that u1≤CL0u2
if u∈H0⊥⊂B1 (orthogonal complement in the Hilbert space B1).
Proof: Suppose the contrary. Then there exists a sequence uj ∈ H⊥0 such that
uj1=1 Luj2≤ 1 j. (4.2)
Consider
P Luj =uj+T uj, where T is compact, Then
T uj1≤ P Luj1+ uj1
≤CLuj2+ uj1
≤C 1
j
+1
≤ ˜C,
where C,C˜ are constants which depend on the operatorP (recall thatP is a continuous operator fromB2toB1). Sinceuj =1, it follows that{T uj} is a sequence of points in a compact subset of B1, and as such, there is a convergent subsequence yj n=T uj n→y0∈B1. Moreover, y0 =0, since limn→∞Luj n=0, by (4.2), and thus
0= lim
n→∞P Luj n= lim
n→∞uj n+y0, which implies that uj n→ −y0 and
y0 = lim
n→∞uj n =1.
However, Ly0= −limn→∞Luj n=0, as above, and this contradicts the fact that y0 (which is the limit of uj n) ∈H0⊥.
Q.E.D.
Proof of Theorem 4.13 continued: LetCbe the constant in Lemma 4.14.
We claim that forδsufficiently small there exists a somewhat larger constant
˜
C such that, foru∈H⊥0,
(4.4) u1≤ ˜CLtu2,
provided that |t|< δ, where C˜ is independent of t. To see this, we write L0=Lt+L0−Lt,
and therefore (using the operator norm)
L0 ≤ Lt + L0−Lt.
For any ǫ >0, there is a δ >0 so that Lt−L0< ǫ,
for |t| < δ, since the coefficients of Lt are continuous functions of the parameter t. Using Lemma 4.13, we have, foru∈H0⊥,
u1≤CL0u2
≤C(Ltu2+ǫu1), which gives
(1−Cǫ)u1≤CLtu2 for |t|< δ. By choosingǫ < C−1, we see that
u1≤C(1−Cǫ)−1Ltu2
≤ ˜CLtu2,
which gives (4.4). But u ∈ H⊥0 by assumption, and it follows from the inequality (4.4) that H⊥0 ∩Ht = 0 for |t| < δ. Consequently, we obtain dimHt≤dimH0.
Q.E.D.