The Jumping Phenomenon of Hodge Numbers
7 The Jumping Phenomenon of Hodge Numbers
October 24, 2018
XuanmingYe
Abstract
Let X be a compact complex manifold, consider a small defor-
mation φ : X → B of X, the dimension of the Dolbeault cohomology
groups Hq(Xt,Ω
) may vary under this defromation. This paper
will study such phenomenons by studying the obstructions to deform
a class in Hq(X,Ω
X) with the parameter t and get the formula for the
obstructions.
1 Introduction
Let X be a compact complex manifold and φ : X → B be a family of
complex manifolds such that φ−1(0) = X . Let Xt = φ
−1(t) denote the
fibre of φ above the point t ∈ B. We denote by OX and ΩpX the sheaves
of germs of X of holomorphic functions and p-forms respectively. Recall
hp,q = dimCH
q(X,Ω
X) and Pm = dimH
0(X, (ΩnX)
m) where n = dimCX .
S.Iitaka proposed a problem whether all Pm are deformation invariants [1].
This problem was solved by Iku Nakamura in his paper [2], and actually
he gave us some examples of small deformations of complex parallelisable
manifold (by a complex parallelisable manifold we mean a compact complex
manifold with the trivial holomorphic tangent bundle) such that the hodge
numbers of the fibre of the family jump in these deformations.
http://arxiv.org/abs/0704.1977v1
In this paper, we will study such phenomenons from the viewpoint of
obstruction theory. More precisely, for a certain small deformation X of X
parametrized by a basis B and a certain class [α] of the Dolbeaut cohomology
group Hq(X,Ω
X), we will try to find out the obstruction to extending it to
an element of the relative Dolbeaut cohomology group Hq(X ,Ωp
). We
will call those elements which have non trivial obstruction the obstructed
elements.
In §2 we will summarize the results of Grauert’s Direct Image Theorems
and we will try to explain why we need to consider the obstructed elements.
Actually, we will see that these elements will play an important role when
we study the jumping phenomenon of Hodge numbers. Because we will see
that the existence of the obstructed elements is a necessary and sufficient
condition for the variation of the Hodge diamond.
In §3 we will get a formula for the obstruction to the extension we men-
tioned above.
Theorem 3.3 Let π : X → B be a deformation of π−1(0) = X, where X is
a compact complex manifold. Let πn : Xn → Bn be the nth order deformation
of X. For arbitrary [α] belongs to Hq(X,Ωp), suppose we can extend [α] to
order n − 1 in Hq(Xn−1,ΩpXn−1/Bn−1). Denote such element by [αn−1]. The
obstruction of the extension of [α] to nth order is given by:
on,n−1(α) = dXn−1/Bn−1 ◦ κnx(αn−1) + κnx◦dXn−1/Bn−1(αn−1),
where κn is the nth order Kodaira-Spencer class and dXn−1/Bn−1 is the relative
differential operator of the n− 1th order deformation.
In §4 we will use this formula to study carefully the example given by Iku
Nakamura, i.e. the small deformation of the Iwasama manifold and discuss
some phenomenons.
Acknowledgement. The research was partially supported by China-France-
Russian mathematics collaboration grant, No. 34000-3275100, from Sun Yat-
sen University. The author would also like to thank ENS, Paris for its hos-
pitality during the academic years of 2005–2007. Last but most, the author
would like to thank Professor Voisin for her patient helps and valuable sug-
gestions.
2 Grauert’s Direct Image Theorems and De-
formation theory
In this section, let us first review some general results of deformation
theory. Let X be a compact complex manifold. The manifold X has an
underlying differential structure, but given this fixed underlying structure
there may be many different complex structures on X . In particular, there
might be a range of complex structures on X varying in an analytic manner.
This is the object that we will study.
Definition 1.0 A deformation of X consists of a smooth proper morphism
φ : X → B, where X and B are connected complex spaces, and an isomor-
phism X ∼= φ−1(0), where 0 ∈ B is a distinguished point. We call X → B a
family of complex manifolds.
Although B is not necessarily a manifold, and can be singular, reducible,
or non-reduced, (e.g. B = SpecC[ε]/(ε2)), since the problem we are going
to research is the phenomenon of the jumping of the Dolbeaut cohomology,
we may assume that X and B are complex manifolds.
In order to study the jumping of the Dolbeaut cohomology, we need the
following important theorem (one of the Grauert’s Direct Image Theorems).
Theorem 1.1 Let X, Y be complex spaces, π : X → Y a proper holomor-
phic map. Suppose that Y is Stein, and let F be a coherent analytic sheaf on
X. Let Y0 be a relatively compact open set in Y . Then, there is an integer
N > 0 such that the following hold.
I. There exists a complex
E · : ...→ E−1 → E0 → ...→ EN → 0
of finitely generated locally free OY0-modules on Y0 such that for any Stein
open set W ⊂ Y0, we have
Hq(Γ(W, E ·)) ≃ Γ(W,Rqπ∗(F)) ≃ Hq(π−1(W ),F) ∀q ∈ Z.
II. (Base Change Theorem). Assume, in addition, that F is π-flat [i.e.
∀x ∈ X, the stalk F
is flat over as a module over OY,π(x)]. Then, there
exists a complex
E · : 0 → E0 → E1 → ...→ EN → 0
of finitely generated locally free OY0-sheaves Ep with the following property:
Let S be a Stein space and f : S → Y a holomorphic map. Let X ′ =
X ×Y S and f
′ → X and π′ : X ′ → S be the two projections. Then, if
T is an open Stein subset of f−1(Y0), we have, for all q ∈ Z
Hq(Γ(T, f ∗(E ·))) ≃ Γ(T,Rqπ′∗(F
)) ≃ Hq(π′−1(T ),F ′)
where F ′ = (f ′)∗(F).
Let X ,Y be complex spaces, π : X → Y a proper map. Let F be a π-flat
coherent sheaf on X . For y ∈ Y , denote by My the OY -sheaf of germs of
holomorphic functions ”vanishing at y”: the stalk of My at y is the maximal
ideal of OY,y; that at t 6= y” is OY,t. We set F(y) = analytic restriction
of F to π−1(y) = F
(OY /My). Since we just need to study the local
properties, we may assume, in view of Theorem 1.0, part II, that there is a
complex
E · : 0 // OP0Y
// OP1Y
// ... d
// OPNY
with the base change property in Theorem 1.0, part II. In particular, if
y ∈ Y , we have
Hq(π−1(y),F(y)) ≃ Hq(E · ⊗ (OY /My)).
Apply what we discussed above to our case φ : X → B, we get the follow-
ing. There is a complex of vector bundles on the basis B, whose cohomology
groups at the point identifies to the cohomology groups of the fiber Xb with
values in the considered vector bundle on X , restricted to Xb. Therefore, for
arbitrary p, there exists a complex of vector bundles (E·, d·), such that for
arbitrary t ∈ B, Hq(Xt,ΩpXt) = H
q(E·t) = Ker(d
q)/Im(dq−1).
Via a local trivialisation of the bundle Ei, the differential of the complex
E· are represented by matrices with holomorphic coefficients, and follows
from the lower semicontinuity of the rank of a matrix with variable coefficients
, it is easy to check that the function dimCKer(d
q) and −dimCIm(dq) are
upper semicontinuous onB. Therefore the function dimCH
q(E·t) is also upper
semicontinuous. It seems that either the increasing of dimCIm(d
q−1) or the
decreasing of dimCKer(d
q) will cause the jumping of dimCH
q(E·t), however,
because of the following exact sequence:
0 → Ker(dq)t → Eqt → Im(dq)t → 0 ∀t,
which means the variation of−dimCIm(dq) is exactly the variation of dimCKer(dq),
we just need to consider the variation of dimCKer(d
q) for all q.
In order to study the variation of dimCKer(d
q), we need to consider the
following problem. Let α be an element of Ker(dq) at t = 0, we try to find
out the obstruction to extending it to an element which belongs to Ker(dq)
in a neighborhood of 0. Such kind of extending can be studied order by
order. Let E q0 be the stalk of the assocaited sheaf of Eq at 0. Let m0 be
the maximal idea of OB,0. For arbitrary positive intergal n, since dq can be
represented by matrices with holomorphic coefficients, it is not difficult to
check dq(E q0 ⊗OB,0 mn0 ) ⊂ E
0 ⊗OB,0 mn0 . Therefore the complex of the vector
bundles (E·, d·) induces the following complex:
0 → E00⊗OB,0OB,0/mn0
d0→ E10⊗OB,0OB,0/mn0
d1→ ... d
→ EN0 ⊗OB,0OB,0/mn0
dN→ 0.
Definition 2.2 Those elements of H ·(E·0) which can not be extended are
called the first class obstructed elements.
Next, we will show the obstructions of the extending we mentioned above.
For simplicity, my may assume that dimCB = 1, suppose α can be extended
to an element αn−1 such that j
q(αn−1))(t) = 0, then αn−1 can be con-
sidered as the n−1 order extension of α. Here jn−10 (dq(αn−1))(t) is the n−1
jet of dq(αn−1) at 0.
Define a map oqn : H
q(E ·0 ⊗OB,0 OB,0/mn0 ) → Hq+1(E·0) by
[αn−1] 7−→ [jn0 (dq(αn−1))(t)/tn].
At first, we need to check oqn is well defined. So we need to show that
[jn0 (d
q(αn−1))(t)/t
n] is dq+1-closed. Via a local trivialization of the bundles
Ei, the differentials of the complex E· are represented by matrices with holo-
morphic coefficients, and from the lower semi-continuity of the rank of a
matrix with variable coefficients, we may assume that there always exists
1 , ..., σ
l ) which are sections of E
q+1 such that (σ
1 |t=0, ..., σ
l |t=0)
form a basis of Ker(dq+1 : E
0 → E
0 ) and Ker(d
q+1 : Eq+1 → Eq+2) ⊂
Span{σq+1j }. So we can write dq(αn−1) =
j fjσ
Since jn−10 (d
q(αn−1))(t)=0, we have fj = 0 and
= 0, i = 1..n− 1.
(dq(αn−1))|t=0 =
j |t=0+...+
j )|t=0 =
j |t=0,
therefore
dq+1(
(dq(α))|t=0) = dq+1(
j |t=0) = 0,
which means ∂
(dq(αn−1))|t=0 is dq+1-closed.
Next we are going to show that the equivalent class of ∂
(dq(αn−1))|t=0
in Hq+1(E·0) depends only on j
0 (αn−1)(t). Let (σ
1, ..., σ
k) be a bases of
Eq, we only need to show that if jn−10 (αn−1)(t) = 0, then
(dq(αn−1))|t=0
belongs to Im(dq : Eq → Eq+1). Indeed, we can write αn−1 =
j fjσ
j while
fj(0) = 0,
= 0, i = 1...n− 1, then,
(dq(αn−1)) =
i )) =
i )+...+
(dq(σ
i )).
Therefore, ∂
(dq(Ω))|t=0 =
i )|t=0, which belongs to Im(dq :
Eq → Eq+1).
At last, we are going to show that the equivalent class of ∂
(dq(Ω))|t=0
in Hq+1(E·0) depends only on the equivalent class of αn−1 in H
q(E ·0 ⊗OB,0
OB,0/mn0 ). Actually, we only need to show that if αn−1 belongs to Im(dq−1 :
E q−10 ⊗OB,0 OB,0/mn0 → E
0 ⊗OB,0 OB,0/mn0 ), we will have ∂
(dq(αn−1))|t=0
belongs to Im(dq : Eq → Eq+1). In fact, let α′n−1 = dq−1(
j fjσ
j ) such
that jn−10 (α
n−1)(t) = j
0 (αn−1)(t). From the discussion above, we have
(dq(αn−1))|t=0 =
(dq(α
n−1))|t=0 =
(dq(dq−1(
j ))) = 0
in Hq(E·).
Remark It seems that jn0 (d
q(αn−1))(t)/t
n depends on the connection of
Eq+1. But, by using an induction argument, it is not difficult to prove that if
ji0(d
q(αn−1))(t) = 0, ∀i < n, then jn0 (dq(αn−1))(t) is independent of the choice
of the connection of Eq+1.
There is natural a map ρ
i : H
q(E·0) → Hq(E ·0 ⊗OB,0 OB,0/m
0 ) given by
[σ] 7−→ [tiσ], ∀[σ] ∈ Hq(E·0).
Denote the map ρ
i ◦oqn : Hq(E ·0⊗OB,0OB,0/mn0 ) → Hq+1(E ·0⊗OB,0OB,0/mi+10 ), ∀i ≤
n by o
Next we will show that, for arbitrary i, 0 < i ≤ n, αn−1 can be extended
to αn which is the nth order extension of α such that j
0 (αn −αn−1)(t) = 0
if and only if o
n,n−i([αn−1]) is trivial. For necessarity, (αn−αn−1)(t)/ti is sup-
posed to be the preimage of o
n,n−i([αn−1]), so o
n,n−i([αn−1]) is trivial. There-
fore we just need to check whether it is sufficient. In fact, if o
n,n−i([αn−1])
is trivial, then there exists a section β of E ·0 ⊗OB,0 OB,0/mi+10 such that
dq(β) = o
n,n−i([αn−1]). Then it is not difficult to check that αn−1 − tiβ̃
is an n th order extension of α that we need, where β̃ is an extension of β in
the neighborhood of 0. Therefore we have the following proposition.
Proposition 2.3 Let αn−1 be an n−1 th order extension of α, for arbitrary
i, 0 < i ≤ n, αn−1 can be extended to αn which is the nth order extension of
α such that ji−10 (αn − αn−1)(t) = 0 if and only if o
n,n−i([αn−1]) = 0.
In the following, we will show that the obstructions oqn([αn−1]) also play
an important role when we consider about the jumping of dimCIm(d
q). Note
that dimCIm(d
q) jumps if and only if there exist a section β of dimCKer(d
q+1),
such that β0 is not exact while βt is exact for t 6= 0.
Definition 2.4 Those nontrivial elements of H ·(E·0) that can always be
extended to a section which is only exact at t 6= 0 are called the second class
obstructed elements.
Note that if α is exact at t = 0, it can be extended to an element which is
exact at every point. So the definition above does not depend on the element
of a fixed equivalent class.
Proposition 2.5 Let [β] be an nontrivial element of Hq+1(E·0). Then [β]
is a second class obstructed element if and only if there exist n ≥ 0 and αn−1
in Hq(E ·0 ⊗OB,0 OB,0/mn0 ) such that oqn([αn−1]) = [β].
Proof. If oqn([αn−1]) = [β], then j
q(αn−1))(t)/t
n is the extension we need.
On the contrary, if [β] is a second class obstructed element. There exist
β̃ such that β̃t, t 6= 0 is exact. Then (dq)−1(β̃) is a meromorphic section
which has a pole at t = 0. Let n be the degree of (dq)−1(β̃). Then let
αn−1 = t
n(dq)−1(β̃). It is easy to check that oqn([αn−1]) = [β].
Proposition 2.6 Let αn−1 be an element of H
q(E ·0 ⊗OB,0 OB,0/mn0 ) such
that oqn([αn−1]) 6= 0. Then there exists n
′ ≤ n and α′ be an element of
Hq(E ·0 ⊗OB,0 OB,0/mn
0 ), such that ρ
◦ oqn([αn−1]) = o
]) 6= 0.
Proof. If o
n,n−1([αn−1]) 6= 0, then n
= n and α
= αn−1. Otherwise, there
exists α
1, such that d
1) = ρ
◦ oqn([αn−1]). Note that o
n−1,n−2([α
1]) =
n−2 ◦ oqn([αn−1]) = o
n,n−2([αn−1]). If we go on step by step as above, we
can always get the n
and α
for there is at least one of the o
n,i([αn−1]) is
nontrivial.
This proposition tells us that althought oqn([αn−1]) 6= 0 does not mean that
n,n−1([αn−1]) 6= 0, we can always find α
such that oqn([αn−1]) comes from
obstuctions like o
n,n−1([α
]). Therefore we can get the following corollary
immediately from Proposition 2.5 and Proposition 2.6.
Corollary 2.7 Let [β] be an nontrivial element of Hq+1(E·0). Then [β] is
a second class obstructed element if and only if there exist n ≥ 0 and αn−1
in Hq(E ·0 ⊗OB,0 OB,0/mn0 ) such that o
n,n−1([αn−1]) = ρ
n−1([β]).
Let us come back to our problem, suppose α can be extended to an
element αn−1 such that j
q(αn−1))(t) = 0, since what we care is whether
α can be extended to an element which belongs to Ker(dq) in a neighborhood
of 0. So, if we have an nth order extension αn of α, it is not necessary that
ji−10 (αn−αn−1)(t) = 0, ∀i, 1 < i < n.What we need is just j00(αn−αn−1)(t) =
0 which means αn is an extension of α. So the “real” obstructions come from
n,n−1([αn−1]). Since these obstructions is so important when we consider the
problem of variation of hodge numbers, we will try to find out an explicit
calculation for such obstructions in next section.
3 The Formula for the Obstructions
We are going to prove in this section an explicit formula (Theorem 3.3) for
the abstract obstructions described above. Let π : X → B be a deformation
of π−1(0) = X , where X is a compact complex manifold. For every integer
n ≥ 0, denote by Bn = SpecOB,0/mn+10 the nth order infinitesimal neighbor-
hood of the closed point 0 ∈ B of the base B. Let Xn ⊂ X be the complex
space over Bn. Let πn : Xn → Bn be the nth order deformation of X . In
order to study the jumping phenomenon of Dolbeaut cohomology groups, for
arbitrary [α] belongs to Hq(X,Ωp), suppose we can extend [α] to order n−1
in Hq(Xn−1,Ω
Xn−1/Bn−1
). Denote such element by [αn−1]. In the following,
we try to find out the obstruction of the extension of [αn−1] to nth order.
Denote π∗(m0) by M0. Consider the exact sequence
0 → Mn0/Mn+10 ⊗ Ω
X0/B0
Xn/Bn
Xn−1/Bn−1
which induces a long exact sequence
0 → H0(X,Mn0/Mn+10 ⊗Ω
X0/B0
) → H0(Xn,ΩpXn/Bn) → H
0(Xn−1,Ω
Xn−1/Bn−1
→ H1(X,Mn0/Mn+10 ⊗ Ω
X0/B0
) → ....
The obstruction for [αn−1] comes from the non trivial image of the connecting
homomorphism δ∗ : Hq(Xn−1,Ω
Xn−1/Bn−1
) → Hq+1(X,Mn0/Mn+10 ⊗Ω
X0/B0
We will calculate it by C̆ech calculation.
Cover X by open sets Ui such that, for arbitrary i, Ui is small enough.
More precisely, Ui is stein and the following exact sequence splits
0 → π∗n(ΩBn)(Ui) → ΩXn(Ui) → ΩXn/Bn(Ui) → 0.
So we have a map ϕi : ΩXn/Bn(Ui) → ΩXn(Ui), such that, ϕi(ΩXn/Bn(Ui))⊕
π∗n(ΩBn)(Ui)
∼= ΩXn(Ui). Denote by ιi, ι−1i the inclusion from π∗n(ΩBn)(Ui)
to ΩXn(Ui) and its inverse. Define d
Xn/Bn
by ϕi ◦ dXn/Bn ◦ ϕ−1i and diBn by
ιi ◦ dBn ◦ ι−1i . Then it determines a local decomposition of the exterior dif-
ferentiation dXn in Ω
dXn = d
+ diXn/Bn .
Denote the set of alternating q-cochains β with values in F by Cq(U,F), i.e.
to each q+1-tuple, i0 < i1... < iq, β assigns a section β(i0, i1, ..., iq) of F over
Ui0 ∩ Ui1 ∩ ... ∩ Uiq .
Let us still using ϕi denote the following map,
ϕi : π
) ∧ Ωp
Xn/Bn
(Ui) → Ωp+rXn (Ui)
ϕi(ωi1 ∧ ... ∧ ωir ∧ βj1 ∧ ... ∧ βjp) = ωi1 ∧ ... ∧ ωir ∧ ϕi(βj1) ∧ ... ∧ ϕi(βjp).
Define ϕ : Cq(U, π∗n(ΩrBn) ∧ Ω
Xn/Bn
) → Cq(U,Ωp+rXn ) by
ϕ(β)(i0, i1, ..., iq) = ϕi0(β(i0, i1, ..., iq)) ∀β ∈ Cq(U, π∗n(ΩrBn) ∧ Ω
Xn/Bn
where i0 < i1... < iq.
Define the total Lie derivative with respect to Bn
LBn : Cq(U,Ω
) → Cq(U,Ωp+1Xn )
LBn(β)(i0, i1, ..., iq) = d
(β(i0, i1, ..., iq)) ∀β ∈ Cq(U,ΩpXn),
where i0 < i1... < iq.
Define, for each Ui the total interior product with respect to Bn, I
(Ui) → ΩpXn(Ui) by
I i(µdg1 ∧ dg2 ∧ ...∧ dgp) = µ
dg1 ∧ ...∧ dgj−1 ∧ diBn(gj)∧ dgj+1 ∧ ... ∧ dgp.
When p = 0, we put I i = 0.
Define λ : Cq(U,ΩpXn) → C
q+1(U,Ω
(λβ)(i0, ..., iq+1) = (I
i0 − I i1)β(i1, ..., iq+1) ∀β ∈ Cq(U,ΩpXn).
Lemma 3.0
λ ◦ ϕ ≡ δ ◦ ϕ− ϕ ◦ δ
mod. π∗n(Ω
) ∧ Ωp−1Xn .
Proof. Define J : Cq(U,Ω
Xn/Bn
) → Cq((U),ΩpXn) by
(J(β))(i0, ..., iq+1) = (−1)(ϕi0 − ϕi1)(β(i1, ..., iq+1),
where i0 < i1 < ... < iq+1. For arbitrary β belongs to C
q(U,Ω
Xn/Bn
(δ ◦ ϕ(β))(i0, ..., iq+1) =
(−1)jϕ(β)(i0, ..., îj, ..., iq+1)
= ϕi1(β)(i1, ..., iq+1)
(−1)jϕi0(β)(i0, ..., îj , ..., iq+1),
while
(ϕ ◦ δ(β))(i0, ..., iq+1) = ϕ(
(−1)j(β)(i0, ..., îj , ..., iq+1))
(−1)jϕi0(β)(i0, ..., îj , ..., iq+1).
So we have δ ◦ ϕ− ϕ ◦ δ = J .
Fix (i0, ..., iq+1) and let ω = β(i1, ..., iq+1). We must show that (I
Ii1)(ϕi1(ω)) = (−1)(ϕi0 −ϕi1)(w) mod π∗n(Ω2Bn)∧Ω
. By linearity, we may
suppose ϕi1(ω) = µdg1 ∧ ... ∧ dgp. Then
ϕi0 = µd
Xn/Bn
(g1) ∧ ... ∧ di0Xn/Bn(gp)
= µ(dg1 − di
Xn/Bn
(g1)) ∧ ... ∧ (dgp − di0Xn/Bn(gp))
= µdg1 ∧ ... ∧ dgp −
µdg1 ∧ ...dgj−1 ∧ di0Bn(gj ∧ dgj+1 ∧ ... ∧ dgp
+terms in π∗n(Ω
) ∧ Ωp−1Xn .
Thus ϕi0 ≡ ϕi1(ω) − I i0 ◦ ϕi1(ω) mod. π∗n(Ω2Bn) ∧ Ω
, and I i1 ◦ ϕi1 = 0.
which means λ ◦ ϕ ≡ J mod. π∗n(Ω2Bn) ∧ Ω
Now we are ready to calculate the formula for the obstructions. Let α̃ be
an element of Cq(U,Ωp
Xn/Bn
) such that its quotient image in Cq(U,Ωp
Xn−1/Bn−1
is αn−1. Then δ
∗([αn−1])= [δ(α̃)] which is an element ofH
q+1(X,Mn0/Mn+10 ⊗
X0/B0
) ∼= mn0/mn+10 ⊗Hq+1(X,Ω
X0/B0
Denote rXn the restriction to the complex space Xn. In order to give the
obstructions an explicit calculation, we need to consider the following map
ρ : Hq(X,Mn0/Mn+10 ⊗ Ω
X0/B0
) → Hq(Xn−1, π∗n−1(ΩBn|Bn−1) ∧ Ω
Xn−1/Bn−1
which is defined by ρ[σ] = [ϕ−1 ◦ rXn−1 ◦ LBn ◦ ϕ(σ)].
Lemma 3.1 The map: ρ : Hq(X,Mn0/Mn+10 ⊗ Ω
X0/B0
Hq(Xn−1, π
n−1(ΩBn|Bn−1) ∧ Ω
Xn−1/Bn−1
) is well defined.
Proof. At first, we need to show that if σ is closed, then ϕ−1 ◦ rXn−1 ◦ LBn ◦
ϕ(σ) is closed, which is equivalent to show that δ ◦ rXn−1 ◦ LBn ◦ ϕ(σ) ≡ 0
mod. π∗n−1(Ω
Bn|Bn−1
) ∧ Ωp−1
Xn|Xn−1
Note that dXn ◦ δ = −δ ◦ dXn . Then
δ ◦ rXn−1 ◦ LBn ◦ ϕ(σ) = rXn−1 ◦ δ ◦ LBn ◦ ϕ(σ)
= −rXn−1 ◦ (δ ◦ d·Xn/Bn + d
Xn/Bn
◦ δ + LBn ◦ δ) ◦ ϕ(σ).
Since
LBn ◦ δ ◦ ϕ(σ) ≡ LBn ◦ (δ ◦ ϕ− λ ◦ ϕ)(σ) ≡ LBn ◦ ϕ ◦ δ(σ) = 0
rXn−1 ◦ (δ ◦ d·Xn/Bn + d
Xn/Bn
◦ δ) ◦ ϕ(σ) = 0,
we have δ ◦ rXn−1 ◦ LBn ◦ ϕ(σ) ≡ 0 mod. π∗n−1(Ω2Bn|Bn−1) ∧ Ω
Xn|Xn−1
Next we need to show that if σ is belongs to Cq(U,Mn0/Mn+10 ⊗Ω
X0/B0
then ϕ−1 ◦ rXn−1 ◦ LBn ◦ ϕ ◦ δ(σ) is exact. In fact, as the calculation above:
rXn−1 ◦LBn ◦ϕ ◦ δ(σ) ≡ −rXn−1 ◦ (δ ◦ d·Xn/Bn + d
Xn/Bn
◦ δ+ δ ◦LBn) ◦ϕ(σ) =
−δ ◦ rXn−1 ◦ LBn ◦ ϕ(σ).
In general, the map ρ is not injective. However, as we mentioned at the
end of the previous section. The “real” obstructions are o
n,n−1([αn−1]), but
not oqn([αn−1]). So we don’t need ρ to be injective. In the following, we will
explain that ρ([δ(α̃)]) is exactly the “real” obstructions we need. In fact,
Hq(Xn−1, π
n−1(ΩBn|Bn−1)∧Ω
Xn−1/Bn−1
) = (ΩBn|Bn−1)⊗OBn−1H
q(Xn−1,Ω
Xn−1/Bn−1
Let m = dimCB, let ti, i = 0...m be the local coordinates of B. Then
ρ([δ(α̃)]) can be written as:
i=0 dti⊗ α̃i, where α̃i ∈ Hq(Xn−1,Ω
Xn−1/Bn−1
For a certain direction ∂
, suppose α̃i 6= 0. Then by a simple calculation, it
is not difficult to check that α̃i = constant[δ(α̃)/ti] in H
q(Xn−1,Ω
Xn−1/Bn−1
While [δ(α̃)/ti] is exactly the obstruction o
n,n−1([αn−1]) in the direction of
we mentioned in the previous section.
Now consider the following exact sequence. The connecting homomor-
phism of the associated long exact sequence gives the Kodaira-Spencer class
of order n [4 1.3.2],
0 → π∗n−1(ΩBn|Bn−1) → ΩXn|Xn−1 → ΩXn−1/Bn−1 → 0.
By wedge the above exact sequence with Ω
Xn−1/Bn−1
, we get a new exact
sequence. The connecting homomorphism of such exact sequence gives us a
map from Hq(Xn−1,Ω
Xn−1/Bn−1
) to Hq+1(Xn−1, π
∗(ΩBn|Bn−1) ∧ Ω
Xn−1/Bn−1
Denote such map by κnx, for such map is simply the inner product with the
Kodaira-Spencer class of order n. By the definition and simply calculation
it is not difficult to proof the following lemma.
Lemma 3.2 Let θ be an element of Hq(Xn−1,Ω
Xn−1/Bn−1
), let θ̃ be an
element of Cq(U,Ωp
Xn/Bn
) such that its quotient image is θ. Then [κnxθ] is
equal to [ϕ−1 ◦ rXn−1 ◦ δ ◦ ϕ(θ̃)].
Let us come back to the problem we discussed, we have
rXn−1 ◦ LBn ◦ ϕ ◦ δ(α̃) ≡ rXn−1 ◦ LBn ◦ (δ ◦ ϕ− λ ◦ ϕ)(α̃)
≡ rXn−1 ◦ LBn ◦ δ ◦ ϕ(α̃)
≡ −rXn−1 ◦ (d·Xn/Bn ◦ δ + δ ◦ d
Xn/Bn
+ δ ◦ LBn) ◦ ϕ(α̃)
≡ −rXn−1 ◦ (d·Xn/Bn ◦ δ + δ ◦ d
Xn/Bn
) ◦ ϕ(α̃)
−δ ◦ rXn−1 ◦ LBn) ◦ ϕ(α̃).
Therefore
[rXn−1 ◦ LBn ◦ ϕ ◦ δ(α̃)] = [−rXn−1 ◦ (d·Xn/Bn ◦ δ + δ ◦ d
Xn/Bn
) ◦ ϕ(α̃)]
= −[d·Xn−1/Bn−1 ◦ rXn−1δ ◦ ϕ(α̃) + rXn−1 ◦ δ ◦ d
Xn/Bn
◦ ϕ(α̃)]
= −[d·Xn−1/Bn−1 ◦ ϕ ◦ ϕ
−1 ◦ rXn−1δ ◦ ϕ(α̃)
+rXn−1 ◦ δ ◦ ϕ ◦ dXn/Bn(α̃)]
= −[ϕ ◦ dXn−1/Bn−1 ◦ ϕ−1 ◦ rXn−1δ ◦ ϕ(α̃)
+rXn−1 ◦ δ ◦ ϕ ◦ ( ˜dXn−1/Bn−1(αn−1))]
= −[dXn−1/Bn−1 ◦ κnxαn−1 + κnx◦dXn−1/Bn−1(αn−1)].
From the discussion above, we get the main theorem of this paper.
Theorem 3.3 Let π : X → B be a deformation of π−1(0) = X, where X is
a compact complex manifold. Let πn : Xn → Bn be the nth order deformation
of X. For arbitrary [α] belongs to Hq(X,Ωp), suppose we can extend [α] to
order n − 1 in Hq(Xn−1,ΩpXn−1/Bn−1). Denote such element by [αn−1]. The
obstruction of the extension of [α] to nth order is given by:
on,n−1(αn−1) = dXn−1/Bn−1 ◦ κnx(αn−1) + κnx◦dXn−1/Bn−1(αn−1),
where κn is the nth order Kodaira-Spencer class and dXn−1/Bn−1 is the relative
differential operator of the n− 1th order deformation.
From the theorem, we can get the following corollary immediately.
Corollary 3.4 Let π : X → B be a deformation of π−1(0) = X, where
X is a compact complex manifold. Suppose that up to order n, the d1 of the
Frölicher spectral sequence vanishes. For arbitrary [α] belongs to Hq(X,Ωp),
it can be extended to order n + 1 in Hq(Xn+1,Ω
Xn+1/Bn+1
4 An Example
In this section, we will use the formula in previous section to study the
jumping of the Hodge numbers hp,q of small deformations of Iwasawa man-
ifold. It was Kodaira who first calculated small deformations of Iwasawa
manifold [2]. In the first part of this section, let us recall his result.
1 z2 z3
0 1 z1
0 0 1
 ; zi ∈ C
∼= C3
1 ω2 ω3
0 1 ω1
0 0 1
 ;ωi ∈ Z+ Z
The multiplication is defined by
1 z2 z3
0 1 z1
0 0 1
1 ω2 ω3
0 1 ω1
0 0 1
1 z2 + ω2 z3 + ω2z1 + ω3
0 1 z1 + ω1
0 0 1
X = G/Γ is called Iwasawa manifold. We may consider X = C3/Γ. g ∈ Γ
operates on C3 as follows:
z′1 = z1 + ω1, z
2 = z2 + ω2, z
3 = z3 + ω1z2 + ω3
where g = (ω1, ω2, ω3) and z
′ = z · g. There exist holomorphic 1-froms
ϕ1, ϕ2, ϕ3 which are linearly independent at every point on X and are given
ϕ1 = dz1, ϕ2 = dz2, ϕ3 = dz3 − z1dz2,
so that
dϕ1 = dϕ2 = 0, dϕ3 = −ϕ1 ∧ ϕ2.
On the other hand we have holomorphic vector fields θ1, θ2, θ3 on X given by
, θ2 =
, θ3 =
It is easily seen that
[θ1, θ2] = −[θ2, θ1] = θ3, [θ1, θ3] = [θ2, θ3] = 0.
in view of Theorem 3 in [2], H1(X,O) is spanned by ϕ1, ϕ2. Since Θ is
isomorphic to O3, H1(X, TX) is spanned by θiϕλ, i = 1, 2, 3, λ = 1, 2.
The small deformation o f X is given by
ψ(t) =
tiλθiϕλt− (t11t22 − t21t12)θ3ϕ3t2.
We summarize the numerical characters of deformations. The deformations
are divided into the following three classes:
i) t11 = t12 = t21 = t22 = 0, Xt is a parallelisable manifold.
ii) t11t22−t21t12 = 0 and (t11, t12, t21, t22) 6= (0, 0, 0, 0),Xt is not parallelisable.
iii) t11t22 − t21t12 6= 0, Xt is not parallelisable.
h1,0 h0,1 h2,0 h1,1 h0,2 h3,0 h2,1 h1,2 h3,0
i) 3 2 3 6 2 1 6 6 1
ii) 2 2 2 5 2 1 5 5 1
iii) 2 2 1 5 2 1 4 4 1
Now let us explain the jumping phenomenon of the Hodge number by
using the obstruction formula. From Corollary 4.3 in [6], it follows that the
Dolbeault cohomology groups are:
H0(X,Ω) = Span{[ϕ1], [ϕ2], [ϕ3]},
H1(X,O) = Span{[ϕ1], [ϕ2]},
H0(X,Ω2) = Span{[ϕ1 ∧ ϕ2], [ϕ2 ∧ ϕ3], [ϕ3 ∧ ϕ1]},
H1(X,Ω) = Span{[ϕi ∧ ϕλ]}, i = 1, 2, 3, λ = 1, 2,
H2(X,O) = Span{[ϕ2 ∧ ϕ3], [ϕ3 ∧ ϕ1]},
H0(X,Ω3) = Span{[ϕ1 ∧ ϕ2 ∧ ϕ3]},
H1(X,Ω2) = Span{[ϕi ∧ ϕj ∧ ϕλ]}, i, j = 1, 2, 3, i < j, λ = 1, 2,
H2(X,Ω1) = Span{[ϕi ∧ ϕ2 ∧ ϕ3], [ϕj ∧ ϕ1 ∧ ϕ3]}, i, j = 1, 2, 3,
H3(X,O) = Span{[ϕ1 ∧ ϕ2 ∧ ϕ3]},
For example, let us first consider h2,0, in the ii) class of deformation. The
Kodaira-Spencer class of the this deformation is ψ1(t) =
λ=1 tiλθiϕλ,
with t11t22−t21t12 = 0. It is easy to check that o1(ϕ1∧ϕ2) = ∂(int(ψ1(t))(ϕ1∧
ϕ2) − int(ψ1(t))(∂(ϕ1 ∧ ϕ2)) = 0, o1(t11ϕ2 ∧ ϕ3 − t21ϕ1 ∧ ϕ3) = ∂((t11t22 −
t21t12)ϕ3 ∧ ϕ2) = 0, and o1(ϕ2 ∧ ϕ3) = −t21ϕ1 ∧ ϕ2 ∧ ϕ1 − t22ϕ1 ∧ ϕ2 ∧ ϕ2,
o1(ϕ1 ∧ ϕ3) = −t11ϕ1 ∧ ϕ2 ∧ ϕ1 − t21ϕ1 ∧ ϕ2 ∧ ϕ2. Therefore, we have shown
that for an element of the subspace Span{[ϕ1∧ϕ2], [t11ϕ2∧ϕ3− t21ϕ1∧ϕ3]},
the first order obstruction is trivial, while, since (t11, t12, t21, t22) 6= (0, 0, 0, 0),
at least one of the obstruction o1(ϕ2 ∧ ϕ3), o1(ϕ1 ∧ ϕ3) is non trivial which
partly explain why the Hodge number h2,0 jumps from 3 to 2. For another
example, let us consider h1,2, in the ii) class of deformation. It is easy to
check that for an element of the subspace (the dimension of such a subspace
is 5) Span{[ϕi∧ϕλ∧ϕ3], [t12ϕ3∧ϕ2∧ϕ3− t11ϕ3∧ϕ1∧ϕ3]}, i = 1, 2, λ = 1, 2,
the first order obstruction is trivial, while at least one of the obstruction
o1(ϕ3 ∧ ϕ2 ∧ ϕ3), o1(ϕ3 ∧ ϕ1 ∧ ϕ3) is non trivial.
Remark 1 It is easy to see that, in the ii) or iii) class of deformation, the
first order obstruction for any element in H1(X,Ω) is trivial. The reason of
Hodge number h1,1’s jumping from 6 to 5 comes from the existence of the
second class obstructed elements o1(ϕ3). After simple calculation, it is not
difficult to get the structure equation of Xt, t 6= 0.
dϕ1 = 0,
dϕ2 = 0,
dϕ3 = −ϕ1 ∧ ϕ2 + to1(ϕ3), i = 1, 2, λ = 1, 2,
which can be considered an example of proposition 2.5.
Remark 2 From the example we discussed above, it is not difficult to
find out the following fact. Let X be an non-Kähler nilpotent complex par-
allelisable manifold whose dimension is more than 2, and φ : X → B be the
versal deformation family of X . Then the Hodge number h1,0 will jump in a
neighborhood of 0 ∈ B. In fact, let ϕi, i = 1...n, n = dimC(X) be the linearly
independent holomorphic 1-forms of X . By the theorem 3 of [2], H1(X,O) is
spanned by a subset of {ϕi}, i = 1..n. So we have ∂ : H1(X,O) → H1(X,Ω)
is trivial, which means one term of the first order obstruction of the holomor-
phic 1-forms vanishes. Let θi, i = 1...n be the dual of ϕi, which are linearly
independent holomorphic vector fields. Since X is non-Kähler, which means
X is not a torus, there exists ϕi such that ∂ϕi 6= 0. Since X is nilpotent, there
exist ϕj such that ∂ϕj = 0. Assume that ∂ϕi = Aϕk ∧ ϕl + ... with A 6= 0.
Consider θkϕj in H
1(X, TX). It is easy to check that o1(∂ϕi, θkϕj) 6= 0.
References
[1] S. Iitaka, Plurigenera and classification of algebraic varieties, Sugaku 24
(1972), 14-27.
[2] Nakamura, I(1975). Complex parallelisable manifolds and their small
deformations, J.Differential Geom. 10, 85-112.
[3] C. Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge
University Press 2002.
[4] C. Voisin, Symétrie miroir, Société Mathématique de France, Paris,
1996.
[5] Bell S. and Narasimhan R., Proper holomorphic mappings of complex
spaces, Encyclopedia of Mathematical Sciences, Several Complex Vari-
ables VI, Springer Verlag, pp. 1-38, 1991.
[6] Cordero, L. A., Fernández, Gray, A. and Ugate, L.(1999). Frölicher
Spectral Sequence of Compact Nilmanifolds with Nilpotent Complex
Structure. New developments in differential geometry, Budapest 1996,
77-102, Kluwer Acad. Publ., Dordrecht.
	Introduction
	Grauert's Direct Image Theorems and Deformation theory
	The Formula for the Obstructions
	An Example
