a) The resolution of a point in a complex vector space.
b) The Hodge theory of the complex (F, ^+ /"^T^).
As pointed out in the Introduction, the Hodge theory of the resolution of the point {0} in a complex Hermitian vector space plays an essential, if modest, role in our whole paper.
The purpose of this Section is in fact to recall the elementary results established in [B3] concerning the Dolbeault resolution of the Koszul complex on a complex Hermitian vector space V. In the next Sections, the results of this Section will be applied to the fibres of the normal bundle N to Y in X.
In a), we describe the Koszul complex of V, and in b), we discuss the Hodge theory of the associated Dolbeault double complex.
We use the notation of Sections la) and 5a). This Section is otherwise self-contained.
a) The resolution of a point in a complex vector space
We use the notation of Sections la) and 5a). In particular VR is a real even dimensional vector space. Also J is a complex structure on VR, V is the associated complex vector space. Let n be the complex dimension ofV. Recall that if zeV, z represents Z = z + z e V^.
Let i be the embedding {0} -> V.
In the sequel, A(V*) will be considered as a Z-graded vector bundle on V. If zeV, let ^ be the interior multiplication operator by z. Then ^ acts on A(V*). We now consider the holomorphic chain complex on V
Let r be the restriction map: aeA°(V*)|^} -> aeC. Then by [GrH, p. 688], we have the exact sequence of sheaves
(7.2) 0 -^ ^ (A" (V*)) ———^ C?y (A""' (V*)). . . ———> ^ ^ C -> 0,
^iz ^iz r
i. e. the complex (AV*, I' —\ i^) resolves the sheaf ^ C.
We can then apply the results of Sections Ib), c), d) in this situation. Let Ny, NH be the operators which define the Z-grading on A(V*), A(V*) respectively. We define the Z-grading on A (V*) (§) A (V*) by the operator Ny — N^.
Let r be the set of smooth sections of A (V*) (8) A (V*) over V. Then F is also Z-graded by the operator NV—NH. Let ~Q be the standard Dolbeault operator acting on r. If zeV, the operator ^ acts naturally on F, with the convention that if ZeV^
is written in the form Z==z+z, zeV, then at ZeV^, ^a is exactly ^(a(Z)). Both operators ~S and / ~ 1 ^ are odd and increase the total degree in Y by one. Also 3+ /— 1 ^ is a chain map i. e.
(7.3) (S+^T^O.
As in (1.31), we extend r to a linear map from F into C. Namely, if a is a smooth section of A^ (V*) ® A9 (V*), i f / ? + ^ > 0 , r a = 0 , and if p=q=0, ra=a(0)eC.
The Dolbeault complex of {0} is simply C——>0. By [B3, Proposition 1.1], which
8W
is a special case of Theorem 1.7, the map r: (F, 3 + / — 1 ^) -> (C, ^{0}) is a quasi-isomorphism of Z-graded complexes. Therefore the cohomology of the complex (r, 3+ / — I ^) is concentrated in degree zero, and is one dimensional.
b) The Hodge theory of the complex (F, 3+ / — I ^)
We now recall the results of [B3, Section 1] which concern the Hodge theory of the complex (F, 3 + / - HJ.
As in Section la), we assume that V is equipped with a Hermitian product. Let dvy (Z) be the volume form on VR. Let r° be the set of the square integrable sections of A (V*) (§) A (V*) over VR. We equip r° with the Hermitian product
/ ^ \dimv r
(7.4) a, P-^<a, ?>= — <a, P>A(V^A(V^V
\2n/ JVR
The adjoint of the operator ^ is the operator i^=z A . Let 3* be the formal adjoint of ~S with respect to the Hermitian product (7.4). Then 3 * — / — 1 ^ is the formal adjoint of 3 + /— 1 ^. Also
(7.5) (^-y^T^O.
Recall that 9 is the Kahler form on Vg, so that if X . Y e V g (7.6) 6 ( X , Y ) = = < X , J Y > .
Then 6 is a (1, 1) form on V or equivalently an element of A1 (V*) ® A1 (V*), whose total degree is zero.
Let L be the operator
(7.7) a e A ( V * ) ® A ( V * ) - ^ L a = e A aeA(V*) ®A(V*).
Let A be the adjoint of L.
Definition 7.1. - S denotes the operator in End^^ACV*) (§) A(V*)) (7.8) S = - ( L + A )
Then S is a self-adjoint operator. It obviously acts on F and r°. Let e^ . . ., e^
be an orthonormal base of Vg. The Laplacian A of VR is given by
In
A=E(V,.)2 1
The operator A also acts on F.
The following result is proved in [B3, Proposition 1.4].
Proposition 7.2. - The following identities hold
^w
(7-9) /^T 2n 2 2
^^—T.^^c(Je,).
^
1Let Cy(Vn) be the set of real smooth functions of VR which have compact support. Let JS? be the differential operator
(7.10) ^ ^ ( - A + I Z l2^ ^ ) .
Then JS? is the harmonic oscillator on VR. By [G1J, Theorem 1.5.1], we know that
^ is essentially self-adjoint on C? (V^), that its closure is nonnegative and has compact resolvent. The spectrum of JS? is the set of nonnegative integers N and the kernel of J5? is one-dimensional and is spanned by the function exp(- (| Z \2/2)).
Let To be the set of C°° sections of A (V*) (§) A (V*) with compact support. Clearly (7.11) Fo = C? (V,) (g) (A (V*) ® A (V*)).
Also oS? acts as the operator JS? (x) 1 on TQ. By Proposition 7.2, we have the identity (7.12) (^+/^,+3*-/^,*)2=^+S+^
Therefore the operator ( 3 + / - - l ^ + 3 * - / - n * )2 is essentially self-adjoint on Fo, its closure is nonegative and has compact resolvent.
By definition, the form exp (6) is given by
n ndimV
(7.13) e x p ( e ) = l + - + . . . + — — - . I! (dimV)!
Since 6 has total degree zero, exp (6) also has total degree zero.
Proposition 7.3. — The lowest eigenvalue of the self-adjoint operator S e End (A (V*) (§) A (V*)) is equal to — n, and the corresponding eigenspace is spanned by the form exp (6). Also
(7.14) ^(e^v.^Atv^"^
Proof. - Proposition 7.3 follows from [B3, Proposition 1.5], and [B3, eq. (1.25) and (1.31)]. D
Theorem 7 . 4 . — Let P e r° be given by (7.15) p ^ e x p ^ e - i ^ V
Then P has total degree zero, and also
/ i \dimV
r-(7.16) — <P,P>^V.)®A(V.)^V=I.
\ 2 n / JVR
Moreover P spans the one-dimensional kernel of the operator
li^-^i!)2.
Finally
(7.17)
10P=0,
(3*-,/^T^)p=o,
<(N^)M>-<(N,-j)M>=0.
Proof. - Theorem 7.4 is proved in [B3, Theorem 1.6]. The fact that P spans the kernel of (3 4- / — \ i - +3*— / — I f * )\ -^ z ^ z /2 can also be derived from the considerations which follow Proposition 7.2 and from Proposition 7.3. Moreover (7.15) is a conse-quence of (7.14) and of the trivial
(7.18) f exp^^l)2^^10^ D
JVR
Remark 7.5.- Since P has total degree zero, then (7.19) (NV-NH)P=O.
So (7.19) is compatible with the last identity in (7.17).
Remark 7.6. - As pointed in [B3, Section Id)], the L^ cohomology of the complex (r°, 3+^/^T^) is concentrated in degree zero and H°(r°, ^ / ^ T ^ ) is spanned by p. Also observe that
(7.20) r p = l .
Since (3+^/~ 1 ^)P=0, P is a representative of the element in H°(r, ~8+
which correspond to l e C via the quasi-isomorphism r: (F, 3+/: rT^)->(C, 7)^).
The form P is in fact the unique harmonic representative in r° of the corresponding cohomology class.
Observe that with the notation of Section 5a) (7.21) ^T(^-^)= V - ( j z ) .
Let D be the operator (7.22) D=3+3*.
We then have the identity
(7.23) 3 + / ^ , + 3 * - / ^ T ^ = D + v ^ - ^ J Z ) .
VIII - A TAYLOR EXPANSION OF THE OPERATOR