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Sections of Hodge bundles II: Deformation of $(p,p)$-classes and applications to K\"ahler geometry

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Pith's one-line read The paper claims that the deformation of (p,p)-classes and of Kähler cones in a family of compact Kähler manifolds is fully controlled by the Beltrami differential on the central fiber through explicit Hodge-bundle sections, yielding upper

desk verdict Fresh and promising framework for deforming (p,p)-classes, but the main cone-containment proof rests on a type-comparison that doesn't survive inspection. read the letter →

arxiv 2602.13951 v2 pith:KSMCORG5 submitted 2026-02-15 math.AG

Kefeng Liu , Yang Shen This is my paper · ORCID
classification math.AG MSC 14D0732G0532Q1514C30
keywords BeltramidifferentialHodgebundlesKählercone(pp)-classesdeformationtheorylocusalgebraicapproximationvariationalconjecture
topics The Hodge Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to show that, for any family of compact Kähler manifolds over an analytic base, the Kähler cones of nearby fibers can be reconstructed from the central fiber alone: one adds to each Kähler class on the central fiber explicit correction terms built from harmonic projections and the Beltrami differential of the deformation. This produces a Hodge map whose values are (p,p)-classes on nearby fibers, and ∇^{1,1}-flat extensions of the Kähler cone that are always contained in the true Kähler cone, equaling it except on a countable union of analytic loci where analytic cycles fail to extend. If correct, the construction gives a uniform mechanism for Kähler stability, valid for obstructed deformations and over large regions of the base, for algebraic approximation criteria for (p,p)-classes, and for an intrinsic description of Hodge loci. The paper matters because it replaces first-order, unobstructed reasoning with explicit all-orders formulas in the Beltrami differential.

What carries the argument

The central object is the Beltrami differential φ(t) of the deformation—an (0,1)-form with values in the holomorphic tangent bundle that records how the complex structure twists—together with the contraction exponential e^{i_φ} = Σ (1/k!) i_φ^k and the harmonic-theoretic operator (I+T i_φ)^{-1} built from T = ∂*G∂ on the central fiber. The key identity (22) equates the period-matrix blocks Φ^{(p,p+k)}(t) with the harmonic projections of (1/k!) i_φ^k (I+T i_φ)^{-1} eη^{(p)}; this identity is what turns abstract period variation into explicit sections of Hodge bundles and makes the Hodge map computable from central-fiber data alone.

What would settle it

For a concrete family with computable periods, such as a two-parameter deformation of a complex torus, pick a harmonic (n-p,p)-form η and compute the period block Φ^{(p,p+1)}(t) by classical period theory; then compare its linear coefficient at t = 0 with H(i_{φ_1} η), where φ_1 = Σ θ_i t_i is the first-order Beltrami term. Any mismatch falsifies identity (22), and with it the Hodge map and cone-extension claims.

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Extended reading notes

Core claim

The central claim is that the deformation of (p,p)-classes along a family of compact Kähler manifolds is governed by explicit sections of Hodge bundles of the form H(e^{i_φ(t)}(I+T i_φ(t))^{-1}eη), where φ(t) is the Beltrami differential realizing the nearby complex structures, T = ∂*G∂ is the Green-operator contraction, and H is harmonic projection on the central fiber. These sections coincide with the period-matrix blocks and give, via an implicit-function argument, a real-analytic Hodge map H(σ,t) that sends any Kähler class σ on the central fiber to a (1,1)-class on X_t with a positive definite representative. From this the authors derive upper semicontinuity of Kähler cones, equality of

Load-bearing premise

The load-bearing premise is identity (22), imported without proof from the authors' companion preprint: the period-matrix blocks equal the harmonic projections of i_φ^k (I+T i_φ)^{-1} applied to harmonic representatives, and if this equality fails, the Hodge map, the Kähler-cone extensions, and every subsequent application lose their foundation.

Editorial extensions

If this is right

  • Kähler cones are upper semicontinuous under parallel transport of (1,1)-classes, with explicit positive representatives for every class in the extension.
  • Away from a countable union of analytic subsets—where some analytic cycle fails to deform—the ∇^{1,1}-flat extension equals the entire Kähler cone of the nearby fiber.
  • All nearby fibers remain Kähler, and the extension exists, on the whole region of the base where the Beltrami differential has operator norm below an explicit constant, with no unobstructedness assumption.
  • Strong algebraic approximation follows from an openness condition on a single higher-order Beltrami map; for (p,p)-classes, the full chain of contractions with φ, not just the first-order term, is the right criterion.
  • The Hodge locus of a rational (p,p)-class is described intrinsically by the vanishing of H(i_φ(t)(I+T i_φ(t))^{-1}eσ), and the variational Hodge conjecture for a smooth subvariety is equivalent to a normal-bundle obstruction statement.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the foundational identity withstands scrutiny, the framework converts period-map computations on nearby fibers into central-fiber harmonic analysis, making Kähler and Hodge-theoretic questions potentially accessible to explicit computation in examples such as tori or Calabi-Yau families where Beltrami differentials can be written down.
  • The all-orders formulation suggests the classical obstruction to algebraic approximation for (p,p)-classes with p ≥ 2—failure of the first-order density criterion—may be overcome by higher-order terms; a natural next step is to find concrete classes satisfying the new openness condition but not the old one.
  • The large-scale stability result hints at a new route to global Kähler rigidity: families whose Beltrami pseudo-distance remains below the threshold c0 can be shown Kähler without elliptic-operator regularity arguments, potentially yielding new proofs that degenerate central fibers in such families are Kähler.
  • The Hodge-locus formula may make the variational Hodge conjecture computationally approachable: one only needs to compare the vanishing locus of the Beltrami normal-bundle obstruction with the vanishing locus of H(i_φ eσ_Z).
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper develops an explicit 'Hodge map' parametrizing nearby (p,p)-classes on deformations of a compact Kähler manifold, using period-matrix blocks and Beltrami differentials from the authors' companion preprint [20]. It defines ∇^{1,1}-flat extensions of Kähler cones, claims upper semicontinuity (K^{∇1,1}_{t0,t} ⊂ K_t) with explicit positive representatives, and a large-scale Kähler stability theorem. It further claims generalizations of Green's density criterion, approximation of real (p,p)-classes by Hodge classes, and a Beltrami-differential criterion for the variational Hodge conjecture. The main theorems are stated for possibly singular Kuranishi bases and obstructed deformations.

Significance. If the central constructions and theorems were correct, the paper would provide a substantial new tool: explicit, higher-order Beltrami expressions for the deformation of (p,p)-classes and Kähler cones, with applications to algebraic approximation and the variational Hodge conjecture. The explicit positivity formula in Proposition 0.2 and the uniform large-scale statement in Theorem 4.2 would go beyond Demailly–Paun's results. However, the paper's key technical steps are not established: the foundational identity (22) is imported from [20], and the proof of the main cone-containment theorem contains a type-comparison gap that appears to invalidate the argument. The later sections repeatedly replace precise local equivalence by 'close to' or '≈' arguments, which are not justified for openness or zero-locus statements. Thus the significance is currently conditional on substantial repair.

major comments (4)
  1. [Theorem 3.4, equations (49)–(52)] The proof of the central containment K^{∇1,1}_{t0,t} ⊂ K_t derives (52) by 'comparison of types' between the ω-harmonic representative (49) and its expansion in the X_t-coframe (50)–(51). This comparison is invalid: the forms eη_ω^(0) and eη_ω^(2) are harmonic with respect to ω on the central fiber, but in the X_t-coframe (dz^i + φ dz̄^i) ∧ (dz̄^j + φ̄ dz^j) they acquire nontrivial (1,1)-components such as φ⌟eη_ω^(0). The displayed (1,1)-component of eH^ω is therefore not simply ω; it contains additional terms from α^(0) and its conjugate. No argument shows these terms vanish or are absorbed into g_{ij}(z,t). Consequently (52) is an assumption, not a consequence, and the subsequent positivity argument does not prove that H(σ,t) is a positive (1,1)-form on X_t. This gap undermines Theorem 0.3(1), Proposition 0.2, and Theorem 4.2.
  2. [Theorem 1.2, identity (22)] The equivalence of the period-matrix sections (14) and the Beltrami-defined sections (21) is stated as Theorem 1.2 with 'by comparing constant terms', but no proof is given; the identity (22) is imported from the companion preprint [20]. This identity is used throughout: it defines the quasi-period maps in (41), (56), (74), underlies the Hodge map equations (31)–(32), the Hodge locus formula (87), and the variational Hodge criterion. If (22) fails, the period-matrix blocks no longer describe harmonic projections of iφ^k(I+T iφ)^{-1} eη, and all subsequent statements lose their foundation. The paper must either prove (22) or state it with a precise theorem and proof in [20] that is accessible to the reader.
  3. [Theorem 5.3, proof around (69)] The proof reduces the openness of the Hodge map to the openness of α^(0)(α^0_(1),·): B→H^{0,2}, asserting that because Φ^{0,2} − Φ^{0,1}Φ^{1,2} = o(Φ^{1,2}), the openness of the former is 'equivalent' to the openness of α^0_(1)Φ^{1,2}. This equivalence is not justified: openness of a map is not invariant under addition of a term that is o of the leading term, without uniform control. Similarly, the step replacing H(iφ(·)(I+T iφ(·))^{-1}ω0) by H(iφ(·)ω0) uses only that the operator norm of (I+T iφ)^{-1} − I tends to 0; small perturbations do not in general preserve openness. The same '≈ implies equivalence' pattern recurs in Theorem 6.3 and Theorem 7.5, where it is load-bearing for the Hodge-locus identification.
  4. [Theorem 7.5, proof of (91)] The implication (91) is derived from Theorem 7.2 and the statement that H(iφ(t)(I+T iφ(t))^{-1}eσ_Z) 'is close to' H(iφ(t)eσ_Z) for small t, so the vanishing of one is equivalent to the vanishing of the other. This is not valid: closeness of functions does not imply equality of their zero loci. The proof needs an exact identity or a precise argument that the zero set is unchanged under the operator (I+T iφ)^{-1}. Without it, the necessary-and-sufficient criterion for the variational Hodge conjecture is unproven. The citation to [6] for the obstruction term H_{N_{Z|X}}(φ(t)|_{N_{Z|X}}) is also vague; the precise definition and deformation-theoretic statement should be included.
minor comments (4)
  1. [Throughout] The notation K^{∇1,1}_{t0,t} is defined only in Definition 0.1 after being used in the abstract; please reorder or add a forward reference.
  2. [Section 4, Theorem 4.2] The constant c0 = min(c1,c2) is not explicit; it depends on the choice of finite cover (53) and on the implicit function theorem radius. The paper should clarify whether c0 is uniform in the initial Kähler form or only in the Beltrami differential.
  3. [References] Reference [24] is listed as 'Rao, Wan, and Zhao' with an apparent typo in the title; please check 'Nagoya Mathematical Journal, 246'. Also [20] is a companion preprint and should include a precise statement of the results used here.
  4. [Equation (18)] The supremum norm ∥φ∥_E is defined via local charts but the maximum over the cover requires a choice of refinements; the dependence on this choice should be stated explicitly, even if the norm is equivalent to ∥φ∥_ω.

Circularity Check

2 steps flagged · score 4.0 of 10

Moderate circularity: the key period-map/Beltrami identity is imported from the authors' own [20], and the VHC 'criterion' restates the defining inclusion.

  1. self citation load bearing [Section 1, Theorem 1.2, Eq. (22); used throughout Sections 2-7]
    "In this section, we recall the global construction of Hodge bundles in [20]... By comparing the constant terms of the expansions of Ω^{(p)}(t) in (14) and eΩ^{(p)}(t) in (21), we have the following theorem... 1/k! H(i^k_φ(t)(I+T iφ(t))^{-1}eη^{(p)}) = Φ^{(p,p+k)}(t)·eη^{(p+k)}."

    This equality is the load-bearing bridge between the period-matrix blocks Φ and the Beltrami/harmonic-projection expression i_φ^k(I+T i_φ)^{-1}eη. The paper labels Section 1 as a recollection from the authors' own companion preprint [20] and does not carry out the proof here. The Hodge map, quasi-period maps, positivity arguments, and Hodge-locus formulas all inherit this unproved same-author identity. If (22) were not established independently, the subsequent 'explicit' formulas are unsupported.

  2. self definitional [Section 7, Definition 7.4 and Theorem 7.5, Eq. (91)]
    "Clearly, Def(X, Z)⊂B^p_σZ. Thus, the variational Hodge conjecture reduces to showing that Z deforms unobstructedly at every point of B^p_σZ. ... Then the variational Hodge conjecture holds for X at Z if and only if the implication H_{N_{Z|X}}(φ(t)|_{N_{Z|X}})≠0 =⇒ H(iφ(t)eσ_Z)≠0 holds for every t∈B."

    By Definition 7.4, VHC at Z is exactly the equality B^p_σZ = Def(X,Z); the text notes Def(X,Z)⊂B^p_σZ is trivial, so all content lies in B^p_σZ⊂Def(X,Z). Theorem 7.2 rewrites B^p_σZ via H(iφ(I+T iφ)^{-1}eσ)=0, and [6] expresses the obstruction to deforming Z as H_N(φ|_N). Substituting these descriptions turns the needed inclusion into (91). Thus the 'criterion' is the defining inclusion in different notation, not an independent derivation of VHC.

full rationale

The paper's main Kähler-cone results are not pure tautologies: Theorem 3.4 contains an independent positivity argument and relies on the external Demailly-Paun characterization [9]. The Section 5/6 openness criteria are sufficient conditions, not reverse implications, so I do not count them as circular. However, the central machinery depends on equation (22), which is explicitly recalled from the same-authors preprint [20] rather than re-derived, and the VHC theorem is a reformulation of the defining inclusion. Because the core cone theorems retain independent content, the overall circularity burden is moderate rather than total.

Assumptions & free parameters 2 free parameters · 5 assumptions · 3 invented entities

The paper's central claims rest on the Kuranishi deformation framework, on the companion-paper identity connecting period matrices to Beltrami differentials, and on several unproved 'closeness implies equivalence' steps. There are no fitted data, but the radius of the polydisk and the constant c0 are implicit existence parameters. The main invented objects are the Hodge map, the ∇^{1,1}-flat extension, and the Beltrami pseudo-distance; all are defined internally and not supported by independent evidence.

free parameters (2)
  • sufficiently small radius ε of the Kuranishi polydisk = arbitrarily small; not quantified
    All theorems hold only for ε small enough; statements depend on this implicit choice (e.g., Theorem 2.4, Theorem 3.4).
  • constant c0 = min(c1,c2) in Theorem 4.2 = existence only
    Chosen so that determinant (60) and Jacobian (62) are non-degenerate and ||φ||<1; no quantitative value is given.
assumptions (5)
  • domain assumption Kuranishi's theorem: every nearby complex structure is X_t = X_{φ(t)} with φ holomorphic in t and H[φ,φ]=0
    Basis for parameterizing deformations; cited from Morrow–Kodaira [23], Section 3, Theorem 3.1.
  • ad hoc to paper Identity (22): Φ^{(p,p+k)}(t)·η^{(p+k)} = (1/k!) H(i^k_φ (I+T i_φ)^{-1} eη^{(p)})
    Bridge between period matrices and Beltrami differentials; proof deferred to companion preprint [20], Section 1, Theorem 1.2.
  • ad hoc to paper H(iφ(I+T iφ)^{-1}eσ) is 'close to' H(iφeσ) and has the same behavior for zero loci and openness on small B
    Used in Theorem 7.5 and Section 6 proofs; closeness of functions does not imply equality of zero loci, so this is an unproved assumption.
  • standard math Griffiths transversality for the quasi-period map
    Used in Lemma 1.4 and Proposition 3.2; relies on Griffiths [13].
  • ad hoc to paper The set of classes ζ for which B→H^{0,2} is not open is an algebraic subset of codimension ≥1
    Assumed in the proof of Theorem 5.3 without proof; openness is not a Zariski condition, so this is likely false in general.
invented entities (3)
  • Hodge map H
    purpose: Real-analytic parametrization of nearby (p,p)-classes from central-fiber classes
    Defined in Theorem 2.4; its existence is the theorem, not an independently evidenced entity.
  • ∇^{1,1}-flat extension of the Kähler cone
    purpose: Relates Kähler cones across deformation
    Defined in Definition 0.1; containment results are the main theorems, not external facts.
  • Beltrami pseudo-distance d_B(t0,t)
    purpose: Measures distance of complex structures in Theorem 4.2
    Defined in Definition 4.1; used to define large-scale stability regions.

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Pith. "Pith review of Sections of Hodge bundles II: Deformation of $(p,p)$-classes and applications to K\"ahler geometry." pith.science (2026). https://pith.science/paper/KSMCORG5

@misc{pith2026260213951,
  author       = {Pith},
  title        = {Pith review of: Sections of Hodge bundles II: Deformation of $(p,p)$-classes and applications to K\"ahler geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KSMCORG5}},
  note         = {Machine review of arXiv:2602.13951}
}
abstract

Let $(X,\omega_0)$ be a compact K\"ahler manifold and $\mathcal X\to B$ its Kuranishi family, where $B$ may be singular and $\dim_{\C}B\ge1$. Using explicit sections of Hodge bundles, we define an intrinsic period map and a Hodge map parametrizing nearby $(p,p)$-classes. For deformations over irreducible analytic bases, we introduce two flat extensions of K\"ahler cones defined by the reference and moving Hodge connections. The extension associated with the reference connection admits explicit positive representatives and yields uniform upper semicontinuity, while that associated with the moving connection identifies the K\"ahler cones away from a countable union of proper analytic subsets and admits an explicit expression in terms of the period map and the Beltrami differential. These constructions provide a description of K\"ahler cones through analytic cycles and yield both local and large-scale K\"ahler stability without assuming unobstructedness. As further applications, we generalize Green's density criterion to strong algebraic approximation and to the approximation of real $(p,p)$-forms. We also obtain an intrinsic analytic description of Hodge loci, leading to a Beltrami-differential criterion for the variational Hodge conjecture.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Higher order isomonodromic deformation of Higgs bundles and a characterization of the non-abelian Noether-Lefschetz locus

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    The non-abelian Noether-Lefschetz locus equals the maximal complex analytic subvariety on which the isomonodromic deformation of Higgs bundles is holomorphic.

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Works this paper leans on

26 extracted references · 1 linked inside Pith · cited by 1 Pith paper

  1. [20]

    Liu and Y

    K. Liu and Y. Shen, Sections of Hodge bundles I: Global theory and applications to period maps, preprint, (2026)

    2026

  2. [6]

    Clemens, Geometry of formal Kuranishi theory,Adv

    H. Clemens, Geometry of formal Kuranishi theory,Adv. Math.,198(2005), pp. 311–365

    2005

  3. [1]

    D. Barlet, Espace analytique r´ eduit des cycles analytiques complexes compacts d’un espace analy- tique complexe de dimension finie,Fonctions de Plusieurs Variables Complexes, II (S´ em. Franois Norguet, 1974–1975), Lecture Notes in Math.,482, Springer-Verlag, New York (1975), pp. 1–158

    1974

  4. [2]

    Bloch, Semi-regularity and de Rham cohomology,Invent

    S. Bloch, Semi-regularity and de Rham cohomology,Invent. Math.,17(1972), pp. 51–66

    1972

  5. [3]

    Buchdahl, Algebraic deformations of compact K¨ ahler surfaces,Mathematische Zeitschrift,253 (2006), pp

    N. Buchdahl, Algebraic deformations of compact K¨ ahler surfaces,Mathematische Zeitschrift,253 (2006), pp. 453–459

    2006

  6. [4]

    Buchdahl, Algebraic deformations of compact K¨ ahler surfaces II,Mathematische Zeitschrift, 258(2008), pp

    N. Buchdahl, Algebraic deformations of compact K¨ ahler surfaces II,Mathematische Zeitschrift, 258(2008), pp. 493–498

    2008

  7. [5]

    Cao, On the approximation of K¨ ahler manifolds by algebraic varieties,Math

    J. Cao, On the approximation of K¨ ahler manifolds by algebraic varieties,Math. Ann.,363(1–2) (2015), pp. 393–422

    2015

  8. [7]

    Dan and I

    A. Dan and I. Kaur, Semi-regular varieties and variational Hodge conjecture,C. R. Acad. Sci. Paris, Ser. I,354(2016), pp. 297–300

    2016

Show all 26 references
  1. [8]

    Debarre, Periods and moduli,Current Developments in Algebraic Geometry,MSRI Publica- tions,59(2011), pp

    O. Debarre, Periods and moduli,Current Developments in Algebraic Geometry,MSRI Publica- tions,59(2011), pp. 65–84. 46 Sections of Hodge bundles II

    2011

  2. [9]

    Demailly and M

    J.-P. Demailly and M. Paun, Numerical characterization of the K¨ ahler cone of a compact K¨ ahler manifold,Ann. of Math.,159(2004), pp. 1247–1274

    2004

  3. [10]

    Graf, Algebraic approximation of K¨ ahler threefolds of Kodaira dimension zero,Math

    P. Graf, Algebraic approximation of K¨ ahler threefolds of Kodaira dimension zero,Math. Ann., (2017)

    2017

  4. [11]

    Griffiths, Periods of integrals on algebraic manifolds I,Amer

    P. Griffiths, Periods of integrals on algebraic manifolds I,Amer. J. Math.,90(1968), pp. 568–626

    1968

  5. [12]

    Griffiths, Periods of integrals on algebraic manifolds II,Amer

    P. Griffiths, Periods of integrals on algebraic manifolds II,Amer. J. Math.,90(1968), pp. 805– 865

    1968

  6. [13]

    Griffiths, On the periods of certain rational integrals: I and II,Annals of Mathematics,90 (1969), pp

    P. Griffiths, On the periods of certain rational integrals: I and II,Annals of Mathematics,90 (1969), pp. 460–495 and 496–541

    1969

  7. [14]

    Kiremidjian, Deformations of complex structures on certain noncompact manifolds,Ann

    G. Kiremidjian, Deformations of complex structures on certain noncompact manifolds,Ann. of Math.,98, No. 3 (1973), pp. 411–426

    1973

  8. [15]

    Kloosterman, Variational Hodge conjecture for complete intersections on hypersurfaces in projective space,Rend

    R. Kloosterman, Variational Hodge conjecture for complete intersections on hypersurfaces in projective space,Rend. Semin. Mat. Univ. Padova,148(2022), pp. 185–201

    2022

  9. [16]

    Kodaira, On compact analytic surfaces

    K. Kodaira, On compact analytic surfaces. II, III,Ann. of Math. (2),77(1963), pp. 563–626;78 (1963), pp. 1–40

    1963

  10. [17]

    Kodaira and D

    K. Kodaira and D. C. Spencer, On deformations of complex analytic structures, III,Annals of Mathematics,Second Series,71(1) (1960), pp. 43–76

    1960

  11. [18]

    Liu and S

    K. Liu and S. Rao, Remarks on the Cartan formula and its applications,Asian J. Math.,16 (2012), pp. 15–169

    2012

  12. [19]

    K. Liu, S. Rao and X. Yang, Quasi-isometry and deformations of Calabi-Yau manifolds.Invent. Math.,199, no. 2 (2015), pp. 423–453

    2015

  13. [21]

    Liu and Y

    K. Liu and Y. Shen, K¨ ahler rigidity for certain family of compact complex manifolds with trivial canonical bundles,preprint, (2026)

    2026

  14. [22]

    Liu and S

    K. Liu and S. Zhu, Solving equations with Hodge theory,arXiv:1803.01272, (2018)

    2018 arXiv

  15. [23]

    Morrow and K

    J. Morrow and K. Kodaira,Complex Manifolds, AMS Chelsea Publishing, Porvindence, RI, (2006), Reprint of the 1971 edition with errata

    2006

  16. [24]

    S. Rao, X. Wan, and Q. Zhao, Power series proofs for local stabilities of K¨ ahler and balanced structures with mild∂ ∂-lemma,Nagoya Mathematical Journal,246(2022), pp. 305–354

    2022

  17. [25]

    Siu, Every K3 Surface is K¨ ahler,Inventiones mathematicae,73(1983), pp

    Y.-T. Siu, Every K3 Surface is K¨ ahler,Inventiones mathematicae,73(1983), pp. 139–150

    1983

  18. [26]

    Voisin,Hodge theory and complex algebraic geometry II, Cambridge Universigy Press, New York, (2003)

    C. Voisin,Hodge theory and complex algebraic geometry II, Cambridge Universigy Press, New York, (2003). Mathematical Sciences Research Center, Chongqing University of Technology, Chongqing 400054, China; Department of Mathematics,University of California at Los Angeles, Los An...

    2003

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