A differential-geometric approach to deformations of pairs $(X,E)$

This article gives an exposition of the deformation theory for pairs $(X, E)$, where $X$ is a compact complex manifold and $E$ is a holomorphic vector bundle over $X$, adapting an analytic viewpoint \`{a} la Kodaira-Spencer. By introducing and exploiting an auxiliary differential operator, we derive the Maurer--Cartan equation and differential graded Lie algebra (DGLA) governing the deformation problem, and express them in terms of differential-geometric notions such as the connection and curvature of $E$, obtaining a chain level refinement of the classical results that the tangent space and obstruction space of the moduli problem are respectively given by the first and second cohomology groups of the Atiyah extension of $E$ over $X$. As an application, we give examples where deformations of pairs are unobstructed.


Introduction
The theory of deformations of pairs (X, E), where X is a compact complex manifold and E is a holomorphic vector bundle over X, has been studied using both algebraic [19,15,9,14] and analytic [20,6] approaches and is well-understood Date: August 27, 2014. among experts. In this paper, we revisit this problem from a viewpointà la Kodaira-Spencer [11,12,10], emphasizing the use of differential-geometric notions such as connections and curvatures of E and the induced differential operators. What we obtain is a Chern-Weil-type refinement of the classical results.
To illustrate our strategy, recall that a family of deformations {X t } t∈∆ of a compact complex manifold X over a small ball ∆ can be represented by elements {ϕ t } t∈∆ ⊂ Ω 0,1 (T X ), where T X is the holomorphic tangent bundle of the complex manifold X. While the Dolbeault operator∂ t : Ω 0 → Ω 0,1 on X t is not easy to write down explicitly, one may consider the more convenient operator ∂ + ϕ t ∂ : Ω 0 → Ω 0,1 which has the same kernel as∂ t and hence also defines the same space of holomorphic functions on X t . Now given a family of deformations {(X t , E t )} t∈∆ of a holomorphic pair (X, E), we obtain a family of elements {ϕ t } t∈∆ ⊂ Ω 0,1 (T X ) since {X t } is in particular a family of deformations of X. By choosing a hermitian metric on E and considering the associated Chern connection, we define a differential operator D t : Ω 0,q (E) → Ω 0,q+1 (E) which satisfies the Leibniz rule andD 2 t = 0 (see Section 3 for details). Whilē D t is not the Dolbeault operator∂ Et on the holomorphic bundle E t , its kernel gives precisely the space of holomorphic sections of E t over X t . Furthermore, it determines a family of elements A t :=D t −∂ E − ϕ t ∇ ∈ Ω 0,1 (End(E)).
Applying this, we derive the Maurer-Cartan equation: 3.20). Given a holomorphic pair (X, E) and a smooth family of elements {(A t , ϕ t )} t∈∆ ⊂ Ω 0,1 (E). Then (A t , ϕ t ) defines a holomorphic pair (X t , E t ) (i.e. an integrable complex structure J t on X together with a holomorphic bundle structure on E over (X, J t )) if and only if the Maurer-Cartan equation Moreover, the triple (Ω 0, * (E),∂ E , [−, −]) forms a differential graded Lie algebra (DGLA) which turns out to be naturally isomorphic to the one obtained via algebraic means [19,15] (see Appendix A).
From this we deduce that the space of first order deformations of (X, E) is given by the first cohomology group H 0,1 ∂E ∼ = H 1 (X, E) (Section 4), and that the obstruction theory is captured by the Kuranishi map Ob (X,E) : H 1 (X, E) → H 2 (X, E), whence obstructions lie inside the second cohomology group H 0,2 ∂E ∼ = H 2 (X, E) (Section 5). We also prove the existence of a locally complete (or versal) family (see Theorem 6.2; cf. [20]) using an analytic method originally due to Kuranishi [13]. Remark 1.3. After we posted an earlier version of this paper on the arXiv, Carl Tipler informed us that the paper [6] of L. Huang had a large overlap with most of our results in Sections 3 to 6. Hence the reader may regard those sections as an exposition of known results. Nevertheless, we would like to point out that we have more detailed expositions of first order deformations (Section 4) and the proof of existence of Kuranishi families (Section 6) than Huang's paper; also we have a comparison with the algebraic approach (Appendix A) showing in particular that the isomorphism class of the DGLA is independent of the choice of hermitian metric on E. For these reasons and also for making our paper more self-contained, we retain these sections in this new version.

Acknowledgment
The authors are grateful to Conan Leung, Si Li and Yi Zhang for various illuminating and useful discussions. Thanks are also due to Carl Tipler for pointing out the paper [6] and to Richard Thomas for interesting comments and suggestions on an earlier draft of this paper. 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).

Connections, curvature and the Atiyah class
In this section, we review some basic notions in the theory of holomorphic vector bundles over complex manifolds and fix our notations. Excellent references for these materials include the textbooks [5,8].
Let E be a complex vector bundle over a smooth manifold X. For k ≥ 0, we denote by Ω k the sheaf of k-forms and by Ω k (E) the sheaf of E-valued kforms over X. Recall that a connection on E is a C-linear sheaf homomorphism ∇ : Ω 0 (E) → Ω 1 (E) satisfying the Leibniz rule: for α ∈ Ω k and any s ∈ Ω 0 (E). The curvature of ∇ can then be regarded as a global End(E)-valued 2-form. Also, ∇ induces a natural connection on End(E) by where A ∈ Ω 0 (End(E)) and s ∈ Ω 0 (E), and we have the Bianchi identity Now suppose that X is a complex manifold. For p, q ≥ 0, we denote by Ω p,q the sheaf of (p, q)-forms and by Ω p,q (E) the sheaf of E-valued (p, q)-forms over X. Recall that a holomorphic structure on a complex vector bundle E over X is uniquely determined by a C-linear operator∂ E : Ω 0 (E) → Ω 0,1 (E) satisfying the Leibniz rule and the integrability condition∂ 2 E = 0. If we further equip E with a hermitian metric h, then there exists a unique connection ∇ on E which is hermitian (i.e. dh(s 1 , s 2 ) = h(∇s 1 , s 2 ) + h(s 1 , ∇s 2 ) for any s 1 , s 2 ∈ Ω 0 (E)) and compatible with the holomorphic structure on E (i.e. ∇ 0,1 =∂ E , where ∇ 0,1 = Π 0,1 • ∇ and Π p,q : Ω p+q (E) → Ω p,q (E) is the natural projection map). ∇ is usually called the Chern connection on (E, h). The curvature F ∇ of the Chern connection on (E, h) is real and of type (1, 1), so the Bianichi identity implies that∂ End(E) F ∇ = 0, and thus this defines a class [F ∇ ] ∈ H 1,1 (X, End(E)), called the Atiyah class of E [1].
Lemma 2.1. The Atiyah class is well defined.
Proof. We need to check that the Atiyah class is independent of the choice of hermitian metrics on E. Let h and h ′ be two hermitian metrics on E, and let ∇ and ∇ ′ be the corresponding Chern connections. Then there exists a ∈ Ω 1,0 (End(E)) such that ∇ ′ = ∇ + a. Then Because F ∇ ′ ∈ Ω 1,1 (End(E)), we have ∇ 1,0 a + a ∧ a = 0 so that and hence [F ∇ ′ ] = [F ∇ ] as cohomology classes.

Maurer-Cartan equations
In this section, we start our study of the deformation theory of pairs (X, E). Our goal is to derive the Maurer-Cartan equation which governs this deformation problem.
3.1. Deformations of complex structures and holomorphic vector bundles. We begin by a brief review of the classical theory of deformations of complex structures and holomorphic vector bundles.
We first recall that a family of deformations π : X → ∆ of a compact complex manifold X can be represented by a family of sections ϕ t ∈ Ω 0,1 (T X ), where T X is the holomorphic tangent bundle of X (or the i-eigenbundle of the almost complex structure defining X), satisfying the Maurer-Cartan equation An essential ingredient in the proof is the Newlander-Nirenberg Theorem [17] which states that any integrable almost complex structure comes from a complex structure.
where∂ t is the∂-operator of the complex manifold X t .
Proof. Let z 1 , . . . , z n be local holomorphic coordinates on X (where n is the complex dimension of X). Then ϕ t is of the form Hence T 0,1 Xt is locally spanned by The result follows.
Notice that∂ + ϕ t ∂ is not the same as∂ t but their kernels coincide. Hencē ∂ + ϕ t ∂ completely determines the local holomorphic functions with respective to the complex structure J t on X t . The same idea will be applied to deformations of holomorphic pairs.
Next we recall the deformation theory of holomorphic vector bundles. Let E → X be a complex vector bundle over a complex manifold X. It is a standard fact in complex geometry that E admits a holomorphic structure if and only if there exists a linear operator∂ E : Ω 0,q (E) → Ω 0,q+1 (E) satisfying∂ 2 E = 0 and the Leibniz rulē for any α ∈ Ω 0,q (E) and smooth section s of E (we call this the linearized version of the Newlander-Nirenberg Theorem; see e.g. [8, Theorem 2.6.26] or [16,Theorem 3.2]). Hence if we have a family of holomorphic vector bundles V → ∆ (or {E t } t∈∆ ) on X, then we have a family of Dolbeault operators∂ Et , whose squares are zero and all satisfy the Leibniz rule.
Proposition 3.2. Given a family of deformations {E t } t∈∆ of E, the element A t := ∂ Et −∂ E ∈ Ω 0,1 (End(E)) and satisfies the Maurer-Cartan equation Proof. Note that The result follows from the linearized version of the Newlander-Nirenberg Theorem.

3.2.
Deformations of holomorphic pairs and the operatorD t . Definition 3.3. A holomorphic pair (X, E) consists of a compact complex manifold X together with a holomorphic vector bundle E over X.
Definition 3.4. Let (X, E) be a holomorphic pair. A family of deformations of (X, E) over a small ball ∆ centered at the origin in C d consists of a proper and submersive holomorphic map π : X → ∆ (a family of deformations of X over ∆) and a holomorphic vector bundle V → X such that π −1 (0) = X and V| π −1 (0) = E. For t ∈ ∆, we denote by (X t , E t ) the holomorphic pair parametrized by t.
By the theorem of Ehresmann, if ∆ is chosen to be small enough, the family X is smoothly trivial, i.e. one can find a diffeomorphism F : X → ∆ × X. Restricting to a fiber X t ⊂ X , one can push forward the complex structure on X t to define J t on X t := {t} × X via F . One can also trivialize V as ∆ × E by a smooth bundle isomorphism P and the holomorphic structure on E t := {t} × E is induced from that on V| Xt via the map P . Hence we can assume that our family is a smoothly trivial family ∆ × E → ∆ × X over a small ball ∆ in C d centered at the origin. Now let {(X t , E t )} t∈∆ be a family of deformations of (X, E). By definition, {X t } t∈∆ is a family of deformations of X, so it can be represented by an analytic family of sections ϕ t ∈ Ω 0,1 (T X ) satisfying the Maurer-Cartan equation (1). Define the operatorD t : Ω 0,q (E) → Ω 0,q+1 (E) bȳ where {e k } is a local holomorphic frame of E t .
Proposition 3.5. The linear operatorD t is well-defined and satisfies the Leibniz ruleD Proof. To prove well-definedness, we need to show thatD t is independent of the choice of a local holomorphic frame {e k } of E t . So suppose {f j } is another local holomorphic frame of E t . Let τ k j be local holomorphic functions on X t such that f j = τ k j e k . Then for a local section s = s k e k = s j f j , we have s j = s k τ j k and thus then P ′ • P −1 is an isomorphism between holomorphic bundles and soD t is also independent of the choice of trivializations.
The Leibniz rule forD t is clear since∂ and ∂ both satisfy the usual Leibniz rule. Finally, for a smooth section s of E, if we write s = s k e k locally with {e k } a local holomorphic frame of E t , then we havē We claim thatD 2 t = 0. By our definition ofD t , for any smooth function f : X → C and local nowhere vanishing holomorphic section e of E t , we havē To compute the right hand side, we need the following Lemma 3.6. For any ϕ ∈ Ω 0,p (T X ) and α ∈ Ω 1 (E), we have the Leibniz rulē To compute the last term, first note that the contraction of ∂ ∂z i with α is taken in the (1, 0)-part, we can therefore assume α = α k i dz i ⊗ e k , where {e k } is a local holomorphic frame of E. So we have , and hence the desired formula.
We can now computeD 2 t . Lemma 3.7. For any smooth function, f : X → C, we have the equality Proof. First, we have By Lemma 3.6, the first term is given bȳ Since ∂∂ = −∂∂, we have∂ For the last term, by writing ϕ t = ϕ l m dz m ⊗ ∂ ∂z l in local coordinates, we have ϕ t ∂f = ϕ l m ∂f ∂z l dz m , and so The result follows.
As {X t } t∈∆ is an honest family of deformations of X, the Maurer-Cartan equation (1) for ϕ t holds. Hence we have Proposition 3.8.D 2 t = 0. From the viewpoint of Proposition 3.1, it is natural to compare the operatorD t with∂ E + ϕ t ∇.
Proof. Let f be a smooth function and s a smooth section of E. Using the Leibniz rules, and the fact that the contraction is only taken in the (1, 0)-part, we have In the other direction, suppose we are now given elements A t ∈ Ω 0,1 (End(E)) and ϕ t ∈ Ω 0,1 (T X ), parameterized by t ∈ ∆, we can then define an operator We extendD t to Ω 0,q (E) in the obvious way, so that the Leibniz rulē Using the Leibniz rule, we have for any smooth function and sections of E that (1) by Lemma 3.7. Therefore, the almost complex structure defined by ϕ t is integrable.
We now need to show that E also admits holomorphic structure over X t . To prove this, we first make the following assertion: Any smooth sections of E can locally be written as s k e k , where {e k } ⊂ ker(D t ). We then define∂ Et bȳ To check that it is well-defined, suppose we have another local basis . This proves well-definedness.
Clearly, it satisfies the Leibniz rulē Hence by the linearized version of the Newlander-Nirenberg Theorem, E t = (E,∂ Et ) is a holomorphic vector bundle over X t .
It remains to prove that our assertion is correct: Proof. Let us first fix a smooth local frame {σ k } of E t over a coordinate neighborhood U ⊂ X t . What we need are coordinate changes This in turn is equivalent to the following system of PDEs We will show that this system is solvable, following the line of proof in [16, Theorem 3.2] (linearized version of the Newlander-Nirenberg Theorem).
First of all we set Then applying the Maurer-Cartan equation (1) gives . Hence by the Newlander-Nirenberg Theorem, we obtain holomorphic coordinates is invertible for all (z, w, t). It follows that (F i l ) is also invertible for all (z, w, t). Applying the exterior differential on N and evaluating at w = 0, we have Together with the formula we just obtained, we arrive at In summary, we have proved the following Theorem 3.12. Given A t ∈ Ω 0,1 (End(E)) and ϕ t ∈ Ω 0,1 (T X ). If the induced differential operatorD t : Ω 0,q (E) → Ω 0,q+1 (E) satisfiesD 2 t = 0 and the Leibniz ruleD 3.3. DGLA and the Maurer-Cartan equation. To simplify notations, from this point on, we will denote the vector bundles End(E) and Hom(T X , End(E)) by Q and H respectively unless specified otherwise.
We are now ready to derive the Maurer-Cartan equation governing the deformations of pairs. Given A t ∈ Ω 0,1 (End(E)), ϕ t ∈ Ω 0,1 (T X ) such that the induced differential operatorD t satisfiesD 2 Let us expand the left hand side: Applying Lemma 3.6 to the term∂ E (ϕ t ∇), we get Since ∇ is the Chern connection, we have F ∇ =∂ E ∇ + ∇∂ E , and so Note that the curvature F ∇ is given by As a whole we obtain But X t is integrable, ϕ t satisfies the Maurer-Cartan equation (1), and so Hence we conclude thatD 2 t = 0 is equivalent to the following two equations The following proposition can be proven by straightforward, but tedious, computations which we omit: On the other hand, we let B ∈ Ω 0,1 (H) be the form defined by Proof. This follows from the Bianchi identity∂ Q F ∇ = 0: For any v ∈ T X , Then we define a differential operator∂ EB : Proof. Clearly∂ EB satisfies the Leibniz rule, so it suffices to prove that∂ 2 EB = 0. But∂ 2 EB = 0 if and only if∂ H B = 0 which holds by Proposition 3.14. This proves the first part of the proposition.
To see the second part, suppose that and extend to E-valued p-forms. We computē Hence F in fact defines a holomorphic bundle isomorphism between (E,∂ EB ) and (E,∂ E B ′ ). Since the curvature F ∇ differs by an exact End(E)-valued 1-form if another metric was used, this shows that the holomorphic structure of E only depends on the class [B] but not the metric. By abuse of notations, we will now write∂ EB as∂ E , keeping in mind that a hermitian metric on E has been chosen.
Definition 3.17. The holomorphic vector bundle (E,∂ E ), which is an extension of Q by T X , is called the Atiyah extension of E.
Again by direct computations, one can prove that the bracket [−, −] and the Dolbeault operator∂ E are compatible with each other:  In the appendix, we will prove that there exists a natural isomorphism between the complex (Ω 0, * (E),∂ E ) and the one obtained using algebraic methods [19,15] intertwining our bracket [−, −] with the algebraic one. This gives alternative proofs of Propositions 3.13 and 3.18, and shows that our DGLA is naturally isomorphic to the one derived using algebraic methods. In particular, the isomorphism class of our DGLA is independent of the choice of the hermitian metric we used to define the Chern connection ∇.
Using the bracket [−, −] and the Dolbeault operator∂ E , we can now rewrite the two equations (2) as the following Maurer-Cartan equation: which governs the deformation of pairs. We summarize our results by the following Theorem 3.20. Given a holomorphic pair (X, E) and a smooth family of elements {(A t , ϕ t )} t∈∆ ⊂ Ω 0,1 (E). Then (A t , ϕ t ) defines a holomorphic pair (X t , E t ) (namely, an integrable complex structure J t on X together with a holomorphic bundle structure on E over (X, J t )) if and only if the Maurer-Cartan equation is satisfied.

First order deformations
The Maurer-Cartan equation (4) implies that a first order deformation (A 1 , ϕ 1 ) (the linear term of the Taylor series expansion of a family (A t , ϕ t )) is∂ E -closed: and hence defines a cohomology class in the Dolbeault cohomology group H 0,1 To determine the space of first order deformations of a holomorphic pair (X, E), it remains to identify isomorphic deformations.
Definition 4.1. Two deformations V → X , V ′ → X ′ of (X, E) are said to be isomorphic if there exists a biholomorphism F : X → X ′ and a holomorphic bundle isomorphism Φ : V → V ′ covering F such that F | X = id X and Φ| E = id E . Proposition 4.2. Suppose V → X and V ′ → X ′ are isomorphic 1-real parameter family of deformations of (X, E). If we denote by (A t , ϕ t ) and (A ′ t , ϕ ′ t ) the elements that represent the families V → X and V ′ → X ′ respectively, then there exists , where the error R depends smoothly on t, A(t), ϕ(t), Θ 1 , v and first partial derivatives of Θ, v. Moreover, R is of order s 2 in the sense that for some map R 1 which depends smoothly (with respect to the Sobolov norm; see Section 6 for its precise definition) in s, (A, ϕ) ∈ Ω 0,1 (E) and (Θ, v) ∈ Ω 0 (E).
Proof. As before, let v ∈ Ω 0 (T X ) be the vector field which generates the 1-parameter family of diffeomorphisms F t : X → X of the underlying smooth manifold X. Since v)), for some Θ 1 ∈ Ω 0 (Q).

Hence it remains to show that
We define an endomorphism of E as follows: Fix p ∈ X and the fiber E p of E. Let P γp(t) : E p → E Ft(p) be the parallel transport along t −→ γ p (t) := F t (p).
. We want to compute the first derivatives of both sides of this equation with respective to t at t = 0.
First note that Φ t = P γ(t) Θ t , so we have Since e = e 0 is holomorphic with respective to V| π −1 (0) , we havē Moreover, since e t is holomorphic with respective to E t , we haveD t e t = 0, that is, Differentiate with respective to t and set t = 0, we obtain As a whole we obtain the formulā . Since e is holomorphic with respective to V| π −1 (0) ,∂ E (Θ 1 (e)) = (∂ Q Θ 1 )(e), so that or in other words, , where R is of order t 2 and depends smoothly on t, A t , ϕ t , Θ 1 , v. Moreover, since the equation∂ depends smoothly on first order partial derivatives of Θ 1 and v, we see that the error R also depends smoothly on first order partial derivatives of Θ 1 and v.
Finally, R satisfies for some map R 1 which depends smoothly in s, (A, ϕ) ∈ Ω 0,1 (E) and (Θ, v) ∈ Ω 0 (E). This follows from the fact that Corollary 4.3. If V → X and V ′ → X ′ are isomorphic deformations of (X, E), then the first order terms (A 1 , ϕ 1 ) and (A ′ 1 , ϕ ′ 1 ) of the corresponding families (A t , ϕ t ) and (A ′ t , ϕ ′ t ) respectively differ by an∂ E -exact form.
We conclude that the space of first order deformations of a holomorphic pair (X, E) is precisely given by the Dolbeault cohomology group H 0,1 ∂E ∼ = H 1 (X, E)

Obstructions and Kuranishi family
Now given a first order deformation [(A 1 , ϕ 1 )] ∈ H 0,1 ∂E ∼ = H 1 (X, E), it is standard in deformation theory to ask whether one can find a family (A t , ϕ t ) integrating (A 1 , ϕ 1 ) to give an actual family of deformations. To study this problem, we use a method due to Kuranishi [13].
We need to review several operators commonly used in Hodge theory. We first choose a hermitian metric g on X and h on E, so that we can define a hermitian product (·, ·) on Ω 0, * (E). Define the formal adjoint of∂ E with respective to (·, ·) by (∂ E α, β) = (α,∂ * E β). Then the Laplacian is defined by This is an elliptic self-adjoint operator and thus has a finite dimensional kernel H p (X, E), consisting of harmonic forms. We have the standard isomorphism from Hodge theory: Take a completion of Ω 0, * (E) with respective to (·, ·) to get a Hilbert space L * , and let H : L * → H * (X, E) be the harmonic projection. The Green's operator G : L * → L * is defined by It commutes with∂ E and∂ * E . Now let η 1 , . . . , η m ∈ H 1 (X, E) be a basis and ǫ 1 (t) := m j=1 t j η j ∈ H 1 (X, E). Consider the equation We denote the Hölder norm by · k,α . The following estimates are obvious: In [3], Douglis and Nirenberg proved the following nontrivial a priori estimate: Applying these and following the proof of [10, Chapter 4, Proposition 2.3], one can deduce an estimate for the Green's operator G: where all C i 's are positive constants which depend only on k and α.
Then by the same argument as in [10, Chapter 4, Proposition 2.4], or alternatively using an implicit function theorem for Banach spaces [13], we obtain a unique solution ǫ(t) which satisfies the equation and is analytic in the variable t. Note that the solution ǫ(t) is always smooth. Indeed, by applying the Laplacian to the above equation, we get Also, the solution ǫ(t) is holomorphic in t, so we have j ∂ 2 ǫ(t) ∂t j ∂t j = 0. Now since the operator ∆ E + j ∂ 2 ∂t j ∂t j is elliptic, we see that ǫ(t) is smooth by elliptic regularity.
Following Kuranishi [13] (see also [10,Chapter 4]), we have the following Proposition 5.1. The solution ǫ(t) that satisfies Conversely, suppose that H[ǫ(t), ǫ(t)] = 0. We must show that Recall that ǫ(t) is a solution to and ǫ 1 (t) is∂ E -closed. By applying∂ E to this equation, we get Hence
In the case when H[ǫ(t), ǫ(t)] vanishes identically (which always holds if H 2 (X, E) = 0), we have the following Proof. If H[ǫ(t), ǫ(t)] = 0 for all t, then ǫ(t) = (A t , ϕ t ) satisfies the Maurer-Cartan equation and so (X t , E t ) is holomorphic for each t. In particular, we obtain a deformation X of X. Let V := ∆ × E. A smooth section σ : X → V of V on X can be written as σ : (t, x) −→ (t, s(x, t)), for some smooth map s : X → E. We define a Dolbeault operator∂ V : Ω 0 ∂ Et s(t, x)). Note that∂ V is well-defined for, if {e k (t, x)} are local holomorphic frame of E t → X t , then we can writē which is a smooth section of Ω 0,1 X (V). Clearly,∂ 2 V = 0 and hence V is a holomorphic vector bundle over X .
In general, the condition H 2 (X, E) = 0 may not be satisfied. But we can define the (singular) analytic space and form a family V → X over S, which is called the Kuranishi family of (X, E). In particular, we see that the obstruction space is precisely given by the Dolbeault cohomology group H 0,2 ∂E ∼ = H 2 (X, E), and the obstructions to deformations of a holomorphic pair (X, E) is captured by the Kuranishi map

A proof of completeness
The goal of this section is to give a proof of the local completeness of a Kuranishi family for the deformation of the pair (X, E). Existence of a locally complete (or versal) family for deformations of pairs was first proved by Siu-Trautmann [20]. Here we give another proof using Kuranishi's method. Definition 6.1. A family V → X over a analytic space S is said to be locally complete (or versal) if for any family V ′ → X ′ over a sufficiently small ball ∆, there exists a analytic map f : ∆ → S such that the family V ′ → X ′ is the pull-back of V → X via f .
Recall that for given ǫ 1 (t) ∈ H 1 (X, E), we have existence of solutions ǫ(t) to and ǫ(t) satisfies the Maurer-Cartan equation if and only if H[ǫ(t), ǫ(t)] = 0. We then obtain an analytic family V → X over The main theorem is as follows: Theorem 6.2. The Kuranishi family V → X over S is locally complete.
Before going into the details of the proof, we first introduce the Sobolev norm: One can endow E a hermitian metric H, induced from that of E and X, and define the inner product One has the estimate for some constant C k > 0. We take a completion of Ω 0, * (E) with respective to (·, ·) k to get a Hilbert space L * k . The harmonic projection H : L * k → H * (X, E) and the Green's operator G : The following lemma will be useful in the proof of the completeness theorem. Lemma 6.3. For fixed ǫ 1 (t) ∈ H 1 (X, E), t ∈ S, the equation has only one small solution.
We are now ready to prove the local completeness theorem.
Proof of Theorem 6.2. Let V ′ → X ′ be a deformation of (X, E). Let ǫ ′ , be the element representing this deformation. We first prove that if∂ * E ǫ ′ = 0, then there exists t ∈ S such that ǫ ′ = ǫ(t).
By the order condition on the error term R, the derivative of F with respective to η at (0, 0) is the identity map. Hence by the implicit function theorem, there is a C ∞ function g such that F (ǫ ′ , η) = 0 if and only if η = g(ǫ ′ ). By the error condition again, the (second order) operator is still a quasi-linear elliptic operator. By elliptic regularity, η is smooth. This completes our proof.

Unobstructed deformations and applications
In this section, we investigate various circumstances under which deformations of holomorphic pairs are unobstructed. We will also apply our results to show that the dimension of the cohomology group H 1 (X, End(T X )) is invariant under small deformations of an algebraic K3 surface, answering a question of Huybrechts [7] in the 2-dimensional case. 7.1. Unobstructed deformations of pairs. To begin with, note that we have an exact sequence of holomorphic vector bundles by the construction of E (which shows that E is an extension of Q = End(E) by T X ). This induces a long exact sequence in cohomology groups: and the first order term (A 1 , ϕ 1 ) defines a class [(A 1 , ϕ 1 )] ∈ H 1 (X, E).
The following proposition, which first appeared in [7,Appendix] without proof, describes the relations between the deformations of a pair (X, E) and that of X and E. . Let X be a compact complex manifold with unobstructed deformations and L be a holomorphic line bundle over X such that the map ∪c 1 (L) : is surjective. Then deformations of the pair (X, L) are unobstructed.
For example, if X is an n-dimensional compact Kähler manifold with trivial canonical line bundle, then X admits unobstructed deformations. If we further assume that H 0,2 (X) = 0 (e.g. when the holonomy of X is precisely SU (n)), then deformations of (X, L) for any line bundle L are unobstructed.
Definition 7.5. A holomorphic vector bundle E over a compact complex manifold X is said to be good if H 2 (X, Q 0 ) = 0, where Q 0 is the trace-free part of Q = End(E).
Proposition 7.6. Let X be a compact complex surface with trivial canonical line bundle (e.g. a K3-surface), and let E be a good bundle over X with c 1 (E) = 0. Then deformations of the pair (X, E) are unobstructed.
In this case, the connecting homomorphism δ : H 1 (X, T X ) ∼ = H 1,1 (X) → H 2 (X, Q) ∼ = C is simply given by When c 1 (E) = 0, δ is a nonzero map and hence surjective. Proposition 7.3 then says that any deformation of (X, E) is unobstructed. [7]. In this section, we apply the theory of deformation of pairs to study the jumping of the dimension of the cohomology group H 1 (X, End(T X )) under small deformations. In the case of projective Calabi-Yau manifolds, this was asked by Huybrechts in [7].

Applications to a question of Huybrechts
To begin with, let Def (X) and Def (X, E) be the deformation spaces of X and of the pair (X, E) respectively. Suppose we have a deformation {(X t , E t )} t∈Def (X,E) of (X, E). Recall that we have a differential operatorD t : Ω 0,• (E) → Ω 0,•+1 (E) satisfying the Leibniz rule andD 2 t = 0. Lemma 7.7. Let∂ Et be the Dolbeault operator of E t → X t and P : Ω 0,p (E) → Ω 0,p t (E) be the natural projection, given by restricting the projection P : Ω p (X) → Ω 0,p t (X) to Ω 0,p (E). Then∂ Et is identified withD t via P i.e., ∂ Et P = PD t .
Proof. We first prove the case when p = 0 and E = O X . In this case, A = 0 and P : Ω 0 (X) → Ω 0 t (X) = Ω 0 (X) is the identity map. We fix x ∈ X and let z j be local complex coordinates around x. Letv j = ∂ ∂z j + ϕ k j ∂ ∂z k andǭ j be its dual vector. Then by the Maurer-Cartan equation satisfied by ϕ, Hence we have complex coordinates ζ j on X t such that ∂ ∂ζ j =v j , dζ j =ǭ j at the point x. Then at x, We need to show that P (dz j ) =ǭ j . We write Hence P (dz j ) =ǭ j . Therefore, This proves the case when p = 0. For p > 0, let α = α J dz J . Then We need to show that∂ t (ǭ j ) = 0 for all j. Sinceǭ j is a local frame of (T 0,1 Xt ) * , dǭ j ∈ Ω 1,1 t (X) ⊕ Ω 0,2 t (X). Hence, to prove that∂ tǭ j = 0, it is equivalent to show that dǭ j (v k ,v l ) = 0 for all k, l. We compute: This proves the case when p > 0. Now for a general holomorphic vector bundle E,D t is locally given bȳ where e j is a local holomorphic frame of E t . In this case, P is given by It is then obvious that the required relation follows from the O X case.
For |t| small, P is an isomorphism, so we obtain the following Corollary 7.8. For any holomorphic vector bundle E t → X t , we have the isomorphism This shows that jumping of the dimension dim C H • (X t , E t ) is equivalent to the jumping of dim C H • (Ω 0,• (E),D t ).
In order to formulate the jumping of dim C H • (Ω 0,• (E),D t ) in a more concrete and rigorous way, we apply Grauert's Direct Image Theorem [4] to obtain a complex of vector bundles (V • , d • ) over the deformation space Def (X, E) such that for t ∈ Def (X, E).
Note that for fixed p, dim C V p−1 t is independent of t. Therefore, the exact sequence and D t α q t = 0. We now focus on the case p = 1. Instead of proving directly that dim C H 1 (X t , End(T Xt )) does not jump at t = 0 for any deformation of X, we first prove that in some nice cases, dim C H 1 (X t , E t ) does not jump at t = 0 for any deformation of (X, E).
To do this, we recall that, by choosing a harmonic basis {(A i , ϕ i )} m i=1 for H 1 (X, E), the obstruction map Ob (X,E) : H 1 (X, E) → H 2 (X, E) of the deformation theory of (X, E) is given by Moreover, (A t , ϕ t ) satisfies the Maurer-Cartan equation if and only if Ob (X,E) = 0. Suppose now that Ob (X,E) = 0. Then we havē Differentiating (A t , ϕ t ) with respect to t i and setting t = 0, we get Hence, for each i = 1, . . . , m, if we define (B t , ψ t ) i to be forms a basis for H 1 (X, E). Note that the differential operatorD Et defined bȳ It follows thatD Et defines a deformation {(X t , E t )} t∈Def (X,E) of the pair (X, E).
In fact, E t is the Atiyah extension of the deformed bundle E t on X t . Proof. Since this is true for the harmonic basis , it is true for any element in H 1 (X, E).
Lemma 7.11. Let X be a compact complex manifold and E → X be a holomorphic vector bundle. Suppose the deformation of the pair (X, E) is always unobstructed and dim C H 0 (X t , E t ) does not jump at t = 0 along any deformations of (X, E). Then dim C H 1 (X t , E t ) does not jump at t = 0 along any deformation of (X, E).
We are now going to prove that under certain assumptions, the dimension dim C H 1 (X t , End(T Xt )) does not jump at t = 0 along any deformation of X t .
First of all, when E = T X , we have a canonical lift L : where T : Ω 0,p (T X ) ⊗ Ω 0,q (T X ) → Ω 0,p+q (T X ) is the graded torsion defined as Moreover, if Ob X = 0, then we have a Maurer-Cartan element ϕ t ∈ Ω 0,1 (T X ) and we obtain a deformation of (X, T X ) bȳ In fact, the deformation induced by this operator is isomorphic to the family {(X t , T Xt )} t∈Def (X) , where T Xt is the holomorphic tangent bundle of X t . Therefore, L induces a natural embedding Def (X) ⊂ Def (X, T X ).
In the following theorem, by a Calabi-Yau n-fold we mean an n-dimensional compact Kähler manifold X with trivial canonical line bundle, i.e. K X ∼ = O X and such that H 0,p (X) = 0 for all p = 0, n. Theorem 7.12. Suppose that X is a Calabi-Yau manifold such that deformations of the pair (X, T X ) are unobstructed, then dim C H 1 (X t , End(T Xt )) does not jump at t = 0 for any deformation of X.
Proof. Since the pair (X, T X ) admits unobstructed deformations, Lemma 7.10 allows us to extend any element in H 1 (X, E) to an element in H 1 (X t , E t ), where t ∈ Def (X, T X ). Consider the Atiyah exact sequence of (T X ) t (caution: this is not the tangent bundle of X t in general!) over X t : which gives rise to the injective map ι * t : H 0 (X t , End((T X ) t )) → H 0 (X t , E t ). Since the tangent bundle of a Calabi-Yau manifold is stable, we have H 0 (X, End 0 (T X )) = 0 and so Also, since the identity map id (TX )t is always a non-zero holomorphic section of H 0 (X t , End((T X ) t )) and ι * t : for |t| small. By Lemma 7.11, we conclude that dim C H 1 (X t , E t ) does not jump at t = 0 along any deformation of (X, T X ).
For t ∈ Def (X) ⊂ Def (X, T X ), we have a family of canonical lifts L t : H 1 (X t , T Xt ) → H 1 (X t , E t ), since E t is the Atiyah extension of T Xt for t ∈ Def (X). So the map π * t : H 1 (X t , E t ) → H 1 (X t , T Xt ) is surjective and we obtain the following exact sequence Since dim C H 1 (X t , T Xt ) = dim C H n−1,1 (X t ) does not jump at t = 0 for t ∈ Def (X) with |t| small, we see that dim C H 1 (X t , End(T Xt )) does not jump at t = 0 for any deformation of X.
Finally we consider algebraic K3 surfaces, i.e. complex algebraic surfaces with trivial canonical line bundle and such that H 1 (X, O X ) = 0. Recall that for every K3 surface X, we have dim C H 1 (X, T X ) = 20 (see e.g. [2]). Theorem 7.13. Suppose that X is an algebraic K3 surface. Then the dimension dim C H 1 (X t , End(T Xt )) does not jump at t = 0 for any small algebraic deformation of X.
Proof. Let L be a line bundle on X with c 1 (L) = 0. Define E := T X ⊗ L and let E be the Atiyah extension of E.
Note that as c 1 (E) = c 1 (L) = 0 and H 2 (X, End 0 (E)) ∼ = H 0 (X, End 0 (T X )) = 0 (T X is stable), we can apply Proposition 7.6 to conclude that Ob (X,E) = 0. Moreover, since any deformation of (X, E) gives rise to a deformation of X via the map π * : H 1 (X, E) → H 1 (X, T X ), we obtain the Atiyah exact sequence of E t on X t : This shows that dim C H 0 (X t , E t ) does not jump at t = 0 along any deformation of (X, E). Hence it follows from Lemma 7.11 that dim C H 1 (X t , E t ) does not jump at t = 0 along any deformation of (X, E).
Recall that the Atiyah exact sequence of L gives Since c 1 (L) = 0, the connecting homomorphism δ : H 1 (X, T X ) → H 2 (X, End(E)) ∼ = C is surjective and so dim C ker(δ) = 19. Since H 0 (X, T X ) = 0, for this 19dimensional space, the map π * : H 1 (X, E L ) → ker(δ) is an isomorphism. Since line bundles are always good, Proposition 7.6 can be applied again to conclude that the pair (X, L) admits unobstructed deformations over a 19-dimensional space Def 19 (X) whose tangent space is given by ker(δ). Hence we obtain a family {(X t , L t )} t∈Def 19 (X) . Note that c 1 (L t ) = 0 for all small |t|. Indeed, since c 1 (L) = 0 and Pic(X) ∼ = H 1,1 (X) ∩ H 2 (X, Z) ⊂ H 2 (X, Z), L is non-trivial as a smooth line bundle over X. Hence L t is also a non-trivial smooth line bundle over X t and so cannot be holomorphically trivial. Since for |t| small, Pic(X t ) ∼ = H 1,1 (X t ) ∩ H 2 (X t , Z), we must have c 1 (L t ) = 0. Now in the 19-dimensional space Def 19 (X), we have a deformation of E given by E t = T Xt ⊗ L t , t ∈ Def 19 (X), where L t is constructed above. This gives an inclusion Def 19 (X) ⊂ Def (X, E). Consider the Atiyah sequence of E t : The vanishing of H 0 (X t , T Xt ) ∼ = H n−1,0 (X t ) and the surjectivity of δ t : H 1 (X t , T Xt ) → H 2 (X t , End(E t )) ∼ = C imply that for all t ∈ Def 19 (X), we have the following exact sequence Finally, it remains to see that Def 19 (X) is precisely the algebraic deformation space Def alg (X) of X. Recall that we have a family of holomorphic line bundles L t over X t with c 1 (L t ) = 0 ∈ H 1,1 (X t ) ∩ H 2 (X t , Z). Hence the rank of the Picard group Pic(X) is bigger than or equal to 1. Therefore, the K3 surfaces X t must all be algebraic. Hence we obtain an embedding Def 19 (X) ⊂ Def alg (X).
Since both Def 19 (X) and Def alg (X) are 19-dimensional, we see that we must have Def 19 (X) = Def alg (X).

Appendix A. Comparison with the algebraic approach
The aim of this appendix is to give an explicit comparison between the analytic approach we adopt here and the classical algebraic approach (see the book [19] for the deformation theory of (X, L) where L is a holomorphic line bundle on X, and the thesis [15] for the general case).
We start with a definition Definition A.1. A differential operator of order 1 on a vector bundle E is a linear map P : Ω 0 (E) → Ω 0 (E) such that locally, with (g ij ) be a matrix with entries in O X (U α ) and h k ij ∈ O X (U α ). A differential operator of order 1 is said to be with scalar principle symbol if h k ij = h k · I.
In the algebraic approach, the role of the Atiyah extension E is replaced by the sheaf D 1 (E) of scalar differential operators of order less than 1 on E, namely, we have an exact sequence where the surjective map σ : D 1 (E) → T X is locally defined by the symbol There is an obvious identification of D 1 (E) with E as smooth vector bundles, but we will see that in fact D 1 (E) can be given a holomorphic structure so that D 1 (E) and E are isomorphic as holomorphic vector bundles. First of all, locally on an open set U α , we can write P | Uα = g α + d α .
Let e α be local sections of E, {f αβ } be holomorphic transition functions of E and P α := P | Uα (e α ). To get a global differential operator, we must have f αβ P β = P α f αβ . Hence Define τ α := g α − d α (h −1 α ∂h α ), where h α is the Hermitian metric on E| Uα .
Using h β = f βα h αfαβ , we compute that as desired. Finally, , since g α is a matrix with entries in O X (U α ), d α is a holomorphic vector and F ∇ = ∂(h −1 α ∂h α ).
Therefore, the map defined by Φ : g α + d α −→ (g α − d α h −1 α ∂h α , d α ) is an isomorphism between D 1 (E) and E. So we can give D 1 (E) a holomorphic structure by pulling back that on E via Φ. Hence we obtain Proposition A.3. D 1 (E) carries a natural holomorphic structure so that it is isomorphic to the Atiyah extension E. In particular, H p (X, D 1 (E)) ∼ = H p (X, E) for any p.
Therefore, their sum equals to Finally, which is the required equality.