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 ahler metric if X is compact and has a
ahler metric. This can be proved using a result of Varouchas [118]. (See [ibid.]
for detail.) First introduce a notion of a ahler morphism. If X point is a
ahler morphism, then X is called a K¨ahler space, and if X is nonsingular, this is
equivalent to the existence of a ahler metric. Then Varouchas showed that
SnX
is a ahler space in this sense. Then apply the following result to the Hilbert-Chow
morphism X[n] SnX: If Y Z is a ahler morphism and Z is a ahler space,
then any relatively compact subset Y of Y is a ahler space. This result can
be applicable since the Hilbert-Chow morphism is projective and any projective
morphism is a 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
.
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