Hermitian Exterior Algebra on a Hermitian Vector Space

Chapter VI Kodaira’s Projective Embedding Theorem 217 1. Hodge Manifolds 217

1. Hermitian Exterior Algebra on a Hermitian Vector Space

C O M PA C T

C O M P L E X M A N I F O L D S

In this chapter we shall apply the differential equations and differential geometry of the previous two chapters to the study of compact complex manifolds. In Sec. 1 we shall present a discussion of the exterior algebra on a Hermitian vector space, introducing the fundamental 2-form and the Hodge

∗-operator associated with the Hermitian metric. In Sec. 2 we shall discuss and prove the principal results concerning harmonic forms on compact manifolds (real or complex), in particular, Hodge’s harmonic representation for the de Rham groups, and special cases of Poincaré and Serre duality.

In Sec. 3 we present the finite-dimensional representation theory for the Lie algebrasl(2,C), from which we derive the Lefschetz decomposition theorem for a Hermitian exterior algebra. In Sec. 4 we shall introduce the concept of a Kähler metric and give various examples of Kähler manifolds (manifolds equipped with a Kähler metric). In terms of a Hermitian metric we define the Laplacian operators associated with the operatorsd,∂, and∂¯ and show that when the metric is Kähler that the Laplacians are related in a simple way. We shall use this relationship in Sec. 5 to prove the Hodge decomposi- tion theorem expressing the de Rham group as a direct sum of the Dolbeault groups (of the same total degree). In Sec. 6 we shall state and prove Hodge’s generalization of the Riemann period relations for integrals of harmonic forms on a Kähler manifold. We shall then use the period relations and the Hodge decomposition to formulate the period mapping of Griffiths. In par- ticular, we shall prove the Kodaira-Spencer upper semicontinuity theorem for the Hodge numbers on complex-analytic families of compact manifolds.

V is a choice of ordering of a basis such as {e1, . . . , ed} up to an even permutation, which is equivalent to a choice of sign for a particulard-form, e.g., e₁∧ · · · ∧e_d.

We now define the Hodge ∗-operator. Choosing an orthonormal basis {e₁, . . . , e_d}forV as above, fix an orientation ofV by specifying thed form e₁∧ · · · ∧e_d which we will denote by vol (for volume element). The Hodge

∗-operator is a mapping

∗: ∧^pV −→ ∧^d−pV defined by setting

∗(e_i1 ∧ · · · ∧e_ip)= ±e_j1∧ · · · ∧e_jd−p,

where {j1, . . . , j_d−p}is the complement of {i1, . . . , i_p}in {1, . . . , d}, and we assign the plus sign if {i1, . . . , i_p, j1, . . . , j_d−p} is an even permutation of {1, . . . , d}, and the minus sign otherwise. In other words∗is defined so that (1.1) e_i1∧ · · · ∧e_ip∧ ∗(e_i1 ∧ · · · ∧e_ip)=e1∧ · · · ∧e_d =vol.

Extending ∗ by linearity to all of ∧^pV we find that if α, β∈ ∧^pV, then

(1.2) α∧ ∗β = α, β vol,

where α, β is the inner product induced on ∧^pV from V. Let us check that (1.2) is valid. Namely, if

α=

|I|=p

′a_Je_J,

and

β=

|J|=p

′b_Je_J, using multi-index notation, then

α∧ ∗β =

|I|=p

|J|=p

′aIbJeI ∧ ∗eJ.

We see that the wedge product in each term of the sum vanishes unless I = {i₁, . . . , i_p} coincides with J = {j₁, . . . , j_p}, and then it follows immediately from (1.1) that

α∧ ∗β =

|I|=p

a_Ib_I vol

= α, β vol.

It is easily checked that the definition of the Hodge∗-operator is independent of the choice of the orthonormal basis, and depends only on the inner product structure of V as well as a choice of orientation.†

†The classical references for the ∗-operator are Hodge [1], de Rham [1], and Weil [1].

We can extend (1.2) easily to complex-valued p-forms. Namely, if α, β∈

∧^pV ⊗C, then β¯ is well defined (cf. Sec. I.3). We write

α=

|I|=p

′α_Ie_I, α_I ∈C,

β =

|I|=p

′β_Ie_I, β_I ∈C,

then we define an Hermitian inner product on ∧^pV ⊗Cby α, β :=

|I|=p

′α_Iβ¯_I.

Ifα, β are real, then we have the original inner product, so we use the same symbol , for this complex extension. It follows then immediately that if

∗ is extended to ∧^∗V ⊗Cby complex linearity, we obtain the relation

(1.3) α∧ ∗ ¯β = α, β vol.

Let _r denote the projection onto homogeneous vectors of degree r, r: ∧V −→ ∧^rV ,

and define the linear mapping w: ∧V → ∧V by setting w=(−1)^dr+r_r.

It is easy to see that ∗∗ =w, and we remark that ifd is even, then we have

(1.4) w=(−1)^r_r.

Let E be a complex vector space of complex dimensionn. LetE^′ be the real dual space to the underlying real vector space of E, and let

F =E^′⊗^RC

be the complex vector space of complex-valued real-linear mappings of E to C. Then F has complex dimension 2n, and we let

∧F = 2n p=0

∧^pF

be the C-linear exterior algebra of F. We will refer to an ω∈ ∧^pF as a p-form or as a p-covector (onE). Now, as before, ∧F is equipped with a natural conjugation obtained by setting, if ω∈ ∧^pF,

ω(v₁, . . . , v_p)=ω(v₁, . . . , v_p), v_j ∈E.

We say that ω∈ ∧^pF isreal if ω=ω, and we will let∧^pRF denote the real elements of ∧^pF (noting that ∧^pE^′∼= ∧^pRF).

Let ∧^1,0F be the subspace of∧¹F consisting of complex-linear 1-forms onE, and let∧^0,1F be the subspace of conjugate-linear 1-forms onE. Then we see that ∧^1,0F = ∧^0,1F and moreover

∧¹F = ∧^1,0F ⊕ ∧^0,1F,

and this induces (as in Sec. I.3) a bigrading on ∧F,

∧F = 2n

r=0

p+q=r

∧^p,qF, and we see that if ω∈ ∧^p,qF, then ω∈ ∧^q,pF.

Now we suppose than our complex vector space is equipped with a Hermitian inner product,. This inner product is a Hermitian symmetric sesquilinear† positive definite form, and can be represented in the following manner. If {z1, . . . , z_n} is a basis for ∧^1,0F, then {¯z1, . . . ,z¯_n} is a basis for

∧^0,1F, and we can write, for u, v∈E, u, v =h(u, v), where

h=

µ,v

hµvzµ⊗ ¯zv,

and (h_αβ) is a positive definite Hermitian symmetric matrix. Now h is a complex-valued sesquilinear form acting on E×E, and we can write

h=S+iA,

where S and Aare real bilinear forms acting on E. One finds that S is a symmetric positive definite bilinear form, which represents the Euclidean inner product induced on the underlying real vector space of E by the Hermitian metric on E. Moreover one can calculate easily that

A= 1 2i

µ,ν

h_µν(z_µ⊗ ¯z_ν− ¯z_ν⊗z_µ)

= −i

µ,ν

h_µνz_µ∧ ¯z_ν. Let us define

(1.5) = i

2

µ,ν

h_µνz_µ∧ ¯z_ν,

the fundamental 2-form associated to the hermitian metric h. One sees immediately that

= −¹2A= −¹2 Im h, and thus

(1.6) h=S−2i.

Moreover ∈ ∧^1,1R F, i.e., is a real 2-form of type (1, 1). We can always choose a basis {z_µ}of ∧^1,0F so that h has the form

(1.7) h=

µ

z_µ⊗ ¯z_µ.

†We recall that a mapping f: E×E→C is sesquilinear if f is real bilinear, and moreover, f (λu, v)=λf (u, v), and f (u, λv)=λf (u, v), λ∈C.

It then follows that, if we let x_µ= z_µ+ ¯z_µ

2 , y_µ=z_µ− ¯z_µ 2i be the real and imaginary parts of {zµ}, then

(1.8) h=

µ

(x_µ⊗x_µ+y_µ⊗y_µ)−2i

µ

(x_µ∧y_µ), and thus from (1.5), with respect to this basis,

S=

x_µ⊗x_µ+y_µ⊗y_µ =

x_µ∧y_µ= i 2

z_µ∧ ¯z_µ. (1.9)

It follows from this that

(1.10) ⁿ=n!x1∧y1∧ · · · ∧xn∧yn.

Thus the fundamental 2-form associated to a Hermitian metric is a real form of type (1, 1) whose coefficient matrix is positive definite, and moreover,ⁿ is a nonzero volume element of E^′. Thusⁿ determines an orientation on E^′, and we see from (1.9) that {xµ, yµ} is an orthonormal basis for E^′ in the induced Euclidean metric of E^′. Thus we see that there is a naturally defined Hodge ∗-operator

(1.11) ∗: ∧^pE^′−→ ∧^2n−pE^′

coming from the Hermitian structure ofE. Namely,E^′ has the dual metric to the real underlying vector space of E, while E^′ is equipped with the orientation induced by the 2n-formⁿcoming from the Hermitian structure of E. We define

(1.12) vol= 1

n!ⁿ,

which, with respect to the orthonormal basis used above, becomes vol=x1∧y1∧ · · · ∧xn∧yn.

Note that the definition (1.12) does not depend on the choice of the basis, and is an intrinsic definition of a volume element on E^′.

We are now interested in defining various linear operators mapping

∧F → ∧F in terms of the above structure. Recall that we already defined w for an even dimensional vector space by (1.4), and this therefore defines

w: ∧E^′−→ ∧E^′ which we extend by complex-linearity to

w: ∧F −→ ∧F where

_r: ∧F −→ ∧^rF

is the natural projection. Similarly, since E has a Hermitian structure, as we saw above, there is a natural ∗-operator

∗: ∧^pE^′−→ ∧^2n−pE^′

which we also extend as a complex-linear isomorphism to

∗: ∧^pF −→ ∧^2n−pF.

Both w and ∗are real operators. Now we let _p,q: ∧F −→ ∧^p,qF be the natural projection, and we define

J: ∧F −→ ∧F by

J =

i^p−q_p,q.

Recall that the real operator J which represents the complex structure of the vector space F has the property that if v∈ ∧^1,0F, thenJ v=iv, and if v ∈ ∧^0,1F, then J v= −iv. Thus we see immediately that J defined above is the natural multilinear extension of the complex structure operator J to the exterior algebra of F. We note also that J²=w as linear operators.

We now define a linear mappingLin terms of , the fundamental form associated to the Hermitian structure of E, namely, let

L: ∧F −→ ∧F be defined by L(v)=∧v. We see that

L: ∧^pF −→ ∧^p+2F so it is homogeneous and of degree 2. Moreover,

L: ∧^p,qF −→ ∧^p+1,q+1F

and L is bihomogeneous of bidegree (1, 1), and it is apparent thatL is a real operator since is a real 2-form. Recall from (1.3) that ∧^pF has a natural Hermitian inner product defined by

α, βvol=α∧ ∗ ¯β,

where vol=(1/n!)ⁿ as before. With respect to this inner product L has a Hermitian adjoint

L∗: ∧^pF −→ ∧^p−2F, 2≤p≤2n, and one finds that

(1.13) L^∗=w∗L∗.

To see that (1.13) holds we compute, for α∈ ∧^pF, β∈ ∧^p+2F, Lα, βvol=∧α∧(∗ ¯β)

=α∧∧(∗ ¯β)

=α∧L∗ ¯β

=α∧ ∗w∗L∗ ¯β

=α∧ ∗w∗L∗ ¯β

= α, w∗L∗βvol

= α, L∗βvol

using the fact that∗w∗ =id, and∗, L, andware real operators. It follows from (1.13) that L^∗ is a real operator, homogeneous of degree −2. It will follow from the next proposition that L^∗ is bihomogeneous of degree (−1,−1).

If M and N are two endomorphisms of a vector space, then we will denote by[M, N] =MN−N Mthe commutator of the two endomorphisms.

We now have a basic proposition giving fundamental relationships between the above operators.

Proposition 1.1: LetEbe a Hermitian vector space of complex dimension nwith fundamental formand associated operatorsw, J, L, andL∗. Then

(a) ^∗_p,q =^∗_n−q,n−p,

(b) [L, w] = [L, J] = [L^∗, w] = [L^∗, J] =0, (c) [L^∗, L] =2n

p=0(n−p)_p.

To prove Proposition 1.1, it is necessary to introduce some notation which will allow us to effectively work with the convectors in ∧F. Let N = {1,2, . . . , n}, and let us consider multi-indices I = (µ₁, . . . , µ_p), where µ₁, . . . , µ_p are distinct elements of N, and set |I| =p. Let{z₁, . . . , z_n} be a basis for ∧^1,0F such that the Hermitian metric h on E has the form

h =

µz_µ⊗ ¯z_µ as in (1.7), with given by (1.9), and with (1/n!)ⁿ = vol=x1∧y1∧ · · · ∧x_n∧y_n where z_µ=x_µ+iy_µ, as in (1.10). The operator

∗is now well-defined in terms of the orthonormal basis {x1, y1, . . . , x_n, y_n}. If I =(µ1, . . . , µ_p), then we let

z_I =z_µ1 ∧z_µ2∧ · · · ∧z_µp x_I =x_µ1∧x_µ2 ∧ · · · ∧x_µp

·· If M is a multiindex, we let·

w_M= 2

µ∈M

z_µ∧ ¯z_µ=(−2i)^|M|2

µ∈M

x_µ∧y_µ.

In this last product it is clear that the ordering of the factors is irrelevant, since the terms commute with one another, and we shall use the same symbol Mto denote the orderedp-tuple and its underlying set of elements, provided

that this leads to no confusion. Any element of ∧F can be written in the

form

A,B,M

′c_A,B,Mz_A∧ ¯z_B∧w_M,

where c_A,B,M ∈C, andA, B, andM are (for a given term) mutually disjoint multiindices, and, as before, the prime on the summation sign indicates that the sum is taken over multiindices whose elements are strictly increasing sequences (what we shall call an increasing multiindex).

We have the following fundamental and elementary lemma which shows the interaction between the∗-operator (defined in terms of the real structure) and the bigrading on ∧F (defined in terms of the complex structure).

Lemma 1.2: Suppose that A, B, and M are mutually disjoint increasing multiindices. Then

∗(zA∧ ¯zB∧wM)=γ (a, b, m)zA∧ ¯zB∧wM′

for a nonvanishing constant γ (a, b, m), where a = |A|, b = |B|, m= |M|, and M^′=N−(A∪B∪M). Moreover,

γ (a, b, m)=i^a−b(−1)^p(p+1)/2+m(−2i)^p−n where p=a+b+2m is the total degree ofz_A∧ ¯z_B∧w_M.

Proof: Letυ=z_A∧ ¯z_B∧w_M. IfA=A1∪A2for some multiindexA, let ǫ_A^A1A2 =

⎧⎪

⎨

⎪⎩

0 if A1∩A2=∅

1 if A₁A₂ is an even permutation of A

−1 if A1A2 is an odd permutation of A.

Using this notation it is easy to see that zA=

A=A1∪A2

′ǫ_A^A1A2i^a2xA₁∧yA₂,

where the sum runs over all decompositions ofAinto increasing multiindices A1∪A2, and a1= |A1|, etc. Thus we obtain

υ=(−2i)^m

A=A1∪A2 B=B1∪B2

′ǫ_A^A1A2ǫ^B_B^1B2i^a2−b2x_A1 ∧y_A2∧x_B1 ∧y_B2 ∧2

µ∈M

x_µ∧y_µ. We now want to compute ∗υ, having expressedυ in terms of a real basis, and we shall do this term by term and then sum the result. To simplify the notation, consider the case where B =∅. We obtain

(1.1) ∗(z_A∧w_M)=(−2i)^m

A=A₁∪A₂

ǫ_A^A1A2i^a2∗{x_A1∧y_A2∧2

µ∈M

x_µ∧y_µ}. It is clear that the result of∗acting on the bracketed expression is of the form

(1.2) ±x_A2 ∧y_A1∧ 2

µ∈M^′

x_µ∧y_µ,

where M^′=N−(A∪M). The only problem left is to determine the sign.

To do this it suffices (because of the commutativity of 2

µ∈Mxµ∧yµ) to consider the product (setting a₂= |A₂|)

x_A1 ∧y_A2∧x_A2∧y_A1 =(−1)^a22x_A1 ∧y_A1∧x_A2∧y_A2. Now, in general,

x_C∧y_C =(−1)^{|C|(|C|−1)/2}x_µ1 ∧y_µ2∧ · · · ∧x_µ|C|∧y_µ|C|,

and applying this to our problem above, we see immediately that the sign in (1.2) is of the form

(−1)^{a22+a1(a1−1)/2+a2(a2−1)/2}=(−1)^r. Putting this into (1.1), we obtain

(1.3) ∗(z_A∧w_M)=(−2i)^m

A=A1∪A2

′ǫ_A^A1A2i^a2(−1)^rx_A2∧y_A1 ∧ 2

µ∈M^′

x_µ∧y_µ. The idea now is to change variables in the summation. We write

ǫ_A^A1A2 =(−1)^a1a2ǫ^A_A^2A1 i^a2 =i^a(−1)^a1i^a1, and substituting in (1.3) we obtain

∗(z_A∧w_M)=i^a(−2i)^m

A=A₁∪A₂

′ǫ^A_A^2A1i^a1{(−1)^r+a1+a1a2}

·x_A2∧y_A1∧ 2

µ∈M′

x_µ∧y_µ,

which is, modulo the bracketed term, of the right form to be const(zA∧w_M).

A priori, the bracketed term depends on the decompositions A=A1∪A2; however, one can verify that in fact

(−1)^r+a1+a1a2 =(−1)^a(a+1)/2=(−1)^p(p+1)/2+m,

and the bracketed constant pulls out in front the summation, and we obtain

∗(zA∧wM)=i^a(−1)^p(p+1)/2+m(−2i)^p−nzA∧wM′. The more general case is treated similarly.

Q.E.D.

Proof of Proposition 1.1: Part (a) follows immediately from Lemma 1.2.

We note that (a) is equivalent to

(a^′) ∗|∧^p,qF: ∧^p,qF −→ ∧^n−q,n−pF is an isomorphism.

Part (b) follows from the fact that L and Λ are homogeneous operators and are real.

We shall show part (c). Using the notation used in Lemma 1.2, we observe that

L(z_A∧ ¯z_B∧w_M)= i 2

3ⁿ

µ=1

z_µ∧ ¯z_µ 4

∧z_A∧ ¯z_B∧w_M

= i

2z_A∧ ¯z_B∧3

µ∈M^′

w_M∪{µ}

4 , (1.4)

where M^′=N −(A∪B∪M), as before. On the other hand, we see that, using Lemma 1.2 and the definition of Λ,

(1.5) Λ(z_A∧ ¯z_B∧w_M)= 2

iz_A∧ ¯z_B∧3

µ∈M

w_M−|µ|4 .

Using these formulas, one obtains easily, assuming that z_A∧ ¯z_B∧w_M has total degree p,

ΛL−LΛ=(n−p)z_A∧ ¯z_B∧w_M, and part (c) of Proposition 1.1 follows immediately.

Q.E.D.