1.5. THE DOUADY SPACE 15

where Δ is the Laplacian and ∇ is the Levi-Civita connection. Integrating both

sides over X, we have ∇ω ≡ 0, which means that ω is parallel. This shows that the

holonomy group is contained in SU(2n) ∩ Sp(n, C) = Sp(n), where n = 1

2

dimC X.

Since a hyper-K¨ ahler manifold can be defined as a Riemannian manifold whose

holonomy group is contained in Sp(n), this completes the proof.

It follows that X[n] has a hyper-K¨ ahler metric if X is a K3 surface or an abelian

surface. Unfortunately, Yau’s solution to the Calabi conjecture is an existence

theorem, so it does not provide us an explicit description of the metric. Compare

with Corollary 3.42.

Exercise 1.20. Compare Beauville’s symplectic form and Mukai’s symplectic

form on the open set

π−1(S(1,...,1)X), n

where S(1,...,1)X

n

is the open stratum of

SnX

and π is the Hilbert-Chow morphism.

1.5. The Douady space

Although we assumed X to be projective, it is known that the Hilbert schemes

can be generalized to the case X is a complex analytic space. This was done by

Douady and the corresponding objects are called Douady spaces [25].

Many results in this chapter can be generalized to Douady spaces. The following

are some of them. Results in later chapters may also generalized, though we may not

mention explicitly. First of all, the Douady space of n points in X, still denoted

by

X[n]

is a complex space. (Since we may not have an ample line bundle and

cannot define the Hilbert polynomial, the definition must be modified. But it is

straightforward.) The Hilbert-Chow morphism π :

X[n]

→

SnX

is still defined as

a holomorphic map. Fogarty’s result (Theorem 1.15) clearly holds from our proof.

Beauville’s symplectic form can be defined for the Douady space of a complex

surface with a holomorphic symplectic structure.

It is also known that

X[n]

has a K¨ ahler metric if X is compact and has a

K¨ ahler metric. This can be proved using a result of Varouchas [118]. (See [ibid.]

for detail.) First introduce a notion of a K¨ ahler morphism. If X → point is a

K¨ ahler morphism, then X is called a K¨ahler space, and if X is nonsingular, this is

equivalent to the existence of a K¨ ahler metric. Then Varouchas showed that

SnX

is a K¨ ahler space in this sense. Then apply the following result to the Hilbert-Chow

morphism X[n] → SnX: If Y → Z is a K¨ ahler morphism and Z is a K¨ ahler space,

then any relatively compact subset Y of Y is a K¨ ahler space. This result can

be applicable since the Hilbert-Chow morphism is projective and any projective

morphism is a K¨ ahler morphism.

In fact, M.A. de Cataldo and L. Migliorini [22] give a construction of the

Douady spaces and the morphism π :

X[n]

→

SnX

(they call the Douady-Barelet

morphism) by a completely different argument based on Theorem 1.9. We would

like to sketch their argument. Consider the bi-disk Δ = {(z1,z2) ∈

C2

| |zα| 1}.

By Theorem 1.9 and the description of the Hilbert-Chow morphism (see Exam-

ple 1.12(4)), we can construct the Douady space

Δ[n]

as

{Z ∈

(C2)[n]

| π(Z) ∈

Sn(Δ)}

= ((B1,B2,i) mod GLn(C)) ∈

(C2)[n]

the absolute values of the eigenval-

ues of B1, B2 are smaller than 1

.

where Δ is the Laplacian and ∇ is the Levi-Civita connection. Integrating both

sides over X, we have ∇ω ≡ 0, which means that ω is parallel. This shows that the

holonomy group is contained in SU(2n) ∩ Sp(n, C) = Sp(n), where n = 1

2

dimC X.

Since a hyper-K¨ ahler manifold can be defined as a Riemannian manifold whose

holonomy group is contained in Sp(n), this completes the proof.

It follows that X[n] has a hyper-K¨ ahler metric if X is a K3 surface or an abelian

surface. Unfortunately, Yau’s solution to the Calabi conjecture is an existence

theorem, so it does not provide us an explicit description of the metric. Compare

with Corollary 3.42.

Exercise 1.20. Compare Beauville’s symplectic form and Mukai’s symplectic

form on the open set

π−1(S(1,...,1)X), n

where S(1,...,1)X

n

is the open stratum of

SnX

and π is the Hilbert-Chow morphism.

1.5. The Douady space

Although we assumed X to be projective, it is known that the Hilbert schemes

can be generalized to the case X is a complex analytic space. This was done by

Douady and the corresponding objects are called Douady spaces [25].

Many results in this chapter can be generalized to Douady spaces. The following

are some of them. Results in later chapters may also generalized, though we may not

mention explicitly. First of all, the Douady space of n points in X, still denoted

by

X[n]

is a complex space. (Since we may not have an ample line bundle and

cannot define the Hilbert polynomial, the definition must be modified. But it is

straightforward.) The Hilbert-Chow morphism π :

X[n]

→

SnX

is still defined as

a holomorphic map. Fogarty’s result (Theorem 1.15) clearly holds from our proof.

Beauville’s symplectic form can be defined for the Douady space of a complex

surface with a holomorphic symplectic structure.

It is also known that

X[n]

has a K¨ ahler metric if X is compact and has a

K¨ ahler metric. This can be proved using a result of Varouchas [118]. (See [ibid.]

for detail.) First introduce a notion of a K¨ ahler morphism. If X → point is a

K¨ ahler morphism, then X is called a K¨ahler space, and if X is nonsingular, this is

equivalent to the existence of a K¨ ahler metric. Then Varouchas showed that

SnX

is a K¨ ahler space in this sense. Then apply the following result to the Hilbert-Chow

morphism X[n] → SnX: If Y → Z is a K¨ ahler morphism and Z is a K¨ ahler space,

then any relatively compact subset Y of Y is a K¨ ahler space. This result can

be applicable since the Hilbert-Chow morphism is projective and any projective

morphism is a K¨ ahler morphism.

In fact, M.A. de Cataldo and L. Migliorini [22] give a construction of the

Douady spaces and the morphism π :

X[n]

→

SnX

(they call the Douady-Barelet

morphism) by a completely different argument based on Theorem 1.9. We would

like to sketch their argument. Consider the bi-disk Δ = {(z1,z2) ∈

C2

| |zα| 1}.

By Theorem 1.9 and the description of the Hilbert-Chow morphism (see Exam-

ple 1.12(4)), we can construct the Douady space

Δ[n]

as

{Z ∈

(C2)[n]

| π(Z) ∈

Sn(Δ)}

= ((B1,B2,i) mod GLn(C)) ∈

(C2)[n]

the absolute values of the eigenval-

ues of B1, B2 are smaller than 1

.