ON THE JUMPING PHENOMENON OF dimC

ON THE JUMPING PHENOMENON OF dim_CH (Xt,Et)

KWOKWAI CHAN AND YAT-HIN SUEN

Abstract. LetX be a compact complex manifold andE be a holomorphic vector bundle onX. Given a deformation (X,E) of the pair (X, E) over a small polydiskB centered at the origin, we study the jumping phenomenon of the cohomology groups dim_CH^q(X_t,E_t) neart= 0. Generalizing previous results of X. Ye [8, 9] (for the tangent bundleE=TX_t and exterior powers of the cotangent bundle E = Ω^p_X

t), we show that there are precisely two cohomological obstructions to the stability of dim_CH^q(X_t,E_t), which can be expressed explicitly in terms of the Maurer-Cartan element associated to the deformation (X,E). As an application, we study the jumping phenomenon of the dimension of the cohomology groupH¹(Xt,End(TX_t)), which is related to a question raised by physicists [5].

Contents

1. Introduction 1

Acknowledgment 3

2. Deformations of Pairs 3

3. An acyclic resolution forE 4

4. Obstructions 8

5. An application: jumping of dim_CH¹(Xt,End(TX_t)) 14

Appendix A. Convergence 17

References 20

1. Introduction

LetX be a compact complex manifold andπ:X →Bbe a small deformation of X =π⁻¹(0) over a small polydiskB centered at the origin in some complex vector space. Suppose thatE is a coherent sheaf on X which is flat overB. Then (X,E) is a deformation of the pair (X,E|X).

It is known by Grauert’s direct image theorem that the dimension dim_CH^q(Xt,Et) is an upper semi-continuous function in t ∈ B. Moreover, we have the following characterization for when the dimension dim_CH^q(Xt,Et) is locally constant, also due to Grauert.

Theorem 1.1 (Grauert [2]). Let π : X → B be a flat proper holomorphic map between complex analytic spacesX, B withB being reduced and connected. Suppose that E is a coherent sheaf on X that is flat over B. Let k(t) := O_B,t/m_t be the residue field at t ∈ B and E_t be the pullback of E to X_t. Then the following are equivalent:

Date: August 23, 2018.

1

(a) The function

t7→dim_CH^q(Xt,Et) is locally constant int∈B.

(b) The sheafR^qπ_∗E is locally free and the natural map R^qπ_∗E ⊗k(t)→H^q(Xt,Et) is an isomorphism.

However, condition (b) in the theorem above is not easy to check in general even when E is locally free. In [8, 9], X. Ye studied the jumping phenomenon of the dimensions dim_CH^q(X,•) under small deformations ofX, where•= Ω^p_X, T_X. He found two explicit obstructions O_n,n−1^q , O^q−1_n,n−1 and proved that the dimension of H^q(X,•) does not jump if and only ifO_n,n−1^q ≡0 andO_m,m−1^q−1 ≡0 for alln, m≥1.

In this paper, we generalize Ye’s results to a much more general setting, namely, whenX is a compact complex manifold andE is an arbitrary holomorphic vector bundle on X. Let (X,E) be a small deformation of (X, E) over a polydisk B centered at the origin in some finite dimensional complex vector space. We assume thatE is flat overB via the proper holomorphic submersionπ:X →B. LetX_t:=

π⁻¹(t) and E_t := E|_Xt. We are interested in characterizing when the dimension dim_CH^q(X_t,E_t) stays constant neart= 0.

Following [8, 9], we formulate the jumping phenomenon of dim_CH^q(Xt,Et) as an extension problem, namely, whether we can extend a nonzero element inH^q(X, E) to one in a nearby fiberH^q(Xt,Et). In general, such extensions may not exist and it suffices to find obstructions to this extension problem. We will see that Ye’s explicit formulae for the obstructions can be generalized to our general setting. On the other hand, while Ye applied a version of Grauert’s direct image theorem, which states thatR^qπ∗E is a quotient of two locally free sheaves of finite ranks overB, thereby allowing him to apply an algebraic approach, here we adapt a differential-geometric approach, following [4, 1].

We will formulate the problem directly as extendingE-valued differential forms overB, which means that, in contrast to [8, 9], we are going to work with sheaves ofinfinite rank. A key step is to obtain an explicit description ofR^qπ_∗E, using an acyclic resolution (D^•,D¯^•) of the sheafEconstructed from the differential operators D¯^• studied in [4, 1] (see Section 3). The operators ¯D^• capture the holomorphic structures of the deformed pairs{(X_t,E_t)}_t∈B(see [1] or Section 2 in this paper). It turns out that essentially the same strategy as in Ye’s proofs works. An advantage of our geometric approach is that the computation of the obstructions becomes much neater and more transparent, as compared to the ˇCech calculations in [8, 9]. Our main result is as follows (see Section 4, in particular, Theorem 4.11 and Equations (1) & (2) for the details):

Theorem 1.2. Let{(A(t), ϕ(t))}_t∈B be the family of Maurer-Cartan elements as- sociated to the small deformation (X,E) of (X, E). We define the n-th order ob- struction maps Oⁱ_n,n−1 : Hⁱ((π_∗D^•)0⊗ OB,0/mⁿ₀) → Hⁱ⁺¹((π_∗D^•)0⊗ OB,0/mⁿ₀), wherei=q, q−1, by

Oⁱ_n,n−1([α_n−1]) =



tⁿ⁻¹

n−1

X

j=0

(ϕ^n−jy∇+A^n−j)α^j_n−1



.

ON THE JUMPING PHENOMENON OF dim_CH (Xt,Et) 3

Then the functiont7→dim_CH^q(Xt,Et)is locally constant if and only ifO^q_m,m−1≡0 andO_n,n−1^q−1 ≡0 for allm, n≥1.

We apply this theorem to study the jumping phenomenon of the dimension dim_CH¹(Xt,End(T_Xt)), which is related to a question raised by physicists [5]. It is conjectured that dim_CH¹(Xt,End(T_Xt)) does not jump along any deformation of a Calabi-YaumanifoldX. What we obtain is the following weaker statement (see Section 5):

Theorem 1.3. (=Theorem 5.3) Suppose thatX is a Calabi-Yau manifold such that the deformation of the pair (X, TX) is unobstructed, then dim_CH¹(Xt,End(T_Xt)) does not jump att= 0for any small deformation of X .

Acknowledgment

We are grateful to Xuanming Ye and Robert Lazarsfeld for various useful discus- sions via emails. We would also like to thank Conan Leung for encouragement and some useful suggestions, and also the referees for valuable comments. The work of the first named author described in this paper was substantially supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK404412 & CUHK400213).

2. Deformations of Pairs

In this section, we briefly review the deformation theory of a pair (X, E), where X is a compact complex manifold and E is a holomorphic vector bundle on X, following the exposition in [1] (cf. [4]), and recall several useful facts.

Definition 2.1. Let B be a small polydisk in some finite dimensional complex vector space containing the origin. A deformation of (X, E) over B consists of a surjective proper holomorphic submersion π:X →B from a complex manifoldX to B, together with a holomorphic vector bundle E on X, such thatπ⁻¹(0) =X andE|_π⁻¹₍₀₎=E.

Given such a deformation of (X, E), we putXt:=π⁻¹(t) andEt:=E|_Xt. Since B is contractible, a theorem of Ehresmann implies that we can choose a diffeomor- phismF :X →X×Band a bundle isomorphismF⁰:E →E×B coveringF such that F, F⁰ are holomorphic with respect tot. Notice that there are two complex structures on X×B: one comes from the push-forward of the complex structure onX and the other comes from the product structure on X×B; we denote these complex structures byJ andJ0, respectively.

Letϕ(t)∈Ω^0,1(TX) be the family of Maurer-Cartan elements which corresponds to the family X → B. In [1], we considered a holomorphic family of differential operators ¯D^q_t : Ω^0,q(E)→Ω^0,q+1(E) defined locally by

D¯_t^q



 X

j

αj⊗ej(t)



:=X

j

( ¯∂+ϕ(t)y∂)αj⊗ej(t),

where {ej(t)} is the push-forward of a local holomorphic frame on Et by F⁰. By choosing a Hermitian metric on E, one can express the operator ¯Dt, in terms of the associated Chern connection∇, as

D¯^q_t = ¯∂_E+ϕ(t)y∇+A(t),

for someA(t)∈Ω^0,1(End(E)).

Then (one direction of) Theorem 1.2 in [1] says that the family of elements (A(t), ϕ(t))∈Ω^0,1(A(E)) satisfies the Maurer-Cartan equation

∂¯A(E)(A(t), ϕ(t)) +1

2[(A(t), ϕ(t)),(A(t), ϕ(t))] = 0

fort∈B; hereA(E) is the Atiyah extension ofE. This in turn is equivalent to the fact that the family of operators{D¯_t^q} satisfies the integrability condition:

D¯^q_tD¯^q−1_t = 0.

Another important feature of the operator ¯Dt, which is going to be useful later, is that its cohomology computes precisely the Dolbeault cohomology of (Xt,Et):

Proposition 2.2 ([1], Proposition 3.13). For each fixedt∈B, we have H^q(Xt,Et)∼=H^q((π_∗D^•)_t⊗k(t))∼=H^q Ω^0,•(E),D¯_t

,

for any q≥0.

3. An acyclic resolution for E

From this point on, for the purpose of simplifying computations and formulae, we will assume that the base B of the deformation is of complex dimension one.

We also abuse notations by writing X for the complex manifold (X×B,J) and E for the vector bundle E×B equipped with the holomorphic structure induced fromE via pushing forward byF⁰:E →E×B.

In this section, we will construct an acyclic resolution of the sheafE in order to get an explicit description of the direct image sheafR^qπ∗E.

To begin with, we define an operator ¯∂_E,B: Ω^0,q_J

0(E)→Ω^0,q+1_J

0 (E) by

∂¯_E,B

X

j

sjej(t)

:=X

j

∂¯Bsj⊗ej(t).

Here Ω^0,•_J

0 is the space of smooth (0,•)-forms on X×B with respect to the prod- uct complex structure J₀. If we choose a different frame f_k(t), then e_j(t) = P

kg^k_j(t)fk(t) for some local smooth functionsg_j^k onX×B which are holomorphic int. Hence ¯∂E,B is well-defined.

For each q ≥ 0, we define a sheaf of OX-modules D^q, which plays a key role throughout this paper: Let π_X :X → X be the projection onto X (which is not necessarily holomorphic). The sheafU 7→Ω^0,k_J

0(U,E) has a typical direct summand given by

De^q,p:=π^∗Ω^0,p_B ⊗π^∗_XΩ^0,q⊗ E, p+q=k,

which carries an OX-module structure via multiplication by J-holomorphic func- tions. The operator ¯∂E,B acts on L

p≥0De^q,p, so we obtain a complex (De^q,•,∂¯_E,B^• ) for eachq. We then define the sheaf ofO_X-modulesD^q by

D^q :U 7→ {s∈Γsmooth(U,De^q,0)|∂¯_E,Bs= 0}.

Clearly, D^• ⊂De^•,0 as OX-submodules. Since ¯Dt varies holomorphically in the variablet, it induces a sheaf map ¯D^q :D^q → D^q+1 for eachq≥0. Moreover, since the kernel of ¯D_t: Ω⁰(E)→Ω^0,1(E) is precisely the space of holomorphic sections

ON THE JUMPING PHENOMENON OF dim_CH (Xt,Et) 5

of Et, the sheaf E, as a sheaf of OX-modules, can be identified with the following sheaf ofOX-modules:

U 7→ {s∈Γsmooth(U,E)|Ds¯ = ¯∂_E,Bs= 0}.

The push-forwards ofDe^q,pand D^q byπ:X →B carry naturalOB-module struc- tures via multiplication.

Lemma 3.1. For eachp, q≥0, the sheafDe^q,pis fine and the complex(π_∗De^q,•, π_∗∂¯_E,B^• ) has no higher cohomology sheaves, i.e.,

H^p(π∗De^q,•) = 0 for allp≥1.

Proof. Fineness is clear because we can apply a partition of unity to conclude that De^q,phas no higher direct images.

To prove that (π_∗De^q,•, π_∗∂¯_E,B^• ) has no higher cohomology, we recall thatH^p(π_∗De^q,•) is the sheafification of

W 7→H^p(Γ(π⁻¹(W),De^q,•)).

It suffices to prove that H^p(Γ(π⁻¹(W),De^q,•)) = 0 for any polydisk W ⊂ B and allp≥1. Let α∈Γ(π⁻¹(W),De^q,p) and{Ui} be a locally finite open covering of X ⊂ π⁻¹(W) by coordinates charts. Let α_i be the restriction of α on U_i×W. Write

α_i=X

I,J

α_IJ,i(z,z, t,¯ ¯t)d¯t^J⊗d¯z^I =:X

I

α_I,i⊗d¯z^I.

Then ¯∂_E,Bα= 0 simply means that, for eachI, 0 = ¯∂_E,B X

J

αIJ,id¯t^J

!

= ¯∂BαI,i.

Hence, for fixedz, we can apply the Dolbeault lemma onW to conclude that αI,i= ¯∂BβI,i,

for some βI,i∈Ω^0,p−1_B (W). SinceαI,i varies smoothly inzand ¯z, we see from the proof of the Dolbeault-Grothendieck lemma thatβI,i can be chosen to be smooth in z as well. Let {ψi} be a partition of unity on X subordinate to the covering {Ui}. Define

β:=X

I,i

ψ_iβ_I,i⊗d¯z^I.

Thenβ∈Γ(π⁻¹(W),De^q,p−1) and

∂¯E,Bβ =X

I,i

ψi( ¯∂BβI,j)⊗d¯z^I =X

I,i

ψiαI,i⊗d¯z^I = X

i

ψi

! α=α.

We have the first equality simply because {ψi} are all independent of t and ¯t, and the second last equality follows from the fact that α is a global section on

π⁻¹(W).

Lemma 3.2. For each q≥0, the sheaf D^q is acyclic with respect to the left-exact functorπ∗.

Proof. Lemma 3.1 shows that (De^q,•,∂¯^•_E,B) is a fine resolution ofD^qand soR^qπ∗D^q∼=

H^p(π_∗De^q,•) = 0 for allq≥1.

Proposition 3.3. The complex of sheaves (D^•,D¯^•) is an acyclic resolution of E with respect to the left-exact functorπ_∗. In particular, we have

R^qπ∗E ∼=H^q(π∗D^•) asOB-modules.

Proof. By Lemma 3.2,R^pπ_∗D^q = 0 for allp≥1. It remains to prove that it defines a resolution of E. We need to show that for any point (x, t)∈ X = X×B, the sequence of stalks

0→ E_(x,t)→ D_(x,t)⁰ → D¹_(x,t)→ · · · is exact. The exactness of

0→ E(x,t)→ D⁰_(x,t)→ D¹_(x,t)

follows from the fact that ¯D⁰ and ¯∂_E share the same kernel. For the remaining exactness, we will focus on the case t = 0; the same argument works for general t∈B.

Recall that ¯Dtis locally defined by D¯_t



 X

j

α_j⊗e_j(t)



:=X

j

( ¯∂+ϕ(t)y∂)α_j⊗e_j(t), so it suffices to prove the exactness for the caseE=O_X.

We would like to first work over C[[t]] instead of C{t} (where the latter is the ring of convergent power series). LetU ⊂X be a polydisk, and denote

Ω^0,•(U){t}:= Ω^0,•(U)⊗_CC{t},

Ω^0,•(U)[[t]] := Ω^0,•(U)⊗_CC[[t]] = Ω^0,•(U){t} ⊗_C{t}C[[t]].

The Maurer-Cartan elementϕ(t) is gauge equivalent to 0 onU. Hence

∂¯+ϕ(t)y∂=e^v(t)∂e¯ ^−v(t)

for somev(t)∈Ω⁰(TU)[[t]], wheree^v(t)acts on Ω^0,q(U)[[t]] by e^v(t)α(t) =

∞

X

n=0

(v(t)y∂)ⁿ n! α(t).

We can then apply the Dolbeault-Grothendieck lemma with analytic parameter (thet-variable) to conclude that (Ω^0,•(U)[[t]],D¯_t^•) is an exact complex.

Now, asC[[t]] is a flat-C{t} module (becauseC[[t]] is torsion free andC{t}is a PID), we have

H^q(Ω^0,•(U)[[t]]) =H^q(Ω^0,•(U){t} ⊗C[[t]])∼=H^q(Ω^0,•(U){t})⊗C[[t]].

But we have shown thatH^q(Ω^0,•(U)[[t]]) = 0. Therefore,H^q(Ω^0,•(U){t})⊗C[[t]] = 0. If we can show thatH^q(Ω^0,•(U){t}) is torsion free, we see thatH^q(Ω^0,•(U){t}) vanishes. Assuming this, we conclude that every ¯D_t-closed (0, q)-form valued power series onU is locally exact.

Now, for any ¯D_(x,0)^q -closed element α ∈ D^q_(x,0), we can represent it by a ¯Dt- closed element α(t) ∈ Ω^0,q(U){t}, for some polydisk U ⊂ X. The vanishing of H^q(Ω^0,•(U){t}) shows thatα(t) = ¯D_tβ(t) for someβ(t)∈Ω^0,q−1(U){t}. Thisβ(t)

ON THE JUMPING PHENOMENON OF dim_CH (Xt,Et) 7

defines an elementβ ∈ D^q−1_(x,0)such that ¯D^q−1β =α. This proves the exactness of the complex (D^•,D¯^•).

To complete the proof of the proposition, we need to prove thatH^q(Ω^0,•(U){t}) is a torsion freeC{t}-module for q > 1. In other words, we need to show that if [α(t)] ∈H^q(Ω^0,•(U){t}) is a nonzero element, then f(t)·[α(t)] is nonzero for all f(t)∈C{t} − {0}. Sincef(t) is invertible iff(0)6= 0, we may assumef(t)∈(t^N) for someN ≥1. We may assumeNis chosen such thatf(t) =t^Ng(t) withg(0)6= 0.

Again, we can invertg(t), so we can further assumef(t) =t^N. Then the vanishing off(t)·[α(t)] = [f(t)·α(t)] means

t^Nα(t) = ¯Dtβ(t) = ( ¯∂+ϕ(t)y∂)β(t),

for someβ(t)∈Ω^0,q−1(U){t}. Since bothα(t) andβ(t) are holomorphic int, the equation shows thatβ(t) is in fact ¯Dt-closed up to orderN−1.

We first prove the following

Lemma 3.4. For any∂-closed¯ β ∈Ω^0,q−1(U),q >1, there existsβ(t)∈Ω^0,q−1(U){t}

such that

β(0) =β andD¯_tβ(t) = 0.

Proof of Lemma 3.4. Since β is ¯∂-closed on the polydisk U, it must be ¯∂-exact.

Writeβ= ¯∂αfor someα∈Ω^0,q−2(U). Define

β(t) :=β+ϕ(t)y∂α∈Ω^0,q−1(U){t}.

Thenβ(0) =β. Since ¯D²_t = 0, we have

D¯tβ(t) = ¯∂ϕ(t)y∂α+ϕ(t)y∂∂α¯ +ϕ(t)y∂β+1

2[ϕ(t), ϕ(t)]y∂α

=

∂α¯ +1

2[ϕ(t), ϕ(t)]

y∂α+ (ϕ(t)y∂∂α¯ −ϕ(t)y∂∂α)¯

= 0,

as desired.

With this lemma in hand, we see that α(t) is ¯Dt-exact and this proves that H^q(Ω^0,•(U){t}) is torsion free.

Sinceβ0 is ¯∂-closed, we can chooseβ1(t)∈Ω^0,q−1(U){t}such that β₁(0) =β₀ and ¯D_tβ₁(t) = 0.

Then we have

t^N−1α(t) = ¯Dt

β(t)−β1(t) t

= ¯Dtγ1(t).

If N = 1, we are done. Otherwise, by evaluating at t = 0, we see that γ₁(0) is

∂-closed. Hence we can find¯ β₂(t) such that

β2(0) =γ1(0) and ¯Dtβ2(t) = 0.

Hence

t^N−2α(t) = ¯D_t

γ₁(t)−β₂(t) t

. Repeating this process, we will arrive at the conclusion that

α(t) = ¯DtγN(t)

for someγN(t)∈Ω^0,q−1(U){t}. This completes the proof of the proposition.

4. Obstructions

In this section, we will find out explicitly the obstruction maps for extending a given element ofH^q(X, E). In [8, 9], X. Ye used Grauert’s direct image theorem to obtain a complex of locally free OB-modules of finite ranks to compute the obstruction maps; here we will instead use the infinite-dimensional complex ofOB- modules (π_∗D^•,D¯^•). We will see that more or less the same strategy of proofs in [8, 9] is going to work in our infinite-dimensional setting as well. We will give most of the details of the proofs in order to make this paper more self-contained.

Recall that Proposition 3.3 gives an isomorphism ofOB-modules:

R^qπ∗E ∼=H^q(π∗D^•).

Together with Proposition 2.2, we see that it is equivalent to work with the sheaf H^q(π_∗D^•) and the cohomology group H^q((π_∗D^•)₀⊗k(0)). Tensoring the stalk (π∗D^•)0 withOB,0/mⁿ⁺¹₀ overOB,0, we obtain a complex

((π_∗D^•)₀⊗ O_B,0/mⁿ⁺¹₀ ,D¯^•_n), where ¯D^•_n is naturally induced from ¯D^•.

Given α ∈ ker( ¯∂_E^q), and supposing that we have a local extension α_n−1 ∈ Γ(U, π∗D^q) ofαsuch that

j₀ⁿ⁻¹( ¯D^qα_n−1)(t) = 0, we define theobstruction map

O^q_n,n−1:H^q((π_∗D^•)0⊗ OB,0/mⁿ₀)→H^q+1((π_∗D^•)0⊗ OB,0/mⁿ₀) by

O^q_n,n−1[jⁿ⁻¹₀ (α_n−1)(t)] := [tⁿ⁻¹·(j₀ⁿ( ¯D^qα_n−1)(t)/tⁿ)]

(1)

Remark 4.1. The(n−1)-st jet can be viewed as an element in(π_∗D^q)₀⊗O_B,0/mⁿ₀. The mapO^q_n,n−1 factors through a map

O^q_n:H^q((π_∗D^•)₀⊗ OB,0/mⁿ₀)→H^q+1((π_∗D^•)₀⊗ OB,0/m₀), given by

O_n^q[j₀ⁿ⁻¹(α_n−1)(t)] := [j₀^q( ¯D^qα_n−1)(t)/tⁿ].

This is well-defined because the cohomology class of j₀^q(d^qα_n−1)(t)/tⁿ only depends on the cohomology class of the (n−1)-st jetjⁿ⁻¹₀ (α_n−1)(t).

For later use, we also define

O^q_n,i[j₀ⁿ⁻¹(α_n−1)(t)] := [tⁱ·(j₀^q( ¯D^qα_n−1)(t)/tⁿ)], fori≥0 andn≥1

The following proposition characterizes when an extension exists up to order n≥1.

Proposition 4.2. For a fixedn≥1, the following are equivalent:

(1) For any local section αn−1 around t = 0 such that j₀ⁿ⁻¹( ¯D^qαn−1)(t) = 0, there exists a local section αn around t = 0 such that j₀⁰(αn−αn−1) = 0 andj₀ⁿ( ¯D^qαn)(t) = 0.

(2) For any cn−1 ∈H^q((π∗D^q)0⊗ OB,0/mⁿ₀), there exists cn ∈H^q((π∗D^q)0⊗ OB,0/mⁿ⁺¹₀ )such thatc_n|t=0 =c_n−1|t=0∈H^q((π_∗D^q)₀⊗k(0)).

ON THE JUMPING PHENOMENON OF dim_CH (Xt,Et) 9

(3) For any local section αn−1 around t = 0 such that j₀ⁿ⁻¹( ¯D^qαn−1)(t) = 0, O_n,n−1^q [j₀ⁿ⁻¹(α_n−1)(t)] = 0.

Proof. We shall prove that (1)⇔(2) and (1)⇔(3).

For (1)⇒ (2) : Letc_n−1 ∈H^q((π_∗D^•)₀⊗ O_B,0/mⁿ₀) andα_n−1 be a local sec- tion aroundt= 0 such that j₀ⁿ⁻¹(α_n−1)(t)∈ker( ¯D_n−1^q ) represents the classc_n−1. Then j₀ⁿ⁻¹( ¯D^qα_n−1)(t) = 0. By assumption, we can extend α_n−1 to a local sec- tion αn around t = 0 such that j₀⁰(αn −α_n−1)(t) = 0 and j₀ⁿ( ¯D^qαn)(t) = 0.

Then ¯D^q_n(j₀ⁿ(α_n)(t)) = 0 ∈ (π_∗D^q+1)₀⊗ O_B,0/mⁿ⁺¹₀ . Set c_n := [j₀ⁿ(α_n)(t)] ∈ H^q((π_∗D^•)0⊗OB,0/mⁿ⁺¹₀ ). Sincej₀⁰(αn−α_n−1) = 0, we havecn|t=0= [j⁰₀(α_n−1)(t)] = c_n−1|t=0= 0.

For (2)⇒(1) : Let α_n−1 be such that j₀ⁿ⁻¹( ¯D^qα_n−1)(t) = 0. Extend c_n−1 :=

[j₀ⁿ⁻¹(α_n−1)(t)] to a classcn∈H^q((π_∗D^•)0⊗ OB,0/mⁿ⁺¹₀ ). Letαnbe local section aroundt = 0 such thatj₀ⁿ(αn)(t) represents the class cn. Then j₀ⁿ( ¯D^qαn)(t) = 0.

Sincecn|t=0=c_n−1|t=0, we have

j₀⁰(αn−αn−1) = ¯D^q−1₀ γ,

for someγ∈(π_∗D^q−1)0⊗k(0). Choose any representativeγ⁰ ofγ and define α⁰_n:=αn−D¯^q−1γ⁰.

Thenj₀ⁿ( ¯D^qα⁰_n)(t) =j₀ⁿ( ¯D^qα_n)(t) = 0 andj₀⁰(α⁰_n−α_n−1) = 0.

For (1)⇒(3) : Letγ:=α_n−1−α_n. Then

D¯_n^qγ=j₀ⁿ( ¯D^q(αn−1−αn))(t) =tⁿ·(j₀ⁿ( ¯D^qαn−1)(t)/tⁿ),

sincej₀ⁿ⁻¹( ¯D^qα_n−1)(t) =j₀ⁿ(d^qα_n)(t) = 0. By assumption,j₀⁰(γ)(t) = 0, soγ=tβ for some local sectionβ aroundt= 0. Hence

D¯^q_n−1j₀ⁿ⁻¹(β)(t) =tⁿ⁻¹·(j₀ⁿ( ¯D^qα_n−1)(t)/tⁿ), which means thatO_n,n−1^q [j₀ⁿ⁻¹(α_n−1)(t)] = 0.

For (3) ⇒ (1) : The vanishing of O^q_n,n−1[jⁿ⁻¹₀ (α_n−1)(t)] gives an element β ∈ (π∗D^q)0⊗ OB,0/mⁿ₀ such that

tⁿ⁻¹·(j₀ⁿ( ¯D^qα_n−1)(t)/tⁿ) = ¯D^q_nβ.

Let β⁰ be a local section around t = 0 representing the germ β and set α_n :=

α_n−1−tβ⁰. Then

j₀ⁿ( ¯D^qαn)(t) =j₀ⁿ( ¯D^qα_n−1)(t)−t·j₀ⁿ⁻¹( ¯D^qβ⁰)(t)

=tⁿ·(j₀ⁿ( ¯D^qα_n−1)(t)/tⁿ)−t·D¯^q_nβ= 0.

Henceαn defines ann-th order extension ofα.

Therefore, if O^q_n,n−1 ≡ 0 for all n ≥ 1, then by (1) above we obtain a formal element α(t) such that ¯Dtα(t) = 0. In Appendix A, we show that after a gauge fixing,α(t) is analytic in a neighborhood around 0∈B.

Remark 4.3. The radius of convergence of each extension α(t) may be different asα=α(0) varies. However, sinceH^q(X, E)is finite dimensional, we can simply choose a basis, for instance, one consisting of harmonic forms with respect to a fixed hermitian metric. Then we obtain a minimum radius of convergence, uniform in all [α]∈H^q(X, E).

Next we shall demonstrate that there is another obstruction for an extension to be nonzero.

Proposition 4.4. A non-exact elementβ ∈ker( ¯∂_E^q)admits a local extensionβ(t)∈ Γ(U, π_∗D^q) such that β(t) is exact for t 6= 0 if and only if there exist n ≥ 1 and [j₀ⁿ⁻¹(α_n−1)(t)]∈H^q−1((π_∗D^•)₀⊗ O_B,0/mⁿ₀)such that

O^q−1_n [j₀ⁿ⁻¹(αn−1)(t)] = [β].

Proof. Suppose thatO_n^q−1[j₀ⁿ⁻¹(αn−1)(t)] = [β]. Then β =j₀ⁿ( ¯D^q−1αn−1)(t)/tⁿ+ ¯∂_E^q−1γ for someγ∈Ω^0,q−1(E). Defineβ(t) by

β(t) := ¯D^q−1(α_n−1(t)/tⁿ) + ¯D^q−1γ(t), t6= 0,

whereγ(t) is any extension of γ. Clearlyβ(t) can be extended through the origin by settingβ(0) =β. Thenβ(t) is a ¯D^q−1-exact class and equalsβ att= 0. Hence β(t) serves as an extension of β which is ¯D^q−1-exact fort6= 0.

Conversely, ifβ(t) is an extension ofβ such that β(t) = ¯D^q−1γ(t)

fort6= 0. Thenγ(t) can be chosen to be meromorphic in twith pole order n≥1 att= 0. Letα_n−1(t) :=tⁿγ(t). Thenα_n−1(t) is holomorphic intand

O^q−1_n [j₀ⁿ⁻¹(α_n−1)(t)] = [j₀ⁿ( ¯D^q−1(tⁿγ(t))/tⁿ] = [j₀ⁿ(tⁿβ(t))/tⁿ] = [β].

This completes the proof.

Proposition 4.5. Let [j₀ⁿ⁻¹(αn−1)(t)] ∈ H^q−1((π∗D^•)0 ⊗ OB,0/mⁿ₀) such that O_n^q−1[j₀ⁿ⁻¹(α_n−1)(t)]6= 0. Then there existn⁰≤nand[jⁿ₀⁰⁻¹(α_n⁰₋₁)(t)]∈H^q−1((π_∗D^•)₀⊗ OB,0/mⁿ₀⁰) such that

O_n,n^q−10−1[j₀ⁿ⁻¹(α_n−1)(t)] =O^q−1_n0,n⁰−1[j₀ⁿ⁰⁻¹(α_n⁰₋₁)(t)]6= 0.

Proof. IfO_n,n−1^q−1 [j₀ⁿ⁻¹(αn−1)(t)]6= 0, we can simply taken⁰ =nandαn⁰−1=αn−1. Otherwise, there existsα⁰₁ such that

D¯_n−1^q−1α⁰₁=O_n,n−1^q−1 [j₀ⁿ⁻¹(α_n−1)(t)].

Then we have

O_n−1,n−2^q−1 [α⁰₁] =O^q−1_n,n−2[j₀ⁿ⁻¹(αn−1)(t)].

SinceO_n^q−1[j₀ⁿ⁻¹(αn−1)(t)]6= 0, we finally arrive at somen⁰ such that O_n,n^q−10−1[j₀ⁿ⁻¹(α_n−1)(t)] =O^q−1_n0,n⁰−1[j₀ⁿ⁰⁻¹(αn⁰−1)(t)]6= 0.

These two propositions together prove the following

Corollary 4.6. Every local extension of every non-exact element β ∈ ker( ¯∂_E^q) is non-exact if and only ifO_n,n−1^q−1 ≡0for all n≥1.

ON THE JUMPING PHENOMENON OF dim_CH (Xt,Et) 11

Proof. For a fixed non-exactβ ∈ker( ¯∂_E^q), if any extension of β is non-exact, then [β]∈/ Im(O^q−1_n,n−1) for all n≥1. HenceO^q−1_n,n−1≡0.

Conversely, if there is an extension ofβ such that it is exact fort6= 0, then there existn≥1 and [j₀ⁿ⁻¹(α_n−1)(t)]∈H^q−1((π_∗D^•)₀⊗ O_B,0/mⁿ₀) such that

O^q−1_n [j₀ⁿ⁻¹(α_n−1)(t)] = [β]6= 0.

But we can also choosen⁰≤nand [jⁿ₀⁰⁻¹(αn⁰−1)(t)]∈H^q−1((π_∗D^•)0⊗ OB,0/mⁿ₀⁰) such that

O_n,n^q−10−1[j₀ⁿ⁻¹(α_n−1)(t)] =O^q−1_n0,n⁰−1[j₀ⁿ⁰⁻¹(αn⁰−1)(t)]6= 0.

This proves the corollary.

Lemma 4.7. For each q≥0,π_∗D^q is a flat OB-module.

Proof. This follows from the fact that (π_∗D^q)tis torsion free andOB,t∼=C{x−t}

is a PID for everyt∈B.

We will need the following fact from homological algebra, whose proof can be found, e.g. in [3].

Proposition 4.8. Let A be a Noetherian ring andC^• be a finite cochain complex of flat A-modules whose cohomology Hⁱ(C^•) is finitely generated for all i. Then there exists a cochain complex of finitely generated flatA-modulesK^•and a cochain map C^• → K^•, which is a quasi-isomorphism. Moreover, for any A-module M, the natural map C^•⊗M →K^•⊗M is a quasi-isomorphism. Furthermore, if the dimension

dimk(p)H^q(K^•⊗k(p))

is locally constant in p ∈Spec(A), then for i =q, q−1, the δ-functors Tⁱ(M) :=

Hⁱ(K^•⊗M) commute with base change.

We apply this proposition to the case A = OB,0, C^• = (π∗D^•)0 to prove the following:

Proposition 4.9. If dim_k(t)H^q((π_∗D^•)_t⊗k(t))is locally constant around0∈B, then the canonical map

H^q((π_∗D^•)₀)⊗k(0)→H^q((π_∗D^•)₀⊗k(0)) is an isomorphism.

Proof. Since (π_∗D^•)₀ is a flat OB,0-module, using Proposition 4.8, we obtain a complex of finitely generated flatO_B,0-modulesK^•such that

H^•((π_∗D^•)₀⊗M)∼=H^•(K^•⊗M) for anyOB,0-moduleM. We claim that the dimension

dim_k(p)H^q(K^•⊗k(p)) is locally constant inp∈Spec(OB,0).

First of all, since dim_k(t)H^q((π_∗D^•)_t⊗k(t)) is locally constant, by Theorem 1.1 and Proposition 3.3, the sheaf H^q(π_∗D^•) ∼= R^qπ_∗E is a locally free OB-module.

Hence

H^q((π_∗D^•)₀)⊗k(0)∼= (R^qπ_∗E)₀⊗k(0)∼=H^q(X, E)∼=H^q((π_∗D^•)₀⊗k(0)).

In particular

dim_k(0)H^q(K^•⊗k(0)) = dim_k(0)H^q((π_∗D^•)0⊗k(0))

= dim_k(0)H^q((π_∗D^•)₀)⊗k(0)

= dim_k(0)H^q(K^•)⊗k(0).

Note that Spec(OB,0) = Spec(C{x}) = {(0),(x)}. Let Q := (OB,0)(0) be the localization of OB,0 at the ideal (0), which is the field of quotients of OB,0. We obtain

H^q(K^•⊗k((0))) =H^q(K^•⊗Q)∼=H^q(K^•)⊗Q,

since localization is flat. On the other hand, as (OB,0)(x)∼=OB,0, we have H^q(K^•⊗k((x)))∼=H^q(K^•⊗ OB,0/m0) =H^q(K^•⊗k(0)), and so

dim_k((x))H^q(K^•⊗k((x))) = dim_k(0)H^q(K^•⊗k(0)) = dim_k(0)H^q(K^•)⊗k(0).

As H^q(K^•) ∼= H^q((π∗D^•)0) is a free OB,0-module and OB,0 is a local integral domain, we have

dimQH^q(K^•)⊗Q= dimk(0)H^q(K^•)⊗k(0).

In summary, we conclude that

dimQH^q(K^•⊗Q) = dim_k((x))H^q(K^•⊗k((x))),

which means that dim_k(p)H^q(K^•⊗k(p)) is constant inp∈Spec(OB,0). HenceT^q commutes with base change. The required isomorphism now follows from taking

M =k(0) in Proposition 4.8.

Remark 4.10. By replacing0∈B by nearbyt∈B, we note that the isomorphism holds in a neighborhood of0.

We are now ready to prove our main result.

Theorem 4.11. dim_k(t)H^q(Xt,Et) is locally constant if and only if O_m,m−1^q ≡ 0 andO_n,n−1^q−1 ≡0 for allm, n≥1.

Proof. If dim_k(t)H^q(Xt,Et) = dim_k(t)H^q((π_∗D^•)_t⊗k(t)) is locally constant, then Proposition 4.9 shows that the natural map

H^q((π∗D^•)0)⊗k(0)→H^q((π∗D^•)0⊗k(0)) is an isomorphism.

Now, let c ∈ H^q((π∗D^•)0⊗k(0)). We can extend it to a nonzero local holo- morphic section of H^q(π∗D^•) since H^q(π∗D^•) is locally free. Denote this ex- tension by ec. Consider the germ of this section ec0 ∈ H^q((π∗D^•)0). Choose a representative αe0 ∈ (π∗D^•)0 in this cohomology class. For each m ≥ 1, αe0

is mapped to (π_∗D^•)0⊗ OB,0/m^m+1₀ via the quotient map pm. Then the class [pm(αe0)] ∈ H^q((π_∗D^•)0⊗ OB,0/m^m+1₀ ) is an m-th order extension of c. Hence O_m,m−1^q ≡0 by Proposition 4.2. Sincemis arbitrary,O^q_m,m−1≡0 for all m≥1.

For the obstruction map O^q−1_n,n−1, if O_n,n−1^q−1 [j₀ⁿ⁻¹(α_n−1)(t)]6= 0 for some n ≥1 and [j₀ⁿ⁻¹(α_n−1)(t)] ∈ H^q((π_∗D^•)0⊗ OB,0/mⁿ₀), then we can find some nonzero [β] ∈ H^q(π_∗D^•₀⊗k(0)) and a local holomorphic extension βe of β such that it is exact only whent6= 0. But sinceH^q(π_∗D^•) is locally free, any extension is locally nonzero by continuity. Therefore,O^q−1_n,n−1≡0 for alln≥1 by Proposition 4.6.

ON THE JUMPING PHENOMENON OF dim_CH (Xt,Et) 13

Conversely, if both obstruction maps vanish, then for each [α]∈H^q((π∗D^•)0⊗ k(0)), we obtain α(t)∈ Γ(U, π∗D^q) such that ¯Dtα(t) = 0 in some neighborhood U ⊂Bcontaining 0 and [α(0)] = [α]. Moreover,α(t) is non-exact sinceO_n,n−1^q−1 ≡0 for alln≥1. Hence for fixedt∈U we obtain an injective linear map

H^q((π∗D^•)0⊗k(0))→H^q((π∗D^•)t⊗k(t)), [α]7→[α(t)].

Therefore,

dim_k(0)H^q((π_∗D^•)0⊗k(0))≤dim_k(t)H^q((π_∗D^•)t⊗k(t)).

By upper semi-continuity, dim_k(t)H^q((π_∗D^•)t⊗k(t)) = dim_k(t)H^q(Xt,Et) is locally

constant.

Recall that by choosing a Hermitian metric onEand using the associated Chern connection, we can write

D¯t= ¯∂+ϕ(t)y∇+A(t),

where {(A(t), ϕ(t))}_t∈B is the family of Maurer-Cartan elements which controls the deformations of (X, E). Hence the n-th order obstruction maps Oⁱ_n,n−1 : Hⁱ((π∗D^•)0⊗ OB,0/mⁿ₀)→Hⁱ⁺¹((π∗D^•)0⊗ OB,0/mⁿ₀), fori=q, q−1, defined in (1) can be rewritten as

(2) Oⁱ_n,n−1([αn−1]) =



tⁿ⁻¹

n−1

X

j=0

(ϕ^n−jy∇+A^n−j)α^j_n−1



.

as claimed in Theorem 1.2.

Example 4.12. We first consider the case whenE=TX, the holomorphic tangent bundle ofX. We deform the pair(X, TX)to(Xt, TX_t), whereTX_tis the holomorphic tangent bundle to Xt (note that TX may have other deformations which are not isomorphic to the holomorphic tangent bundle onXt). In this case, the End(TX)- part of the Maurer-Cartan element(A(t), ϕ(t))is given by

A(t) =−T(ϕ(t),•)− ∇•ϕ(t),

whereT : Ω^0,•(TX)×Ω^0,•(TX)→Ω^0,•(TX) is the graded torsion onTX defined by T(ϕ, ψ) :=ϕy∇ψ−(−1)^|ϕ||ψ|ψy∇ϕ−[ϕ, ψ].

So we have

D¯^•_t = ¯∂_T^•_X + [ϕ(t),−].

Forαn−1∈Ω^0,q(TX)⊗ OB,0 such that D¯^q_tαn−1= 0 modtⁿ⁻¹, we have tⁿ⁻¹(j₀ⁿ( ¯D^q_tα_n−1)/tⁿ) =tⁿ⁻¹



∂¯T_Xαⁿ_n−1+

n−1

X

j=0

[ϕ^n−j, α^j_n−1]



.

As a class in H^q+1(Ω^0,•(TX)⊗ OB,0/mⁿ₀,D¯^•_n−1), it is equal to



tⁿ⁻¹

n−1

X

j=0

[ϕ^n−j, α_n−1^j ]



.

Hence the obstruction is given by

O^q_n,n−1[j₀ⁿ⁻¹(α_n−1)(t)] =



tⁿ⁻¹

n−1

X

j=0

[ϕ^n−j, α_n−1^j ]



.

Example 4.13. For the case E=T_X^∗, we have D¯^•_t = ¯∂_T^•∗

X+ [ϕ(t),−]^∗, where[ϕ(t),−]^∗: Ω^0,q(T_X^∗)→Ω^0,q+1(T_X^∗)is given by

[ϕ(t), η]^∗(v) := [ϕ(t), η(v)]−(−1)^qη([ϕ(t), v]) =ϕ(t)y∂(η(v))−(−1)^qη([ϕ(t), v]) forv∈Ω⁰(TX). Since

ϕ(t)y∂(η(v))−(−1)^qη([ϕ(t), v]) = (ϕ(t)y∂η)(v) +vy∂(ϕ(t)yη), the obstruction is given by

O_n,n−1^q [j₀ⁿ⁻¹(αn−1)(t)] =



tⁿ⁻¹

n−1

X

j=0

(ϕ^n−jy∂α^j_n−1+∂(ϕ^n−jyα^j_n−1))



.

ForE=∧^qT_X^∗, we have D¯t(α1∧ · · · ∧αp) =

p−1

X

j=1

(−1)^j−1α1∧ · · · ∧D¯tαj∧ · · · ∧αp

= ¯∂(α₁∧ · · · ∧α_p) +

p

X

j=1

(−1)^j−1α₁∧ · · · ∧[ϕ(t), α_j]^∗∧ · · · ∧α_p, whereα_j∈Ω⁰(T_X^∗). Then

p−1

X

j=1

(−1)^j−1α1∧ · · · ∧(ϕ(t)y∂αj)∧ · · · ∧αp+

p−1

X

j=1

(−1)^j−1α1∧ · · · ∧∂(ϕ(t)yαj)∧ · · · ∧αp

=ϕ(t)y(∂(α₁∧ · · · ∧α_p)) +∂(ϕ(t)y(α₁∧ · · · ∧α_p)).

Hence the obstruction map is given by

O_n,n−1^q [j₀ⁿ⁻¹(α_n−1)(t)] =



tⁿ⁻¹

n−1

X

j=0

(ϕ^n−jy∂α^j_n−1+∂(ϕ^n−jyα^j_n−1))



.

These two examples recover the obstruction formulae in [8, 9].

5. An application: jumping ofdim_CH¹(Xt,End(T_Xt))

Physicists are interested in knowing whether the dimension of the cohomology group H¹(Xt,End(T_Xt)) is locally constant under small deformations X [5]. The expectation is that dim_CH¹(Xt,End(T_Xt)) does not jump alongany deformation of a Calabi-Yau manifoldX.

In this section, we apply our results to prove a weaker statement, namely, the constancy of this dimension when the Calabi-Yau manifold X satisfies an extra unobstructedness assumption. We will first prove that, in some nice (but restrictive) cases, the dimension dim_CH¹(Xt, A(Et)) does not jump att= 0 for any deformation of the pair (X, E).