Mathematische Annalen (2026) 394:68 https://doi.org/10.1007/s00208-026-03410-y Mathematische Annalen Logarithmic @@ -lemma and several geometric applications (with an appendix joint with Sheng Rao) Runze Zhang 1,2 Received: 13 November 2025 / Revised: 23 January 2026 / Accepted: 31 January 2026 © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2026 Abstract In this paper, we prove a ∂∂ -type lemma on compact Kähler manifolds for logarith- mic differential forms valued in the dual of a certain pseudo-effective line bundle, thereby confirming a conjecture proposed by Wan. We then derive several appli- cations, including strengthened results by Esnault–Viehweg on the degeneracy of the spectral sequence at the E 1 -stage for projective manifolds associated with the logarithmic de Rham complex, as well as by Katzarkov–Kontsevich–Pantev on the unobstructed locally trivial deformations of a projective generalized log Calabi–Yau pair with some weights, both of which are extended to the broader context of compact Kähler manifolds. Furthermore, we establish the Kähler version of an injectivity the- orem originally formulated by Ambro in the algebraic setting. Notably, while Fujino previously addressed the Kähler case, our proof takes a different approach by avoiding the reliance on mixed Hodge structures for cohomology with compact support. Mathematics Subject Classification 58A14 · 32G05 · 32Q25 · 58A10 · 58A25 · 18G40 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Backgrounds and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Idea of the proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Overview and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Logarithmic connection and logarithmic complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Warm-up: all q i > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Hodge decomposition: conic version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Tribute to Professor Kefeng Liu: On the Occasion of His 60th Birthday. B Runze Zhang runze.zhang@unice.fr 1 School of Mathematics and statistics, Wuhan University, Wuhan 430072, China 2 Université Côte d’Azur, CNRS, Laboratoire J.-A. Dieudonné, Parc Valrose, 06108 Nice Cedex 2, France 0123456789().: V,-vol 123 68 Page 2 of 65 R. Zhang 3.2 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 General case: not all q i > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 de Rham–Kodaira decomposition for conic currents . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Log conic current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Proof of Theorem B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Closedness of twisted logarithmic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Degeneracy of spectral sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Injectivity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Unobstructed deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Deformations of pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Differentiable graded Batalin–Vilkovisky algebra . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Proof of Theorem A: algebraic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix I. Proof of Theorem A: analytic approach (joint with Sheng Rao) . . . . . . . . . . . . . . . Appendix II. Proof of King’s quasi-isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix III. Directed limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction 1.1 Backgrounds and main results One of the strongest results in deformation theory is the Bogomolov–Tian–Todorov (BTT) theorem [7, 84, 85], which, using differential geometric methods, establishes that Calabi–Yau manifolds have unobstructed deformations. This means that the Kuranishi space of deformations of complex structures on such manifolds is smooth. We should remark here that while many authors consider Calabi–Yau manifolds to be projec- tive, we make no projectivity assumptions. Specifically, in this paper, a Calabi–Yau manifold X is defined as a compact Kähler manifold with trivial canonical bundle K X O X Algebraic proofs of the BTT theorem have also been provided using the degeneration of the Hodge-to-de Rham spectral sequence plus the nowadays called T 1 -lifting technique [26, 52, 71], see also [46]. In fact, the Kähler condition in this theorem can be relaxed. For instance, it applies to a compact complex manifold X with trivial canonical bundle that satisfies the ∂∂ -lemma, see e.g. [43, the proof in Chapter 6 after minor changes] and [70, Theorem 1.3]. Recall that the (standard) ∂∂ -lemma refers to: for every pure-type d -closed form on a compact complex manifold, the prop- erties of d -exactness, ∂ -exactness, ∂ -exactness, and ∂∂ -exactness are equivalent. More generally, it also holds for such manifolds satisfying a weak version of the ∂ ̄ ∂ -lemma, 1 as shown by the proofs in [62], see also [76] for a CR-version. Moreover, the BTT theorem holds for such manifolds if the Frölicher spectral sequence degenerates at the first page, see for example, [49, the proof in Theorem 4.18] and [2, Theorem 3.3] (this case is even true when K X is a torsion bundle, see e.g. [45, Corollary 4.1]). Noteworthy to mention that the BTT theorem has various extensions, which include results on the unobstructedness of deformations of the following: ( D 1 ) (generalized) log Calabi–Yau pairs [44, 49, 62, 72, 79, 86]; 1 The weak version of the standard ∂ ̄ ∂ -lemma was first introduced in [34] while studying deformations of balanced manifolds. 123 Logarithmic ∂∂ -lemma and several geometric applications... Page 3 of 65 68 ( D 2 ) (weak) Fano varieties [66, 71, 72, 79]; ( D 3 ) (weak) Poisson structures [32, 38, 42, 72, 73]; ( D 4 ) Landau–Ginzburg models [50]; ( D 5 ) a pair ( X , L ) (resp. ( X , F ) ), where L (resp. F ) is a line bundle (resp. coherent sheaf) on a smooth projective variety X over an algebraically closed field of characteristic zero, with a trivial canonical bundle [47, 48]. Notice that direction ( D 1 ) is particularly interesting from a mirror symmetry per- spective, as the deformations of a (generalized) log Calabi–Yau pair are expected to mirror to the deformations of the corresponding complexified symplectic form on the mirror Landau–Ginzburg model, see [4, 5, 49, 58] for further details. Additionally, the BTT theorem has applications in the context of logarithmic geometry, see for instance, [13, 27–31, 33, 53]. In this paper, we focus our attention on the direction ( D 1 ), specifically considering the pair ( X , D ) , where X is a compact Kähler manifold and D is a simple normal cross- ing divisor. One of our main theorems, stated below, strengthens all the corresponding results in this direction. Theorem A Let X be an n-dimensional compact Kähler manifold, and let D = ∑ s i = 1 D i be a simple normal crossing divisor on X . Assume that there exist ratio- nal weights { a i } 1 ≤ i ≤ s ⊂ [ 0 , 1 ] ∩ Q such that s ∑ i = 1 O X ( a i D i ) = − K X ∈ Pic ( X ) ⊗ Z Q =: Pic Q ( X ), (1.1) which means that s ∑ i = 1 O X ( N a i D i ) = − N K X ∈ Pic ( X ), (1.1 ′ ) for some positive integer N . Then, the locally trivial (infinitesimal) deformations of the pair ( X , D ) are unob- structed. That is, for such deformations, the pair admits a smooth Kuranishi space (see Sect. 5.4.1 for further details). Remark 1.1 • The locally trivial deformations of a pair ( X , D ) can be roughly under- stood as deformations of X in which D deforms along with it in a locally trivial manner, so in particular, keeping the analytic singularity types. Therefore, con- sidering locally trivial deformations is not a restriction but rather has geometric significance. Notice that if D is smooth, then every deformation of the pair is locally trivial, see for example [64, Lemmata 4.3.4 & 4.3.5 & Theorem 4.4.3]. Con- sequently in this case, the pair ( X , D ) in Theorem A has unobstructed deformations. • Note that if the locally trivial assumption is dropped in the above theorem (i.e., if partial smoothings of D are allowed), the conclusion may fail, as illustrated by the examples in [31] due to Felten–Petracci–Robin. 123 68 Page 4 of 65 R. Zhang Remark 1.2 ( i ) Theorem A for the case of projective manifolds was proved by Katzarkov–Kontsevich–Pantev [49, § 4.3.3 (iii) ] (see also [58, § 3, Case 3] for a special case). In the projective setting, the equality (1.1) can be rephrased using the terminology and concepts from algebraic geometry as the Q -divisor K X + s ∑ i = 1 a i D i being Q -trivial . Here we also denote by K X the canonical divisor on a projective manifold X . We give a brief overview of the proof in [49], which proceeds in three main stages. 1. Degeneration property of a DGBVA. Katzarkov–Kontsevich–Pantev con- structed a differential graded Batalin–Vilkovisky algebra (DGBVA), together with an associated differential graded Lie algebra (DGLA) ( g , d g ) . This DGLA contains a direct summand ( g ′ , d g ′ ) that governs the locally trivial deformations of the pair ( X , D ) 2 They then prove that this DGBVA satisfies the degeneration property via the theory of mixed Hodge structures, see [49, Lemma 4.21]. 2. Homotopy abelianity and formal smoothness. The degeneration property implies that ( g , d g ) is homotopy abelian , and hence the same holds for ( g ′ , d g ′ ) . As a consequence, the formal moduli space associated with the DGLA ( g ′ , d g ′ ) is smooth. This establishes the formal smoothness of the Kuranishi space, meaning that the completion of its local ring at the maxi- mal ideal is isomorphic to a ring of formal power series. 3. From formal to analytic smoothness via Artin’s theorem. To conclude that the Kuranishi space B is smooth as an analytic germ, one can appeal to M. Artin’s fundamental approximation theorem [3] on the existence of conver- gent solutions for analytic equations from the existence of formal solutions, which implies that this formally smooth analytic germ is in fact analytically smooth, see e.g. [78, Theorem B.1] for a detailed proof. ( ii ) Notably, the setup (1.1) encompasses two important endpoint situations (the corresponding projective cases were also presented in [49, § 4.3.3 (i) & (ii) ], respectively), namely: (a) the log Calabi–Yau case , where the logarithmic canonical bundle n X ( log D ) K X ⊗ O X ( D ) is trivial (i.e., all a i = 1 and N = 1 in (1.1 ′ )); 2 Roughly speaking, the underlying philosophy is encapsulated in Deligne’s principle from his 1986 letter to J. Millson: “ in characteristic 0 , a deformation problem is controlled by a DGLA, and quasi-isomorphic DGLAs give rise to the same deformation theory ”, see [21]. 123 Logarithmic ∂∂ -lemma and several geometric applications... Page 5 of 65 68 (b) the Calabi–Yau case with a simple normal crossing divisor , where K X is trivial (i.e., all a i = 0 and N = 1). Therefore, we can regard the pair ( X , D ) in Theorem A as a Kähler generalized log Calabi–Yau pair Based on the theory of DGLA and the Cartan homotopy construction, Iacono proved cases (a) and (b) in the algebraic setting over an algebraically closed field of characteristic zero [44, Corollaries 5.5 & 5.8 & Remark 5.6] (see also [45, § 4.2] from the perspective of abstract BTT theorem), and later extended the case (a) to the context of the compact complex manifolds where the logarithmic Hodge-to-de Rham spectral sequence degenerates at the E 1 -level [45, Corollary 4.5]. Additionally, see Sano’s work employing the T 1 -lifting technique [79, Remark 2.5] and Ran’s approach via Poisson deformations [72, Theorem 12]. More recently, Liu–Rao–Wan proved cases (a) and (b) using a power series method, more faithfully following Tian–Todorov’s approach [62, Theorems 0.8 & 0.9]. Furthermore, Iacono proved the case (also in the algebraic setting) if any a i is equal to 1 N for some positive integer N in (1.1), via a cyclic covering construc- tion, see [44, Proposition 6.4]. Wan also contributed to this line of research by settling the case of N = 2 (when X is a compact Kähler manifold), with his proof also relying on a cyclic cover trick, as shown in [86, Theorem 0.5]. Indeed, inspired by the approach in [49] (as outlined in Remark 1.2 (i)), we provide two distinct proofs of Theorem A: – a proof based on DGLA theory, following the more algebraic and abstract frame- work of [49], presented in Sect. 5.4.3; – a purely analytic proof using power series method in the spirit of [62], which is more visualizable and geometric . The proof, jointly with Sheng Rao, is given in Appendix I. In both cases, the key step is to solve a certain ̄ ∂ -equation for logarithmic forms twisted by the dual of a pseudo-effective line bundle, originally conjectured by Wan [86]. Theorem B [86, Conjecture 0.8] Let ( X , D ) be as in Theorem A, and let L be a holomorphic line bundle over X . Assume that there exist rational weights { q i } 1 ≤ i ≤ s ⊂ [ 0 , 1 ] ∩ Q such that L = s ∑ i = 1 O X ( q i D i ) ∈ Pic Q ( X ). (1.2) Then for any α ∈ A 0 , q ( X , p − 1 X ( log D ) ⊗ L ∗ ) satisfying ̄ ∂ D ′ h ∗ α = 0 pointwise on X ◦ := X \ Supp D , 123 68 Page 6 of 65 R. Zhang the following twisted logarithmic ̄ ∂ -equation ̄ ∂χ = D ′ h ∗ α pointwise on X ◦ ( ) admits a solution χ ∈ A 0 , q − 1 ( X , p X ( log D ) ⊗ L ∗ ) Here D ′ h ∗ represents the ( 1 , 0 ) -part of the integrable logarithmic connection ∇ h ∗ along D, induced by the singular metric h ∗ on the dual bundle L ∗ (see Sect. 2 for more details). Remark 1.3 Theorem B revisits two extreme cases: L O X and L O X ( D ). Both of which were proved by Liu–Rao–Wan [62, Theorem 0.1 & 0.2]. It also revisits the case where all q i = 1 N for some positive integer N , and D is a smooth divisor, as proved by Wan [86, Theorem 0.1]. See also [77, § 4] for some discussions related to Wan’s conjecture from the perspective of double complexes. Remark 1.4 Strictly speaking, Theorem B is weaker than the “genuine” logarithmic version of the (standard) ∂∂ -lemma aforementioned, as we cannot guarantee that the solution χ obtained in ( ) is moreover D ′ h ∗ -exact. 3 It would be very interesting to know if this can be achieved. As is widely recognized, the E 1 -degeneration of certain spectral sequences is a useful tool in algebraic geometry and complex geometry, such as being used to imply several injectivity, vanishing and torsion-free theorems. For a more comprehensive discussion, we refer the reader to [17, 25, 35], along with the references therein. Following the approach in [62, Theorem 3.2] and utilizing the logarithmic counter- part of the general description of terms in the Frölicher spectral sequence as presented in [16, Theorems 1 & 3], we obtain the expression expression E p , q r Z p , q r B p , q r with the differential map d p , q r : E p , q r → E p + r , q − r + 1 r . The subgroups Z p , q r and B r p , q are detailed in Sect. 5.2. As a direct corollary of Theorem B, we deduce that d p , q r = 0 for all r ≥ 1 and all p , q . This leads to the following result, which extends the work of H. Esnault–E. Viehweg on projective manifolds [25, Theorem 3.2 & Remarks 3.3] to the broader setting of compact Kähler manifolds. Theorem C With the same setting (1.2) as in Theorem B, the following spectral sequence E p , q 1 = H q ( X , p X ( log D ) ⊗ L ∗ ) ⇒ H p + q ( X , • X ( log D ) ⊗ L ∗ ) associated to the logarithmic de Rham complex (see Sect. 2) ( • X ( log D ) ⊗ L ∗ , ∇ h ∗• ) 3 Following the terminology of [75, Notation 3.5] (see also [74] for the foliated setting), we can analogously say that the pair ( X , D ) satisfies S p , q with respect to ( L ∗ , h ∗ ) 123 Logarithmic ∂∂ -lemma and several geometric applications... Page 7 of 65 68 degenerates at the E 1 -level. Here H p + q ( X , • X ( log D ) ⊗ L ∗ ) denotes the hyperco- homology. Building on this, we establish the Kähler version of Ambro’s injectivity theorem [1, Theorem 2.3 & Remark 2.6], originally proved in the algebraic setting for proper, non-singular varieties over an algebraically closed field of characteristic zero, see also the recent work of Murayama [68]. The proof of Theorem D is detailed in Sect. 5.3. Notably, the case of Theorem D where X is a compact complex manifold in Fujiki’s class C (i.e., bimeromorphic to a compact Kähler manifold) was established by Fujino [36, Theorem 1.2], utilizing the theory of mixed Hodge structures for cohomology with compact support. Our approach is quite different from Fujino’s (see Remark 5.4). Theorem D Let ( X , D ) be as in Theorem A, and let L be a holomorphic line bundle over X such that L = K X ⊗ s ∑ i = 1 O X ( b i D i ) ∈ Pic Q ( X ) with b i ∈ ( 0 , 1 ] ∩ Q for all i (1.3) Then the restriction homomorphism H q ( X , L ) j −→ H q ( X ◦ , L | X ◦ ) (1.4) is injective, for all q. Equivalently, for every effective Cartier divisor ̂ D with Supp ̂ D ⊂ Supp D , the natural homomorphism H q ( X , L ) j ′ −→ H q ( X , L ⊗ O X ( ̂ D )) (1.5) induced by the inclusion O X ⊂ O X ( ̂ D ) is injective, for all q. In addition to Theorems A, C, and D, we present another application of Theorem B: the closedness of twisted logarithmic forms, see Sect. 5.1. 1.2 Idea of the proofs We briefly outline the basic strategies behind the proofs of Theorem B and Theorem A. For Theorem B, let us first consider the special case: 0 < q i ≤ 1 for every 1 ≤ i ≤ s , which corresponds to Theorem 3.1. The first observation is that any α ∈ A 0 , q ( X , p X ( log D ) ⊗ L ∗ ) , and consequently D ′ h ∗ α , is in fact smooth in conic sense (Proposition 3.12). By utilizing the Hodge decomposition for forms that are smooth in conic sense , as established in [11], we can deduce that [ D ′ h ∗ α ] ∂ = 0 in the L ∗ -valued ( p , q ) -conic Dolbeault cohomology group (Notation 3.14). Finally, to ensure the existence of a solution χ (with at most logarithmic poles) in Eq. (3.1), we require an acyclic resolution of the sheaf p X ( log D ) ⊗ L ∗ by sheaves of germs of L ∗ -valued ( p , • ) -forms that are smooth in conic sense (Proposition 3.15). 123 68 Page 8 of 65 R. Zhang In the more general setting of Theorem B, D ′ h ∗ α is no longer smooth in conic sense; however, the good news is that it remains a conic current with values in ( L ∗ , h ∗ ) , denoted by T D ′ h ∗ α . We then decompose D into E and F (Dec.), where, roughly speak- ing, E (resp. F ) represents the q i = 0 part (resp. q i > 0 part). By using the de Rham–Kodaira decomposition for conic currents [11], we can decompose T D ′ h ∗ α into two parts. One part falls in the image of ∂ , and the other one is a residue term Inspired by [62, Lemma 2.3] and also [11, Lemma 4.3], the key Lemma 4.27 shows that the residue term can also be expressed as ∂ ̂ T , where ̂ T is a conic current, modulo the space of on-E conic currents valued in ( L ∗ , h ∗ ) on X (where, roughly speaking, any element in this space annihilates all test forms valued in L that are smooth in conic sense and moreover vanish on E ; consequently, they will vanish on D as shown in (4.13)). This concept is motivated by the work of King [55]. The final step to find the solution χ in equation ( ) follows a similar approach. The main difference is that this time we need to replace the protagonists in the corresponding acyclic resolution (Proposition 4.22) with the sheaf of log-E conic currents with values in ( L ∗ , h ∗ ) , which is defined as the quotient sheaf of the sheaf of ( L ∗ , h ∗ ) -conic currents by the sheaf of on- E conic currents valued in ( L ∗ , h ∗ ) , see Definition 4.20. Remark 1.5 • Note that Theorem 3.1 can also be derived using the conic current approach, see Corollary 4.24. Moreover, our method provides part of a new proof of [62, Theorem 0.1], see Remark 4.28 (b). • We would also like to emphasize that, much recently, a general logarithmic type ∂∂ -lemma (for top-degree) proved by Cao–P ̆ aun [11, Theorem 1.1] has played a significant role in proving Fujino’s conjecture on a Kähler injectivity theorem [11, Theorem 1.2] (see also the recent independent work of Chan–Choi–Matsumura [12]). Moreover, their logarithmic ∂∂ -lemma provides new results concerning the invariance of plurigenera in the Kähler setting [10]. Our goal is also to establish a logarithmic ∂∂ -lemma, but our approach differs from that of [11], especially in the construction of the acyclic resolution, while still relying on the conic Hodge theory they developed. For Theorem A, – ( Algebraic approach ) As noted in Remark 1.2, the key requirement is the degen- eration property (Definition 5.16) for a suitable DGBVA. Using Theorem B (or more practically, Theorem C), we can achieve this by considering the DGBVA ( A , d , Δ ) , where A := A 0 , • ( X , ∧ • T 1 X ( − log D ) ) , d := ̄ ∂, and Δ := i − 1 ◦ D ′ h ∗ ◦ i Here T 1 X ( − log D ) denotes the logarithmic tangent bundle (Definition 5.7). The map i : ∧ • T 1 X ( − log D ) −→ n −• X ( log D ) ⊗ L ∗ 123 Logarithmic ∂∂ -lemma and several geometric applications... Page 9 of 65 68 is the isomorphism given by contraction with a nowhere-vanishing section ∈ A 0 , 0 ( X , n X ( log D ) ⊗ L ∗ ) , where the holomorphic line bundle L := n X ( log D ) K X ⊗ O X ( D ) satisfies L = s ∑ i = 1 O X ( q i D i ) ∈ Pic Q ( X ), q i := 1 − a i ∈ [ 0 , 1 ] The essential point is that only the existence of logarithmic solutions is required. – ( Analytic approach ) Following Kawamata [51], the Kuranishi family K of locally trivial deformations of the pair ( X , D ) (Definition 5.8) can be realized concretely inside the space of logarithmic Beltrami differentials real analytic ( X , T 1 X ( − log D ) ⊗ 0 , 1 T ∗ X ) ⊂ A 0 , 1 ( X , T 1 X ( − log D ) ) , whose elements satisfy the integrability condition ̄ ∂φ = 1 2 [ φ, φ ] By constructing explicit solutions to the twisted logarithmic ̄ ∂ -equation (Lemma I.1), we follow the iterative method in [62] to produce a family of integrable logarithmic Beltrami differentials over a sufficiently small disk. This yields the unobstructedness in a manner that aligns closely with the classical Tian–Todorov philosophy. 1.3 Overview and outlook We prove Theorem B via an analytic approach, and Theorem A is an application of it, generalizing [49, § 4.3.3 (iii) ], and consequently [49, § 4.3.3 (i) & (ii) ], to the context of compact Kähler manifolds. Indeed, by constructing again a DGBVA and employing the mixed Hodge theory, Katzarkov–Kontsevich–Pantev provided another generalization of the previous results, that is [49, § 4.3.3 (iv) ], specifically when X is a projective normal-crossing Calabi–Yau . More precisely, assume that X is a strict normal crossings variety with irreducible components X = ⋃ j ∈ J X j equipped with a holomorphic volume form X on X − X sing . This form X satisfies the condition that its restriction to each X j has a logarithmic pole along X j ∩ ( ∪ k = j X k ), and the residues of these restricted forms cancel along each X j ∪ X k . Motivated by this, the following question naturally arises, which we will consider in a forthcoming paper. 123 68 Page 10 of 65 R. Zhang Question Could further analytic methods be developed to generalize the variety X in Theorem B to the setting of normal-crossing Calabi–Yau varieties that are Kähler but not necessarily projective? If so, it may also be promising to extend the variety X in Theorem A to this broader context without relying on the mixed Hodge theory. Notations and conventions. Throughout this paper, we work over the field of complex numbers. – Any (compact) complex manifold X in this paper is assumed to be connected. – Denote by A X (resp. O X ) the sheaf of germs of C ∞ differentiable functions (resp. holomorphic functions) over X – Locally free sheaves of A X -modules (resp. O X -modules) and C ∞ complex (resp. holomorphic) vector bundles are considered synonymous. – The terminology Cartier divisors , invertible sheaves , and holomorphic line bundles are used interchangeably. – A sheaf F is called flabby if for every open subset V of X , the restriction map F ( X ) → F ( V ) is onto, i.e., if every section of F on V can be extended to X – A flabby sheaf F is acyclic on all open sets V ⊂ X , meaning that H q ( V , F ) = 0 for any q ≥ 1. – We use additive notation for tensor products and powers of line bundles, and multiplicative notation (resp. additive notation) for hermitian metrics (resp. local weights) of line bundles. For example, ( L 1 , h 1 , φ 1 ), ( L 2 , h 2 , φ 2 ), and ( L 1 + L 2 , h 1 · h 2 , φ 1 + φ 2 ) . Here h 1 = e − φ 1 (resp. h 2 = e − φ 2 ) is a (possibly singu- lar) metric on L 1 (resp. L 2 ). – For ∈ א Q , א denotes the integral part of א , defined as the only integer such that א ≤ א≤ א + 1. 2 Logarithmic connection and logarithmic complex In this section, we will recall some basic notions regarding the logarithmic connection (induced by the singular metric on a holomorphic line bundle) and the logarithmic de Rham complex. For more details refer to [40, Chapter 3.5] and [25, Chapter 2]. Let X be a compact complex manifold of dimension n and D = ∑ s i = 1 D i be a simple normal crossing divisor ( i.e., a divisor with non-singular components D i intersecting each other transversally) on X . Let p X ( D ) = lim − → v p X (v D ) be the sheaf of germs of p -meromorphic forms which are holomorphic on X − D but possibly with arbitrary orders of poles along D . Obviously, ( • X ( D ), d ) is a complex. Deligne introduced the sheaf of germs of logarithmic p-forms [19] p X ( log D ), 123 Logarithmic ∂∂ -lemma and several geometric applications... Page 11 of 65 68 which is defined as the subsheaf of p X ( D ) with logarithmic poles along D , i.e., if V ⊂ X is open, then ( V , p X ( log D )) = { α ∈ ( V , p X ( D )) | α and d α both have simple poles along D } It turns out that there exists a subcomplex ( • X ( log D ), d ) ⊂ ( • X ( D ), d ) , see for example [20, II, 3.1 − 3.7] or [25, Properties 2.2]. Furthermore, p X ( log D ) = p ∧ 1 X ( log D ) is locally free. More precisely, for any z ∈ X , suppose z ∈ D i for any 1 ≤ i ≤ d and z / ∈ D i for d + 1 ≤ i ≤ s . We can then choose local coordinates { z 1 , . . . , z n } in a small neighborhood V of z = ( 0 , . . . , 0 ) such that D i ∩ V = { z i = 0 } for 1 ≤ i ≤ d One writes δ j = { dz j z j if j ≤ d ; dz j if j > d , and for J = { j 1 , . . . , j p } ⊂ { 1 , . . . , n } with j 1 < j 2 < . . . < j p , δ J = δ j 1 ∧ . . . ∧ δ j p Then { δ J | # J = p } (2.1) forms a basis of p X ( log D ) as a free O X -module over V . Furthermore, we denote by A 0 , q ( p X ( log D )) the sheaf of germs of ( 0 , q ) -forms valued in p X ( log D ) , which is a locally free sheaf of A X -modules. Elements in A 0 , q ( X , p X ( log D )), i.e., the global sections of A 0 , q ( p X ( log D )) , are called logarithmic ( p , q ) -forms Next, we recall the definition of (integrable) logarithmic connection along D with respect to a holomorphic vector bundle. Definition 2.1 ([25, Definition 2.4]). Let E be a locally free sheaf of O X -modules and let ∇ : E −→ 1 X ( log D ) ⊗ E be a C -linear map satisfying ∇ ( f · e ) = f · ∇ ( e ) + d f ⊗ e 123 68 Page 12 of 65 R. Zhang One defines ∇ p : p X ( log D ) ⊗ E −→ p + 1 X ( log D ) ⊗ E by the rule ∇ p (ω ⊗ e ) = d w ⊗ e + ( − 1 ) p ω ∧ ∇ ( e ). We assume that ∇ p + 1 ◦ ∇ p = 0 Such ∇ will be called an integrable logarithmic connection along D , or just a con- nection. The complex ( • X ( log D ) ⊗ E , ∇ • ) is called the logarithmic de Rham complex of ( E , ∇ ) Let L be a holomorphic line bundle over X satisfying L = s ∑ i = 1 O X ( q i D i ) ∈ Pic Q ( X ) with q i ∈ Q (note that here we do not restrict the sign of q i ). Then there naturally exists a singular metric h := h L = e − φ L on L , where φ L is a collection of functions defined on small open sets, called the local weight , locally can be written as φ L = s ∑ i = 1 q i log | z i | 2 , with z i = 0 ( i = 1 , . . . , s ) representing the local equations of components of D , see e.g. [22, § 2]. We then obtain that the curvature current with respect to h is i h ( L ) = 2 π i s ∑ i = 1 q i [ D i ] , thanks to the Lelong–Poincaré formula, where [ D i ] is the current of integration over the irreducible ( n − 1 ) -dimensional analytic set D i for any i . In particular, L is pseudo- effective if all q i are non-negative. One can verify without difficulty that there exists a (global) integrable logarithmic connection along D induced by the metric h , denoted by ∇ h , which has a decomposition ∇ h = D ′ h + ̄ ∂. Here D ′ h is the ( 1 , 0 ) -part of ∇ h , which has the local expression D ′ h := ∂ − ∂φ L = ∂ − s ∑ i = 1 q i dz i z i 123 Logarithmic ∂∂ -lemma and several geometric applications... Page 13 of 65 68 More explicitly, following the notations as in Definition 2.1, we have ∇ h (ω ⊗ e ) = d ω ⊗ e + ( − 1 ) p ω ∧ ( s ∑ i = 1 q i dz i z i ) ⊗ e (2.2) One then can check that ∇ 2 h = 0 3 Warm-up: all q i > 0 As a warm-up, we first consider a simple case of Theorem B. In this section, we are devoted to proving: Theorem 3.1 Under the same settings as in Theorem B, assume further that all rational numbers q i in (1.2) lie in ( 0 , 1 ] . Then, for any α ∈ A 0 , q ( X , p X ( log D ) ⊗ L ∗ ) satisfying ∂ D ′ h ∗ α = 0 pointwise on X ◦ , the logarithmic ∂ -equation: ∂χ = D ′ h ∗ α pointwise on X ◦ (3.1) has a solution χ ∈ A 0 , q − 1 ( X , p X ( log D ) ⊗ L ∗ ) 3.1 Hodge decomposition: conic version In this subsection, we will state the Hodge decomposition for metric with conic sin- gularities, following [11, § 2.2 & 2.3]. The setting of this subsection is as follows. Let X be a compact Kähler manifold with dimension n , and let D = ∑ s i = 1 D i be a simple normal crossing divisor on X Let L be a holomorphic line bundle over X admitting a singular metric h := h L = e − φ L which has logarithmic poles , i.e., its local weight can be written as φ L = s ∑ i = 1 q i log | z i | 2 + φ L , 0 , ( ♥ ) where q i ∈ Q , z i = 0 represent the local equations of components of D and φ L , 0 is a smooth function. Note that the q i are not necessarily positive. The condition ( ♥ ) implies the corresponding curvature current is given by i h ( L ) = 2 π i s ∑ i = 1 q i [ D i ] + θ L , (3.2) 123 68 Page 14 of 65 R. Zhang where θ L is a smooth function on X . Choose a positive integer m such that for each q i ∈ Q \ Z , mq i ∈ Z and q i − 1 m = q i both hold true. ( ♠ ) One then denotes by ω c a metric (on X ◦ = X \ Supp D ) with conic singularities along the Q -divisor D m := s ∑ i = 1 ( 1 − 1 m ) D i By this we mean that if z 1 · · · z s = 0 is the local equation of the divisor D , then ω c s ∑ j = 1 idz j ∧ d ̄ z j | z j | 2 − 2 m + n ∑ j = s + 1 idz j ∧ d ̄ z j =: ω model , (3.3) that is to say, ω c is quasi-isometric with ω model , i.e., C − 1 · ω model ≤ ω c ≤ C · ω model for some constant C > 0. Notice that ω c is a closed positive ( 1 , 1 ) -current (smooth away from the support of D ) on X . For the existence and the explicit constructions of such metric, refer to e.g. [15, Proposition 2.1]. Now let ( V i ; z 1 i , . . . , z n i ) i ∈ I be a finite cover with coordinate charts such that z 1 i · · · z s i = 0 is the local equation of the divisor D when restricted to the set V i . We next consider the local ramified maps π i : U i → V i , π i (w 1 i , . . . , w n i ) := ((w 1 i ) m , . . . , (w s i ) m , w s + 1 i , . . . , w n i ). It defines the orbifold structure corresponding to ( X , D m ) Then, introducing the following definition is natural in our context, as it accounts for the singularities of the metric h on L . This can be seen as a generalization of the usual definition of “orbifold differential forms”, see e.g. [63, § 5.4] and [8, § 2]. Definition 3.2 ([11, Definition 2.14]). Let φ be a smooth form of ( p , q ) -type with values in L defined on the open set X ◦ We say that φ is smooth in conic sense if the quotient of the local inverse images ̃ φ i := 1 w qm i π ∗ i (φ | V i ) (3.4) 123 Logarithmic ∂∂ -lemma and several geometric applications... Page 15 of 65 68 admits a smooth extension to U i . Here in (3.4) we are using the notation w qm i := (w 1 i ) q 1 m · · · (w s i ) q s m in order to simplify the writing. Remark 3.3 Obviously, the notion of “smooth in conic sense” from the above definition can be applied to general (not necessarily compact) complex manifolds. The following proposition plays an important role, as it builds a correspondence between the intrinsic differential operators and the local ones associated to the data (ω c , h ) Proposition 3.4 ([11, Proposition 2.17]) . Let φ be an L-valued ( p , q ) -form, smooth in conic sense. Then its “natural” derivatives D ′ h , D ′∗ h , ∂, ∂ ∗ , are also smooth in conic sense. Furthermore, one has ( a ) sup X ◦ | φ | h ,ω c < ∞ , i.e., forms which are smooth in conic sense are bounded. ( b ) The following equalities hold true π ∗ i ( D ′ φ) = w qm i D ′ ̃ φ i , π ∗ i (∂φ) = w qm i ∂ ̃ φ i ; (3.5) together with π ∗ i ( D ′∗ φ) = w qm i D ′∗ ̃ φ i , π ∗ i (∂ ∗ φ) = w qm i ∂ ∗ ̃ φ i Here the notation D ′ on the left-hand side (resp. the right-hand side) of the first equality in (3.5) refers to D ′ h (resp. is defined by D ′ ς := ∂ς − ∂(φ L , 0 ◦ π i ) ∧ ς ). We recall that φ L = ∑ s i = 1 q i log | z i | 2 + φ L , 0 ( c ) Let ′′ := [ ∂, ∂ ∗ ] be the Laplace operator with respect to (ω c , h ). Then one has π ∗ i ′′ φ = w qm i · ′′ sm ̃ φ i , (3.6) where ′′ sm is the Laplace operator for the local, non singular setting (π ∗ i ω c , φ L , 0 ◦ π i ) As a result, we know that forms which are smooth in conic sense behave well by integrations by parts. Proposition 3.5 ([11, Corollary 2.19]) . Let α and β be an L-valued ( p , q ) -form and an L ∗ -valued ( n − p − 1 , n − q ) -form, respectively, both of which are smooth in conic sense. Then the usual integration by parts formula holds true ∫ X D ′ h α ∧ β = ( − 1 ) p + q + 1 ∫ X α ∧ D ′ h ∗ β. Furthermore, one has the following regularity theorem. 123 68 Page 16 of 65 R. Zhang Proposition 3.6 ([11, Corollary 2.20]) . Suppose that ζ is an L-valued L 2 form on X such that ′′ (ζ ) = φ holds in the sense of currents (see Definition 4.1) on X for some φ, smooth in conic sense. Then ζ is also smooth in conic sense. In particular, any L-valued L 2 form which is ′′ -harmonic is smooth in conic sense. Setting ′ := [ D ′ h , D ′∗ h ] , 4 one gets the conic version of Bochner–Kodaira–Nakano formula. Proposition 3.7 ([11, Proposition 2.21]) . Let φ be an L-valued ( p , q ) -form, which is smooth in conic sense. Then the equality ′′ φ = ′ φ + [ θ L , c ] φ (3.7) holds pointwise on X ◦ (for the notation θ L see (3.2)), where c is the adjoint of the Lefschetz operator L c := ω c ∧• . Furthermore, we have the following Bochner formula ∫ X | ∂φ | 2 e − φ L d V ω c + ∫ X | ∂ ∗ φ | 2 e − φ L d V ω c = ∫ X | D ′ h φ | 2 e − φ L d V ω c + ∫ X | D ′∗ h φ | 2 e − φ L d V ω c + ∫ X 〈[ θ L , c ] φ, φ 〉 e − φ L d V ω c (3.8) Indeed, the special choice of the curvature of the line bundle L will broaden the range of validity for (3.7). We will use the following useful proposition later. Proposition 3.8 Assume further that q i ≥ 0 for every i in ( ♥ ). Let φ be an L-valued ( p , q ) -form, smooth in conic sense. Then the equality ′′ φ = ′ φ + [ θ L , c ] φ holds pointwise on the whole of X . Proof On each local ramified cover π i : U i → V i , the relationship (3.6), combined with the usual Bochner equality ′′ sm = ′ sm + [ i φ L , 0 ◦ π i , π ∗ i ω c ] , where π ∗ i ω c is the adjoint of the Lefschetz operator L π ∗ i ω c := π ∗ i ω c ∧ • , implies that, on U i , we have the following as smooth forms (noting that w qm i does not introduce poles along the divisors since q i ≥ 0 for every i ): π ∗ i ′′ φ = w qm i · ′′ sm ̃ φ i = w qm i · ( ′ sm ̃ φ i + [ i φ L , 0 ◦ π i , π ∗ i ω c ] ̃ φ i ) = π ∗ i ( ′ φ + [ θ L , c ] φ). This leads to the desired result. 4 We leave out the subscript here for simplicity, as long as it does not cause confusion. 123 Logarithmic ∂∂ -lemma and several geometric applications... Page 17 of 65 68 For certain bidegree types, we obtain the following straightforward corollary, which has already been established in [11, Corollary 2.23]. This result will play a role later in establishing the closedness of twisted logarithmic forms of bidegree ( p , 0 ) , see Theorem 5.1. Corollary 3.9 Assume further that q i ≥ 0 for every i and φ L , 0 is a smooth plurisub- harmonic function in ( ♥ ). Let h be a ′′ -harmonic ( n − p , n ) -form with values in L. Then D ′ h h = D ′∗ h h = 0 Similarly to the smooth case, one defines the Hodge operators ∗ and in our setting, namely, given an L -valued ( p , q ) -form t , there exists a unique L ∗ -valued ( n − p , n − q ) - form, denoted by t , such that for any L -valued ( p , q ) -form s , we have 〈 s , t 〉 h d V ω c = s ∧ t , (3.9) where s ∧ t is calculated via the natural pairing L ⊗ L ∗ → C . One then can derive: Proposition 3.10 ([11, Propositions 2.24 & 2.25]) ( 1 ) Let t be an L-valued ( p , q ) -form, smooth in conic sense. Then t is an L ∗ -valued ( n − p , n − q ) -form, also smooth in conic sense (with respect to ( h ∗ , ω c ) ). ( 2 ) Let t be a ′′ -harmonic form with values in L. Then t is also a ′′ -harmonic form with values in L ∗ Cao–P ̆ aun also obtained the conic version of Gårding and Sobolev inequalities together with the Rellich embedding theorem . As a consequence, they finally got the “conic Hodge decomposition” as follows. Theorem 3.11 ([11, Theorem 2.28]) . Let ( L , h ) be a line bundle on X endowed with a metric h such that the requirements ( ♥ ) and ( ♠ ) are satisfied. Let ω c be a Kähler metric with conic singularities as in (3.3). Then we have the following Hodge decomposition A p , q co ( X , L ) = Ker ′′ h ⊕ Im ′′ h , (3.10) and L 2 p , q ( X , L ) = Ker ′′ h ⊕ Im ∂ ⊕ Im ∂ ∗ , where A p , q co ( X , L ) (resp. L 2 p , q ( X , L ) ) is the space of L-valued ( p , q ) -forms, which are smooth in conic sense (resp. in the L 2 sense) with respect to ( h , ω c ). One can also easily check that A p , q co ( X , L ) ⊂ L 2 p , q ( X , L ) 3.2 Proof of Theorem 3.1 We first give the following simple but important observation: Proposition 3.12 Let α ∈ A 0 , q ( X , p X ( log D ) ⊗ L ∗ ). Suppose that q i > 0 (not nec- essarily equal to or less than 1) for every i in ( ♥ ). Then α is smooth in conic sense. Proof This conclusion is based on a local computation. Without loss of generality, we only consider the one-variable case, where w qm · d (w m ) w m is smooth. It’s worth noting that here qm is a positive integer. 123 68 Page 18 of 65 R. Zhang Remark 3.13 Noteworthy to mention that the converse of Proposition 3.12 is in general false when the antiholomorphic degree q ≥ 1. For example, let X be the unit disc with coordinate z , D be a simple normal crossing divisor on it defined by the equation z = 0, and L be the trivial bundle over X , endowed with the metric φ L = 1 2 log | z | In this case, the ( L ∗ -value