THE KURANISHI SPACE OF JOINT DEFORMATIONS OF CALABI–YAU MANIFOLDS AND HOLOMORPHIC VECTOR BUNDLES (WITH AN APPENDIX JOINT WITH HISASHI KASUYA) RUNZE ZHANG Abstract. In this paper, we study joint deformations of pairs ( X, E ) , where X is a compact complex manifold and E → X is a holomorphic vector bundle. We prove that ( X, E ) has unobstructed deformations if X is a Fujiki manifold with torsion canonical bundle and H 2 ( X, End 0 E ) = 0 , where End 0 E denotes the trace-free endomorphism bundle. When the canonical bundle is trivial, the Fujiki assumption can be replaced by three weak ∂ ̄ ∂ -conditions. We construct examples satisfying these conditions whose Frölicher spectral sequences do not degenerate at E 1 . These results provide non-Kähler extensions of the theorems of Li–Pan and Iacono–Manetti. Without this vanishing assumption, we obtain both unobstructed and obstructed pairs. In Thomas’s example, we prove unobstructedness of the pair, although the bundle has obstructed deformations with the manifold fixed. On the other hand, we prove that every strict projective Calabi–Yau manifold of dimension at least three admits a simple bundle with obstructed joint deformations. We also construct obstructed pairs with Hermitian flat bundles on complex tori. These results answer two questions raised by Felten. 1. Introduction 1.1. Background and motivation. For any compact complex manifold X , Kuranishi [Kur65] constructed a semi-universal deformation which, in his words, parametrizes all sufficiently small deformations of X in “ the most economical ” way. Its analytic base is called the Kuranishi space of X and is denoted by Kur ( X ) . In general, Kur ( X ) may be highly singular, even when X is a surface [Vak06]. Building on Bogomolov’s work [Bog78] on holomorphic symplectic manifolds, Tian [Tia87] and Todorov [Tod89] established that every compact Kähler manifold X with trivial canonical bundle has unobstructed deformations , namely, that its Kuranishi space is smooth. This foundational result is now known as the Bogomolov–Tian–Todorov (BTT) theorem , and its original proofs are differential-geometric in nature. Later, Ran, Kawa- mata and Fantechi–Manetti developed the T 1 -lifting method [Ran92, Kaw92, FM99], which gives algebraic proofs of the BTT theorem, including the case where the canonical bundle is torsion; see also [IM10]. Under the assumptions above, the Kähler condition can also be weakened. It suffices that X satisfy the ∂ ̄ ∂ -lemma [Huy05, Pop19], meaning that d -exactness, ∂ -exactness, ̄ ∂ - exactness and ∂ ̄ ∂ -exactness are equivalent for every d -closed form of pure type (such an Date : October 7, 2026. 2020 Mathematics Subject Classification. Primary 32G05, 32G08; Secondary 32Q25, 32L05, 58A14. Key words and phrases. Joint deformations, holomorphic vector bundles, Kuranishi spaces, Calabi– Yau manifolds, ∂ ̄ ∂ -manifold, Fujiki manifold, weak ∂ ̄ ∂ -lemma, logarithmic deformations. 1 2 RUNZE ZHANG X is called a ∂ ̄ ∂ -manifold ). In fact, the weaker assumption that the Frölicher spectral se- quence degenerates at E 1 already suffices for unobstructedness [KKP08, Iac17, ACRT18]; see also [WX25] for a further weakening. In this paper, we study the Kuranishi space Kur ( X, E ) of joint deformations of a com- pact complex manifold X and a holomorphic vector bundle E over X . Such a deformation varies the complex structure of X and the holomorphic structure of E simultaneously. For line bundles, Li–Pan and Iacono–Manetti independently proved the following re- markable results: Theorem 1.1 ([LP19, Theorem 1.2]) Let X be a smooth proper variety over C with torsion canonical bundle, and let L be a line bundle on X . Then the deformations of the pair ( X, L ) are unobstructed. Theorem 1.2 ([IM21, Theorem 1.1]) Let L be a line bundle on a smooth projective variety X defined over an algebraically closed field K of characteristic 0 . If X has trivial canonical bundle, then the DG-Lie algebra controlling the deformations of the pair ( X, L ) is homotopy abelian. In particular, ( X, L ) has unobstructed deformations. (1) For smooth projective varieties, Li–Pan’s geometric proof used the Beauville– Bogomolov decomposition [Bog74, Bea83] to pass to a finite étale cover ̃ X ' A × Y × Z, where A is an abelian variety, and Y and Z are products of simply connected Calabi–Yau manifolds and irreducible holomorphic symplectic manifolds, respec- tively. They reduced the problem to proving unobstructedness for ( A, L 1 ) , ( Y, L 2 ) and ( Z, L 3 ) separately, where the L i are the corresponding restrictions of the pull- back of L . The main difficulty was the abelian case, which they handled through the smoothness of the relevant Hodge locus and a formal extension theorem for line bundles [LP19, §§6–7]. For the initial decomposition step, the classical theorem was already available in the compact Kähler setting, with A a complex torus. Analogous decompositions were established by Biswas–Cao–Dumitrescu–Guenancia for Moishezon manifolds in arbitrary dimension and Fujiki class C manifolds of dimension at most four [BCDG25, Theorem A]. Very recent advances in the Kähler MMP [HX26, MZ26] now yield the arbitrary-dimensional Fujiki case; see [Gue26, Theorem 9.5]. Here Moishezon and Fujiki (class C ) manifolds are compact complex manifolds bimero- morphic to projective and compact Kähler manifolds, respectively; both satisfy the ∂ ̄ ∂ -lemma [DGMS75, Corollary 5.23]. (2) Iacono–Manetti gave two approaches to the problem. The first used the P 1 -bundle P ( O X ⊕ L ) with its two distinguished sections to reduce the problem to the unob- structedness of log Calabi–Yau pairs. Prior to the present work, such results were known in the projective setting [KKP08, Iac15] and the compact Kähler setting [Iac17, Ran17, LRW19, Wan18, Zha26]. Felten adapted this construction to prove unobstructedness for proper log toroidal Calabi–Yau families equipped with a line bundle [Fel25, Theorem 16.2]. Their second approach, outlined over C , used T 1 - lifting, deriving the required lifting property from the constancy of the ranks of certain cup-product maps [IM21, Remark 2.6]. In the complex analytic setting, both of the above unobstructedness results extend to pairs ( X, E ) with H 2 ( X, End 0 E ) = 0 , since the problem then reduces to that for KURANISHI SPACE OF JOINT DEFORMATIONS OF ( X, E ) 3 ( X, det E ) [IM19, Fel25]; see Proposition 3.8. Here End 0 E denotes the trace-free sub- bundle of End E (see the beginning of § 3 for more details). The first motivation of this paper is to develop a direct differential-geometric approach to joint deformations based on the Tian–Todorov method, different from those described in (1) and (2). By solving the coupled obstruction equations recursively, we seek non- Kähler extensions of joint unobstructedness, including cases where the Frölicher spectral sequence does not degenerate at E 1 We also investigate the applicability of the two approaches of Iacono–Manetti in this setting. The second motivation is to understand whether joint unobstructedness persists with- out the trace-free vanishing assumption. We construct obstructed pairs in two settings: • simple bundles on strict projective Calabi–Yau manifolds; • Hermitian flat bundles on complex tori. 1.2. Main results. Throughout the paper, a Calabi–Yau manifold means a connected compact complex manifold with torsion canonical bundle; no Kähler assumption is in- cluded in this terminology. We call it K -trivial when its canonical bundle is holomor- phically trivial. An n -dimensional K -trivial Calabi–Yau manifold X is called strict if H i ( X, O X ) = 0 for every 0 < i < n Our first main result concerns Fujiki manifolds. Theorem A. Let X be a Calabi–Yau Fujiki manifold. Let E be a holomorphic vector bundle of rank r ≥ 1 over X satisfying H 2 ( X, End 0 E ) = 0 . Then the joint deformations of ( X, E ) are unobstructed. Note that for Fujiki manifolds, the torsion assumption on K X in Theorem A is equiva- lent to Tosatti’s Calabi–Yau condition that the first Bott–Chern class c BC 1 ( X ) vanish, or to c 1 ( X ) = 0 in H 2 ( X, R ) [Tos15, Theorem 1.5]. When the canonical bundle is trivial, we can replace the Fujiki assumption by weak versions of the classical ∂ ̄ ∂ -lemma. Such conditions have been studied by Fu–Yau [FY11], Angella–Ugarte [AU16, AU17], Popovici–Ugarte [PU18], Rao–Zhao [RZ18], Rao–Wan– Zhao [RWZ19], Hu–Xia [HX25] and Rao–Zhang [RZ26], among many others; see also a survey by Alessandrini [Ale19]. We use the following notation from [RZ18]. Notation. Let 0 ≤ p, q ≤ n . We say that an n -dimensional compact complex manifold X satisfies S p,q (resp. B p,q ) if for every ∂α ∈ A p,q ( X ) with ∂ ̄ ∂α = 0 , the equation ̄ ∂β = ∂α admits a solution (resp. a ∂ -exact solution). Clearly, (1.1) ∂ ̄ ∂ -lemma = ⇒ B p,q = ⇒ S p,q , for every p, q. The point of our second main result is that only three weak ∂ ̄ ∂ -conditions suffice Theorem B. Let X be an n -dimensional K -trivial Calabi–Yau manifold. Suppose that X satisfies S n, 1 , S n, 2 and B n − 1 , 2 If E → X is a holomorphic vector bundle with H 2 ( X, End 0 E ) = 0 , then Kur( X, E ) is smooth. Theorem B applies even when the Frölicher spectral sequence does not degenerate at E 1 We construct two such K -trivial Calabi–Yau examples in Propositions 3.11 and 3.12. In particular, neither is a ∂ ̄ ∂ -manifold, so the implication from the full ∂ ̄ ∂ -lemma to these three weak conditions in (1.1) cannot be reversed. 4 RUNZE ZHANG Here A ( E ) denotes the Atiyah extension of E , and H 2 ( X, A ( E )) is the obstruction space for joint deformations; see §2. • The first is a compact complex threefold X for which H 2 ( X, A ( E )) = 0 when- ever H 2 ( X, End 0 E ) = 0 Thus joint unobstructedness already follows from the vanishing of the obstruction space. • The second is a compact complex sixfold X admitting a rank-two simple holo- morphic vector bundle E satisfying H 2 ( X, End 0 E ) = 0 , whereas H 2 ( X, End E ) ' C , H 2 ( X, A ( E )) ' C 2 Thus our Tian–Todorov argument applies even though the obstruction space does not vanish. For a line bundle, the trace-free endomorphism bundle is zero. We therefore obtain: Corollary C. Let X be as in Theorem A or Theorem B. Then Kur( X, L ) is smooth for every L ∈ Pic( X ) In dimension two, every ∂ ̄ ∂ -manifold has even first Betti number and is therefore Kähler; see [Buc99, Theorem 11] or [BHPVdV04, Theorem IV.3.1]. Among the three weak conditions in Theorem B, S 2 , 2 and B 1 , 2 hold automatically on every compact complex surface, whereas S 2 , 1 is equivalent to the Kähler condition; see Proposition 3.10. Recall also that E is simple if H 0 ( X, End E ) = C · id E ; see, e.g., [Kob87, §5.7]. Using Serre duality, we obtain the following corollary, which removes the assumption c 1 ( E ) 6 = 0 from [CS16, Proposition 7.7]. Corollary D. Let X be a Kähler K -trivial Calabi–Yau surface, and let E → X be a simple holomorphic vector bundle. Then Kur( X, E ) is smooth. It is natural to ask: does the unobstructedness conclusion of Corollary C extend to higher-rank bundles even when H 2 ( X, End 0 E ) 6 = 0 ? Thomas [Tho99] constructed a stable (hence simple) rank-two bundle E on a smooth hypersurface X ⊂ P 2 × P 2 of bidegree (3 , 3) , which is a strict projective Calabi–Yau threefold. The deformations of E with X fixed are obstructed. In particular, the forgetful morphism Def( X, E ) → Def( X ) is not smooth; see also [Fel25, Example 11.14]. Here Def( X, E ) and Def( X ) denote the deformation functors of ( X, E ) and X , respectively. This led Felten to ask the following questions. Question ([Fel25, Remark 11.15]) (i) Is the functor Def( X, E ) (equivalently, the Kuranishi space Kur( X, E ) ) smooth in Thomas’s example? (ii) If so, does there exist a (projective) Calabi–Yau manifold carrying a holomorphic vector bundle of rank at least two with obstructed joint deformations? We answer both questions affirmatively. Theorem 4.2 answers Question (i). 1 The Serre construction relates Thomas’s bundle E to a rational curve C ⊂ X . In our proof, we extend this correspondence to joint deforma- tions, reducing the problem to lifting ( X, C ) across Artin extensions. We lift the deformed 1 In contrast, for the non-Kähler sixfold in Proposition 3.12, both fixed-base and joint deformations are unobstructed, but the forgetful morphism Def( X, E ) → Def( X ) is not smooth. KURANISHI SPACE OF JOINT DEFORMATIONS OF ( X, E ) 5 curve inside P 2 × P 2 and correct a lift of the bicubic equation to vanish on it, without changing the given deformation over the smaller Artin algebra. The correspondence then recovers a lift of the bundle. Our next main result answers Question (ii) even for simple bundles Theorem E. Let X be a strict projective Calabi–Yau manifold of dimension n ≥ 3 . Then there exists a simple holomorphic vector bundle E → X of rank at least two such that Kur( X, E ) is singular. Thus the higher-dimensional analogue of Corollary D fails. Our proof starts with a length-four subspace of X whose ideal sheaf has a nonzero quadratic obstruction. Following Joyce–Song [JS12, §8], we use Seidel–Thomas twists [ST01] to construct a simple vector bundle while preserving this obstruction. We then show that the resulting bundle extends over every first-order deformation of X This makes the natural map from the fixed-base obstruction space to the joint obstruction space injective, so the obstruction persists when X varies. By contrast, Verbitsky [Ver04] showed that generic complex tori of dimension at least three admit no simple holomorphic vector bundles of rank at least two. Nevertheless, the following result, proved jointly with Hisashi Kasuya in Appendix I, gives obstructed pairs with Hermitian flat bundles on every complex torus of dimension at least two. Theorem F (=Theorem I.1) Let X be a complex torus of dimension n ≥ 2 , and let L ∈ Pic 0 ( X ) . Set E := L ⊕ r , r ≥ 2 Then Kur( X, E ) is singular. Since E is Hermitian flat (i.e., it admits a Hermitian metric with vanishing Chern cur- vature), the splitting theorem of Kasuya–Purho [KP26, Proposition 1.1] gives an isomor- phism of analytic germs Kur( X, E ) ' Kur( X ) × Kur X ( E ) , where Kur X ( E ) parametrizes deformations of E with X fixed. We identify this factor with the germ at the origin of the variety of n pairwise commuting r × r complex matrices, which is singular there. Remark. While polishing this manuscript, we became aware of Wan’s preprint [Wan26]. Together with the splitting theorem of Kasuya–Purho, his Theorem 1.5 also answers Question (ii) affirmatively; see Remark I.3 for more details. 1.3. Proof strategies for Theorems A and B. We give three proofs of Theorem A, none of which uses the Beauville–Bogomolov decomposition or the minimal model pro- gram. The Tian–Todorov approach also yields Theorem B. (1) Tian–Todorov approach. By the analytic description of Huang [Hua95], Chan–Suen [CS16] and Liu–Tu [LT18], together with Proposition 2.8, it suffices to extend any first- order deformation to a formal solution of the Maurer–Cartan system ̄ ∂φ ( t ) = 1 2 [ φ ( t ) , φ ( t )] , ̄ ∂ End E A ( t ) = φ ( t ) y R E + φ ( t ) y D 1 , 0 End E A ( t ) − 1 2 [ A ( t ) , A ( t )] Here φ ( t ) and A ( t ) are formal series with values in A 0 , 1 ( X, T 1 , 0 X ) and A 0 , 1 ( X, End E ) , respectively. A fixed Hermitian metric on E determines its Chern curvature R E and the (1 , 0) -part D 1 , 0 End E of the induced connection on End E Unlike the first equation, the second involves both φ ( t ) and A ( t ) 6 RUNZE ZHANG By [IM19, Fel25] (see Proposition 3.8), the assumption H 2 ( X, End 0 E ) = 0 reduces the problem to ( X, det E ) We carry out this reduction analytically by taking the trace of the second equation and setting a ( t ) := tr A ( t ) ∈ A 0 , 1 ( X )[[ t ]] First consider the torsion case. Choose l ≥ 1 with lK X ' O X and set F := ( l − 1) K X Then F is Hermitian flat, with flat Chern connection ∇ F = D 1 , 0 F + ̄ ∂ F . Fix a nowhere- vanishing section Ω 0 of K X ⊗ F , viewed as an F -valued holomorphic ( n, 0) -form. Following an elegant idea of Liu–Rao–Wan [LRW19], further developed in [Zha26, Appendix I], we introduce an “ auxiliary ” formal power series Ω( t ) = Ω 0 + ∑ k ≥ 1 Ω k t k , Ω k ∈ A n, 0 ( X, F ) In those works, an SNC divisor D deforms locally trivially with X , and the deformation of the pair is governed by a single Maurer–Cartan equation for φ ( t ) ∈ A 0 , 1 ( X, T 1 , 0 X ( − log D )) Our construction takes D = ∅ but requires an additional equation for a ( t ) , which must be solved simultaneously with the complex-structure equation. The bundle-valued Tian–Todorov identity by Liu–Rao [LR12] (see Lemma 3.1) reduces this task to the following three equations: ( ̄ ∂ F + D 1 , 0 F ◦ φ ( t ) y ) Ω( t ) = 0 , ( ̄ ∂ F + 1 2 D 1 , 0 F ◦ φ ( t ) y )( φ ( t ) y Ω( t ) ) = 0 , ( ̄ ∂a ( t ) − φ ( t ) y tr R E − φ ( t ) y ∂a ( t ) ) ∧ Ω( t ) = 0 , where y denotes the contraction map. It is well-known that Fujiki manifolds satisfy the flat-twisted ∂ ̄ ∂ -lemma for every Hermitian flat bundle (see e.g., [HZ26, Proposition A.1] for a proof). This allows us to solve the system order by order and prove Theorem A. However, on an arbitrary ∂ ̄ ∂ -manifold, the flat-twisted ∂ ̄ ∂ -lemma need not hold for every Hermitian flat bundle; see [HZ26, Theorem A]. For Theorem B, the K -trivial assumption allows us to take F = O X , and the same construction requires only the three weak ∂ ̄ ∂ -conditions S n, 1 , S n, 2 and B n − 1 , 2 (2) The approaches of Iacono–Manetti. We also adapt both approaches of Iacono–Manetti in (2) to prove Theorem A. For the P 1 -bundle argument, the key is to extend the weak ∂ ̄ ∂ -lemma for weighted logarithmic forms in [Zha26, Theorem B] to Fujiki manifolds. We achieve this using Kähler modifications and logarithmic Serre duality (Theorem 3.5). This yields unobstructedness of locally trivial deformations of log Calabi–Yau pairs in the Fujiki setting (Theorem 3.6). As an application of independent interest, we also obtain a new analytic proof of Fujino’s injectivity theorem [Fuj17, Theorem 1.2]; see Theorem 3.7. The T 1 -lifting approach proves the K -trivial ∂ ̄ ∂ -manifold subcase of Theorem B. Pass- ing to the canonical finite étale cover and applying Wells’s pull-back injectivity theorem [Wel74] (see Lemma 3.9) then yields Theorem A. For Theorem B in full, the T 1 -lifting property follows a posteriori from our Tian–Todorov proof. However, an independent proof through the cup-product argument seems difficult: it remains unclear how to es- tablish the required lifting properties for the relevant kernels and cokernels using only the three weak ∂ ̄ ∂ -conditions. This paper is organized as follows. KURANISHI SPACE OF JOINT DEFORMATIONS OF ( X, E ) 7 Contents 1. Introduction 1 1.1. Background and motivation 1 1.2. Main results 3 1.3. Proof strategies for Theorems A and B 5 2. Joint deformations of ( X, E ) 7 2.1. Operators D t and A t 7 2.2. Maurer–Cartan system and DGLA for the pair 9 2.3. Kuranishi family 10 3. Theorems A, B and Corollary D: proofs and related examples 12 3.1. Tian–Todorov proofs of Theorem A and Theorem B 12 3.2. P 1 -bundle proof of Theorem A 19 3.3. T 1 -lifting proofs of Theorem A and the ∂ ̄ ∂ -manifold case of Theorem B 23 3.4. Proof of Corollary D 25 3.5. Examples satisfying the three weak ∂ ̄ ∂ -conditions without E 1 -degeneration 27 4. Simultaneous deformations in Thomas’s example 32 5. Simple bundles with obstructed joint deformations 35 Appendix I. Obstructed joint deformations of Hermitian flat bundles on complex tori (joint with Hisashi Kasuya) 37 References 40 2. Joint deformations of ( X, E ) In this section, following [Hua95, CS16, LT18], we review the deformation theory of a pair ( X, E ) , where X is an n -dimensional compact complex manifold and E → X is a holomorphic vector bundle of rank r We mainly follow the notation and analytic presentation of Chan–Suen [CS16]. A joint deformation of ( X, E ) can be viewed as a simultaneous deformation of the holomorphic structures on X and E . More precisely: Definition 2.1. A family of (joint) deformations of the pair ( X, E ) over a small disc D m ⊂ C m centred at the origin consists of the following data: a proper smooth morphism π : X → D m with an isomorphism π − 1 (0) ' X ; a holomorphic vector bundle E → X such that E| π − 1 (0) ' E For any t ∈ D m , we denote by ( X t , E t ) the fibre π − 1 ( t ) together with the restriction E| π − 1 ( t ) , respectively. Then in particular, E t is a holomorphic vector bundle of rank r over X t 2.1. Operators D t and A t Let { ( X t , E t ) } t ∈ D m be a family of deformations of the pair ( X, E ) . By Definition 2.1, it in particular induces a family of deformations of the com- pact complex manifold X , which can be described by an analytic family of the so-called Beltrami differentials φ t ∈ A 0 , 1 ( X, T 1 , 0 X ) satisfying the integrability condition (also called the Maurer–Cartan equation for the complex structures): ̄ ∂φ t = 1 2 [ φ t , φ t ] Furthermore, by Ehresmann’s theorem, after possibly shrinking the disc, we may iden- tify X with X × D m diffeomorphically. Let f be a smooth function locally defined on X 0 8 RUNZE ZHANG Then by e.g. [MK71, § 4, Propsition 1.2], [RZ18, Proposition 2.7] and [CS16, Proposition 3.1], (2.1) f is holomorphic w.r.t. the complex structure of X t ⇐⇒ ( ̄ ∂ − φ t y ∂ ) f = 0 , where y denotes the contraction operator. Let { e i ( t ) } 1 ≤ i ≤ r be local holomorphic frames of E t . Thanks to Ehresmann’s theorem again, these can be regarded as local smooth frames of the fixed bundle E over X . We define an operator D t : A 0 ,q ( X, E ) → A 0 ,q +1 ( X, E ) by D t ( r ∑ i =1 α i ⊗ e i ( t ) ) := r ∑ i =1 ( ̄ ∂ − φ t y ∂ ) α i ⊗ e i ( t ) , where α i are local smooth (0 , q ) -forms. We have the following properties thanks to the criterion (2.1): Proposition 2.2 ([CS16, § 3.2]) (1) D t is a well-defined linear operator, i.e., it is independent of the choice of the local holomorphic frames of E t . Furthermore, it satisfies the Leibniz rule: D t ( α ⊗ s ) = ( ̄ ∂ − φ t y ∂ ) α ⊗ s + ( − 1) | α | α ∧ D t s for any s ∈ A 0 ( X, E ) and α ∈ A 0 ,? ( X ) . Here | α | denotes the degree of α (2) For any s ∈ A 0 ( X, E ) , D t ( s ) = 0 ⇐⇒ ̄ ∂ E t ( s ) = 0 , where ̄ ∂ E t is the Dolbeault operator induced by the holomorphic structure of E t (3) D 2 t = 0 Fix an arbitrary smooth Hermitian metric h E on E . Let ∇ E be the Chern connection on ( E, h E ) , which decomposes into its (1 , 0) - and (0 , 1) -parts as ∇ E = D 1 , 0 E + ̄ ∂ E In the same sprit of (2.1), one naturally to compare the operator D t and ̄ ∂ E − φ t y D 1 , 0 E Proposition 2.3 ([CS16, Proposition 3.9]) A t := D t − ̄ ∂ E + φ t y D 1 , 0 E ∈ A 0 , 1 ( X, End E ) Proof. It is easy to show that A t ( f s ) = f A t ( s ) , for f ∈ A 0 ( X ) and s ∈ A 0 ( X, E ) For the converse direction, assume that we are given elements A t ∈ A 0 , 1 ( X, End E ) together with φ t ∈ A 0 , 1 ( X, T 1 , 0 X ) parametrized by t ∈ D m . One then defines an operator D t : A 0 , 0 ( X, E ) → A 0 , 1 ( X, E ) by setting D t := ̄ ∂ E − φ t y D 1 , 0 E + A t We also extend D t : A 0 ,q ( X, E ) → A 0 ,q +1 ( X, E ) by D t ( α ⊗ s ) := ( ̄ ∂ − φ t y ∂ ) α ⊗ s + ( − 1) | α | α ∧ ( D t s ) , which satisfies the Leibniz rule. We then have D 2 t = 0 = ⇒ the data ( A t , φ t ) defines a holomorphic-pair ( X t , E t ) , thanks to [CS16, Theorem 3.12]. KURANISHI SPACE OF JOINT DEFORMATIONS OF ( X, E ) 9 2.2. Maurer–Cartan system and DGLA for the pair. Let R E ∈ A 1 , 1 ( X, End E ) be the Chern curvature of ( E, h E ) . The Chern connection ∇ E induces a connection on End E , defined by ( ∇ End E A )( s ) = ∇ E ( As ) − A ( ∇ E s ) We decompose (2.2) ∇ End E = D 1 , 0 End E + ̄ ∂ End E By the Bianchi identity (see, e.g. [Dem-eBook, Chapter V, Theorem 4.2]), (2.3) ∇ End E R E = 0 In particular, ̄ ∂ End E R E = 0 , and hence [ R E ] ̄ ∂ End E ∈ H 1 , 1 ( X, End E ) This class is inde- pendent of the choice of h E and is called the Atiyah class of E , denoted by At( E ) By the computations in the previous subsection, expanding D 2 t leads to: Proposition 2.4 ([Hua95, p. 68, (16)-(17)], [CS16, §3.3 (2)], [LT18, Theorem 4.9]) Let ( X, E ) be a holomorphic pair and let { ( A t , φ t ) } t ∈ D m ⊂ A 0 , 1 ( X, End E ) ⊕ A 0 , 1 ( X, T 1 , 0 X ) be a family depending holomorphically on t . Then ( A t , φ t ) defines a holomorphic family of pairs ( X t , E t ) if and only if it satisfies the following Maurer–Cartan system, consisting of two equations: (2.4) ̄ ∂φ t = 1 2 [ φ t , φ t ] , ̄ ∂ End E A t = φ t y R E + φ t y D 1 , 0 End E A t − 1 2 [ A t , A t ] Recall that (see e.g. [Man22, p. 145]) a differential-graded Lie algebra (DGLA) is the data of a differential-graded vector space ( L , d ) together with a bilinear map [ · , · ] : L × L → L (called bracket ) of degree 0 , such that the following conditions hold: (i) (graded skew-symmetry) [ a, b ] = − ( − 1) | a || b | [ b, a ] ; (ii) (graded Jacobi identity) [ a, [ b, c ]] = [[ a, b ] , c ] + ( − 1) | a || b | [ b, [ a, c ]] ; (iii) (graded Leibniz rule) d [ a, b ] = [ d a, b ] + ( − 1) | a | [ a, d b ] By [Hua95, § 1.1] and also [CS16, § 3], set A ( E ) := End E ⊕ T 1 , 0 X as a smooth complex vector bundle over X For ( A, φ ) ∈ A 0 ,p ( X, A ( E )) and ( B, ψ ) ∈ A 0 ,q ( X, A ( E )) , define a bracket [ · , · ] : A 0 ,p ( X, A ( E )) × A 0 ,q ( X, A ( E )) −→ A 0 ,p + q ( X, A ( E )) by [( A, φ ) , ( B, ψ )] := ( φ y D 1 , 0 End E B − ( − 1) pq ψ y D 1 , 0 End E A − [ A, B ] , [ φ, ψ ] ) Define ̄ ∂ = A ( E ) : A 0 ,p ( X, A ( E )) −→ A 0 ,p +1 ( X, A ( E )) on A ( E ) by ̄ ∂ = A ( E ) := ( ̄ ∂ End E = 0 ̄ ∂ T X ) , where = ∈ A 0 , 1 ( X, Hom( T 1 , 0 X , End E ) ) is defined by = ∧ φ := ( − 1) p φ y R E , φ ∈ A 0 ,p ( X, T 1 , 0 X ) 10 RUNZE ZHANG One checks that ( A ( E ) , ̄ ∂ = A ( E ) ) defines a holomorphic vector bundle over X , and that its holomorphic structure depends only on the Dolbeault cohomology class [ = ] , hence only on At( E ) , the Atiyah class of E Thus we will simply write ̄ ∂ A ( E ) and refer to ( A ( E ) , ̄ ∂ A ( E ) ) as the Atiyah extension of E We can verify that ( A 0 , • ( X, A ( E )) , ̄ ∂ A ( E ) , [ · , · ]) is a DGLA. Moreover, the Maurer–Cartan system (2.4) can be packaged into a single Maurer–Cartan equation in A ( E ) : (2.5) ̄ ∂ A ( E ) ( A t , φ t ) = 1 2 [( A t , φ t ) , ( A t , φ t )] Remark 2.5. In the algebraic setting, the role of the Atiyah extension A ( E ) is played by the sheaf D 1 ( E ) of scalar differential operators of order at most one on E (see [Ser06] for the line bundle case and [Mar09] for the general case). Recall that a differential operator of order 1 on a holomorphic vector bundle E of rank r is a linear map P : Ω 0 ( E ) → Ω 0 ( E ) which, locally, takes the form P = ( g ij ) + n ∑ k =1 ( h k ij ) ∂ ∂z k , where ( g ij ) and ( h k ij ) are r × r matrices of holomorphic functions. Such an operator is said to have scalar principal symbol if ( h k ij ) = h k · I r × r for some local holomorphic functions h k . There exists an exact sequence 0 −→ End E −→ D 1 ( E ) σ − − → T 1 , 0 X −→ 0 , where the symbol map σ is locally given by σ ( P ) = ∑ n k =1 h k ∂ ∂z k It was shown in e.g. [CS16, Proposition A.2] that D 1 ( E ) carries a natural holomorphic structure, under which it is isomorphic to the Atiyah extension A ( E ) . Therefore, the DGLA ( A 0 , • ( X, A ( E )) , ̄ ∂ A ( E ) , [ · , · ]) above is naturally isomorphic to the one obtained via the algebraic construction. 2.3. Kuranishi family. For a holomorphic pair ( X, E ) , the existence of the correspond- ing Kuranishi family was proved in [Hua95, Theorem 1.8] and [CS16, §§5-6]. We now follow the analytic construction of Chan–Suen in this subsection. Choose any smooth Hermitian metrics ω on X and g on A ( E ) . By Hodge theory on A 0 , • ( X, A ( E )) , let H (resp. G , ̄ ∂ ∗ A ( E ) ) denote the harmonic projection (resp. the Green operator, the formal adjoint of ̄ ∂ A ( E ) ) with respect to ω and g Let { η 1 , . . . , η m } be a basis of H 1 ( X, A ( E )) ' H 1 , and set (2.6) 1 ( t ) := m ∑ i =1 t i η i ∈ H 1 Consider the non-linear equation (2.7) ( t ) = 1 ( t ) + 1 2 ̄ ∂ ∗ A ( E ) G ( [ ( t ) , ( t )] ) Then there exists a unique canonical formal power series solution of (2.7), (2.8) can ( t ) = ∑ j ≥ 1 can j ( t ) ∈ A 0 , 1 ( X, A ( E )) , KURANISHI SPACE OF JOINT DEFORMATIONS OF ( X, E ) 11 where each can j ( t ) is homogeneous of degree j in t. Indeed, this is clear since (2.9) can 2 ( t ) = 1 2 ̄ ∂ ∗ A ( E ) G [ 1 ( t ) , 1 ( t ) ] , can 3 ( t ) = 1 2 ̄ ∂ ∗ A ( E ) G ([ 1 ( t ) , can 2 ( t )] + [ can 2 ( t ) , 1 ( t )]) , can N ( t ) = 1 2 ̄ ∂ ∗ A ( E ) G ([ 1 ( t ) , can N − 1 ( t ) ] + [ can N − 1 ( t ) , 1 ( t ) ] + N − 2 ∑ k =2 [ can k ( t ) , can N − k ( t ) ]) , N ≥ 4 Another important property is that, by standard elliptic regularity, can ( t ) depends ana- lytically on t for sufficiently small | t | . Moreover, can ( t ) satisfies the Maurer–Cartan equation ̄ ∂ A ( E ) can ( t ) = 1 2 [ can ( t ) , can ( t )] ⇐⇒ H ( [ can ( t ) , can ( t )] ) = 0 The associated Kuranishi obstruction map is Ob ( X,E ) ( t ) : U ⊂ H 1 ( X, A ( E )) −→ H 2 ( X, A ( E )) , t 7 −→ H ( [ can ( t ) , can ( t )] ) , defined on a suitable small neighborhood U of 0 ∈ H 1 ( X, A ( E )) . The Kuranishi space of the pair ( X, E ) is defined to be the analytic subspace Kur( X, E ) := Ob − 1 ( X,E ) (0) = { t ∈ U ∣ ∣ H ( [ can ( t ) , can ( t )] ) = 0 } For each s ∈ Kur( X, E ) , the element can ( s ) = ( A can s , φ can s ) satisfies the Maurer–Cartan system (2.4) and hence, by Proposition 2.4, determines a holomorphic pair ( X s , E s ) . As s varies, these pairs form the Kuranishi family { ( X s , E s ) } s ∈ Kur( X,E ) Remark 2.6. The above construction involves the choice of Hermitian metrics, and hence the resulting analytic space Kur( X, E ) is not uniquely defined . Nevertheless, the germs at the origin arising from different choices of Hermitian metrics are (non-canonically) isomorphic. We denote this Kuranishi germ by ( Kur( X, E ) , 0 ) Definition 2.7. The pair ( X, E ) is said to have unobstructed joint deformations if its Kuranishi germ at the origin is smooth. Proposition 2.8. The joint deformations of ( X, E ) are unobstructed if and only if for 1 ( t ) defined in (2.6), the obstruction equation ̄ ∂ A ( E ) ( t ) = 1 2 [ ( t ) , ( t )] , or equivalently, writing ( t ) = ( A t , φ t ) , (MC1) (MC2) ̄ ∂φ t = 1 2[ φ t , φ t ]; ̄ ∂ End E A t = φ t y R E + φ t y D 1 , 0 End E A t − 1 2 [ A t , A t ] , can be solved order-by-order formally. 12 RUNZE ZHANG Proof. The “only if” part is clear by setting ( t ) := can ( t ) as in (2.8). For the converse, the degree-two part of the Maurer–Cartan equation reads ̄ ∂ A ( E ) 2 ( t ) = 1 2 [ 1 ( t ) , 1 ( t )] In particular, [ 1 ( t ) , 1 ( t )] is ̄ ∂ A ( E ) -exact. By bundle-valued Hodge decomposition, [ 1 ( t ) , 1 ( t )] = H [ 1 ( t ) , 1 ( t )] + ̄ ∂ A ( E ) ̄ ∂ ∗ A ( E ) G [ 1 ( t ) , 1 ( t )] + ̄ ∂ ∗ A ( E ) ̄ ∂ A ( E ) G [ 1 ( t ) , 1 ( t )] , and exactness forces H [ 1 ( t ) , 1 ( t )] = 0 , hence [ 1 ( t ) , 1 ( t )] = ̄ ∂ A ( E ) ̄ ∂ ∗ A ( E ) G [ 1 ( t ) , 1 ( t )] We then replace 2 ( t ) by the degree-two term of the canonical solution of (2.8), namely can 2 ( t ) = 1 2 ̄ ∂ ∗ A ( E ) G [ 1 ( t ) , 1 ( t )] presented in (2.9), which still solves the Maurer–Cartan equation in degree-two. At each higher order N ≥ 3 , a slightly more involved argument shows that one may replace N ( t ) by the “better” term can N ( t ) , so that the full (convergent) power series can ( t ) as in (2.8) still satisfies the Maurer–Cartan equation over a sufficiently small disc. This yields the desired unobstructedness. 3. Theorems A, B and Corollary D: proofs and related examples 3.1. Tian–Todorov proofs of Theorem A and Theorem B. For simplicity, we will always assume m = dim C H 1 ( X, A ( E )) = 1 in what follows, so that there is only one deformation parameter t := t 1 . The case m > 1 can be treated in a similar way. Let E be a holomorphic vector bundle of rank r over X The trace map in linear algebra gives rise to an O X -linear trace map tr : End E −→ O X , satisfying tr(id E ) = r Denote the kernel of the trace map by End 0 E , the trace-free part of the endomorphism bundle. The trace map splits holomorphically via the scaled diagonal embedding O X −→ End E, λ 7 −→ λ r id E , so that End E = End 0 E ⊕ O X as holomorphic vector bundles. Consequently, (3.1) H k ( X, End E ) = H k ( X, End 0 E ) ⊕ H k ( X, O X ) for any k Fix a Hermitian metric h E on E and let ∇ E = D 1 , 0 E + ̄ ∂ E be its associated Chern connection. Denote by R E the curvature of ∇ E . In the sequel we will always adopt the following notation. For any A ∈ A 0 , 1 ( X, End E ) , write A = A 0 + a r id E , tr A 0 = 0 , a = tr A ∈ A 0 , 1 ( X, O X ) Correspondingly, split the curvature: R E = R 0 E + 1 r (tr R E ) id E , tr R 0 E = 0 KURANISHI SPACE OF JOINT DEFORMATIONS OF ( X, E ) 13 Here tr R E is the curvature of det E , which is (1 , 1) -form satisfying (3.2) d tr R E = 0 by the Bianchi identity (2.3). Proof of Theorem A. The proof will be divided into two steps. Step ( I ) Reduction on the obstruction equations: Calabi–Yau property & assumption H 2 ( X, End 0 E ) = 0 Under the assumption that K X is torsion, i.e., lK X is holomorphically trivial for some integer l ≥ 1 , write (3.3) lK X = K X + ( l − 1) K X and set F := ( l − 1) K X Since F is also torsion, it is Hermitian flat. More precisely, we can equip F with a Hermitian metric h F whose Chern connection ∇ F = D 1 , 0 F + ̄ ∂ F is flat, meaning its curvature R F vanishes. Moreover, there exists a nowhere vanishing F -valued holomorphic ( n, 0) -form Ω ∈ H 0 ( X, lK X ) ' H n, 0 ( X, F ) We will need the following bundle-valued version of the Tian–Todorov lemma, in which the flatness of F is not required — it will be needed later; see Lemma 3.4. Lemma 3.1 ([LR12, Corollary 3.4]) For any φ, ψ ∈ A 0 , 1 ( X, T 1 , 0 X ) and any η ∈ A • , • ( X, F ) , one has [ φ, ψ ] y η = − D 1 , 0 F ( ψ y ( φ y η )) + ψ y D 1 , 0 F ( φ y η ) + φ y D 1 , 0 F ( ψ y η ) − ψ y ( φ y D 1 , 0 F η ) Using the Calabi–Yau property of X and following the insightful idea dating back to [LRW19] (also developed in [Zha26]), we will need another auxiliary formal power series Ω( t ) with Ω(0) = Ω to further reduce Proposition 2.8 to the following: Proposition 3.2. In the setting of Theorem A, for any first-order deformation 1 = ( A 1 , φ 1 ) , suppose that there exist formal power series φ ( t ) = ∑ i ≥ 1 φ i t i ∈ A 0 , 1 ( X, T 1 , 0 X )[[ t ]] , a ( t ) = ∑ i ≥ 1 a i t i ∈ A 0 , 1 ( X, O X )[[ t ]] , a 1 = tr A 1 , and Ω( t ) = Ω 0 + ∑ i ≥ 1 Ω i t i ∈ A n, 0 ( X, F )[[ t ]] , Ω 0 = Ω , satisfying the system formally: ( † ) ( ̄ ∂ F + D 1 , 0 F ◦ φ ( t ) y ) Ω( t ) = 0 , ( ̄ ∂ F + 1 2 D 1 , 0 F ◦ φ ( t ) y )( φ ( t ) y Ω( t ) ) = 0 , ( ̄ ∂a ( t ) − φ ( t ) y tr R E − φ ( t ) y ∂a ( t ) ) ∧ Ω( t ) = 0 Then the Kuranishi space of the pair ( X, E ) is smooth. 14 RUNZE ZHANG Proof. The first two equations in ( † ) together with Lemma 3.1 (noting that Ω( t ) is auto- matically D 1 , 0 F -closed for type reasons) imply ̄ ∂ ( φ ( t ) y Ω( t ) ) = − 1 2 D 1 , 0 F ( φ ( t ) y ( φ ( t ) y Ω( t ) )) = 1 2 [ φ ( t ) , φ ( t )] y Ω( t ) − φ ( t ) y D 1 , 0 F ( φ ( t ) y Ω( t ) ) = 1 2 [ φ ( t ) , φ ( t )] y Ω( t ) + φ ( t ) y ̄ ∂ Ω( t ) Hence (3.4) ( ̄ ∂φ ( t ) ) y Ω( t ) = ̄ ∂ ( φ ( t ) y Ω( t ) ) − φ ( t ) y ̄ ∂ Ω( t ) = 1 2 [ φ ( t ) , φ ( t )] y Ω( t ) Set V ( t ) := ̄ ∂φ ( t ) − 1 2 [ φ ( t ) , φ ( t )] = ∑ i ≥ 2 v i t i , with v i ∈ A 0 , 2 ( X, T 1 , 0 X ) . Then (3.4) becomes V ( t ) y Ω( t ) = 0 Locally write Ω = f ( z ) dz 1 ∧ · · · ∧ dz n ⊗ e with f ( z ) nowhere vanishing and e a local frame of F Any v ∈ A 0 , 2 ( X, T 1 , 0 X ) can be expressed locally as v = n ∑ j =1 g j ∂ ∂z j , where g j are smooth (0 , 2) -forms Contracting with Ω gives Ω y v = f ( z ) n ∑ j =1 ( − 1) j − 1 g j dz 1 ∧ · · · ∧ ̂ dz j ∧ · · · ∧ dz n ⊗ e. The ( n − 1 , 2) -forms appearing are linearly independent, hence the map v 7 → Ω y v is injective. Expanding V ( t ) y Ω( t ) = ∑ k ≥ 2 ( ∑ i + j = k v i y Ω j ) t k = 0 , we obtain: For k = 2 , v 2 y Ω = 0 , so v 2 = 0 by injectivity. Assume v 2 = · · · = v N = 0 for some N ≥ 2 . Then the coefficient of t N +1 reduces to v N +1 y Ω = 0 , hence v N +1 = 0 Therefore V ( t ) = 0 , i.e., (3.5) ̄ ∂φ ( t ) = 1 2 [ φ ( t ) , φ ( t )] , namely the first obstruction equation (MC1). Via a similar argument, the third equation in ( † ) is equivalent to (3.6) ̄ ∂a ( t ) = φ ( t ) y tr R E + φ ( t ) y ∂a ( t ) This is obtained by taking the trace of the second obstruction equation (MC2) wherein the following identities are used: – tr( ̄ ∂ End E A ( t )) = ̄ ∂ (tr A ( t )) ; KURANISHI SPACE OF JOINT DEFORMATIONS OF ( X, E ) 15 – tr( φ ( t ) y R E ) = φ ( t ) y tr R E ; – tr([ A ( t ) , A ( t )]) = 0 ; – For the term tr ( φ ( t ) y D 1 , 0 End E A ( t ) ) , we compute locally. In a local trivialisation, D 1 , 0 End E A ( t ) = ∂A ( t ) + [ ω 1 , 0 , A ( t )] , where ω 1 , 0 is a matrix-valued (1 , 0) -form. Then this term equals φ ( t ) y tr ( D 1 , 0 End E A ( t ) ) = φ ( t ) y ( tr( ∂A ( t )) + tr([ ω 1 , 0 , A ( t )]) ) = φ ( t ) y ∂a ( t ) Consequently, the trace-free part of (MC2) becomes (TF) ̄ ∂ End E A 0 ( t ) = φ ( t ) y R 0 E + φ ( t ) y D 1 , 0 End E A 0 ( t ) − 1 2 [ A ( t ) , A ( t )] Expanding in t , the N -th order obstruction equation for N ≥ 2 takes the form ̄ ∂A 0 N = C tf N , where C tf N is ̄ ∂ End E -closed, provided that all obstructions up to order N − 1 in (TF) and all obstructions up to order N in (3.5) vanish . Since H 2 ( X, End 0 E ) = 0 , we can solve for A 0 N inductively. Hence A ( t ) := A 0 ( t ) + a ( t ) r id E satisfies (MC2). This completes the proof. Remark 3.3. The main differences between Proposition 3.2 and [LRW19, Proposition 4.10], as well as [Zha26, Proposition I.3] (both with D = ∅ ), are two-fold: (i) An additional obstruction equation (the third equation in ( † )) needs to be solved simultaneously; (ii) The solutions φ ( t ) , a ( t ) and Ω( t ) are only required to be formal. (a) In the arguments of [LRW19, Zha26], the smoothness of the corresponding Ω( t ) was used to ensure that Ω( t ) has no zeros for small t (since Ω(0) is nowhere vanishing). However, formal solutions suffice as shown above. (b) As for φ ( t ) and a ( t ) , formal solutions are also enough: once they are ob- tained, we can then choose particularly nice solutions whose smoothness is very standard (see Proposition 2.8). This omits the regularity arguments as in [LRW19, proof of Theorem 4.12, pp. 36-39] and [Zha26, Appendix I, Step (III)]. Step ( II ) Solve the obstruction equations: iteration By ( † ) k we mean the part of ( † ) formed by the first equation at order k together with the second and third equations at order k + 1 . We now adopt an iterative approach to solve ( † ). Before proceeding with the proof, we recall the following standard fact; see, for example, [HZ26, Proposition A.1] for a proof. Lemma 3.4. Let X be a Fujiki manifold, and let ( F, h F ) be a Hermitian flat holomorphic vector bundle on X . Write its Chern connection as ∇ F = D 1 , 0 F + ̄ ∂ F Then ( X, F ) satisfies the D 1 , 0 F ̄ ∂ F -lemma. More precisely, for every ∇ F -closed F -valued ( p, q ) -form α on X , the following conditions are equivalent: ∇ F -exactness ⇐⇒ D 1 , 0 F -exactness ⇐⇒ ̄ ∂ F -exactness ⇐⇒ D 1 , 0 F ̄ ∂ F -exactness 16 RUNZE ZHANG For k = 1 , the system ( † ) 1 reads as (3.7) ̄ ∂ F Ω 1 + D 1 , 0 F ( φ 1 y Ω 0 ) = 0 ,