Pith. sign in

REVIEW 5 major objections 3 minor 2 cited by

Spencer Differential Degeneration Theory and Its Applications in Algebraic Geometry

T0 review · 5 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Kernel condition turns Spencer differentials into exterior ones.

desk verdict The one correct result is a definition-level tautology, and the K3 application is vacuous because no parallelizable K3 surface exists. read the letter →

arxiv 2506.07410 v2 pith:CUDFXH3R submitted 2025-06-09 math.GM

Dongzhe Zheng This is my paper
classification math.GM MSC 14J2858A1458J10
keywords SpencerdifferentialdegenerationdeRhamcohomologycompatiblepairsmirrorsymmetryK3surfacesHodgeclassesprincipalbundles
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

This paper tries to show that a side condition on symmetric tensors, δ^λ_g(s)=0, collapses the Spencer differential to the ordinary exterior differential, joining two cohomology theories that normally live on different sides of gauge theory and algebraic geometry. If the collapse holds, degenerate Spencer cocycles become just closed differential forms tensored with kernel elements, and the paper builds a canonical projection from degenerate Spencer cohomology to de Rham cohomology. The paper also claims the collapse is mirror-stable under λ↦−λ, and applies the projection on K3 surfaces to argue that every (1,1)-Hodge class can be reached from degenerate Spencer cocycles. The reward, if correct, is a new constraint-geometric route to algebraic cycle problems.

What carries the argument

The load-bearing object is the Spencer prolongation operator δ^λ_g: Sym^k(g)→$Sym^{{k+1}}$(g), a graded derivation encoding how the constraint function λ twists symmetric tensors, together with its kernel K^k_g(λ)=ker δ^λ_g. The paper's degeneration theorem is the observation that the Spencer differential D^k(ω⊗s)=dω⊗s+(−1)^k ω⊗δ^λ_g(s) loses its second term exactly on Ω^k(M)⊗K^k_g(λ). That single identity carries the argument: it converts Spencer cocycles into de Rham cocycles, makes the projection to de Rham cohomology well-defined, and, through the sign-reversal $δ^{{−λ}}$_g = −δ^λ_g, yields mirror stability of the kernel.

What would settle it

A direct calculation of dim $K^{2}$_{su(2)}(λ) for nonzero λ from Definition 2.8, together with the Euler-characteristic obstruction to a parallelizable K3 surface, would settle whether Section 5 has a valid domain and a nonempty kernel; either computation would determine whether the paper's application claims hold.

Watch

Extended reading notes

Core claim

The core claim is Theorem 3.2: for any α=ω⊗s in D^k_{D,λ}, the Spencer differential satisfies D^k_{D,λ}(ω⊗s)=dω⊗s. This identifies degenerate Spencer cocycles with de Rham cocycles, and Proposition 3.5 characterizes them as Z^k_{dR}(M)⊗K^k_g(λ). Theorem 3.7 turns the projection ω⊗s↦ω into a map on cohomology, and Theorem 6.2 shows K^k_g(λ)=K^k_g(−λ), so the degeneration condition is mirror invariant. On K3 surfaces, Section 5 composes this projection with the Hodge projection to define Φ_{D,λ}: $Z^{2}$_deg(D,λ)→$H^{{1,1}}$(X,C), and the paper claims this map is surjective with image of dimension $h^{{1,1}}$=20.

Load-bearing premise

The load-bearing premise is that the base manifold is compact, orientable, and parallelizable while also being a K3 surface, but parallelizable compact manifolds have zero Euler characteristic and K3 surfaces have Euler characteristic 24, so no such manifold exists.

Editorial extensions

If this is right

  • Computing degenerate Spencer cocycles splits into computing de Rham cohomology and computing the kernel K^k_g(λ).
  • The projection gives a canonical map from degenerate Spencer cohomology to de Rham cohomology, so constraint-geometric invariants can be read in classical differential-topological terms.
  • The degeneration locus is mirror-stable: s lies in the kernel for λ exactly when it lies in the kernel for −λ.
  • On K3 surfaces, the composite map is claimed to hit every (1,1)-class, making degenerate Spencer cocycles a generating source for candidates of algebraic Hodge classes.
  • This provides a framework for approaching algebraic-cycle questions, with K3 serving as the test case for higher-dimensional Calabi-Yau manifolds.

Reading between the lines

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

  • The degeneration identity itself is an immediate consequence of the definition once δ^λ_g(s)=0; the paper's real empirical content is the dimension and structure of the kernel spaces K^k_g(λ), so the K3 conclusions should be read as depending entirely on those computations.
  • The K3 application as written has an empty domain if Assumption 2.1(1) is retained: a compact parallelizable manifold has zero Euler characteristic, while every K3 surface has Euler characteristic 24; extending the theory would require dropping parallelizability or changing the base manifold.
  • Surjectivity onto H^{1,1}(X,C) alone would not establish algebraicity; a class is algebraic only if it is integral, and the paper does not verify integrality of the projected classes, so the 'algebraicity conditions' claim is at best a program.
  • A testable next step is to compute K^2_{su(2)}(λ) for λ≠0: a zero kernel would trivialize the K3 construction, while a positive-dimensional kernel would make the fiber structure of the projection a concrete algebraic invariant.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 3 minor

Summary. The paper develops a 'Spencer differential degeneration theory' for the Spencer complexes attached to compatible pairs (D, λ) of principal-bundle constraint systems. It defines degenerate kernel spaces K^k_g(λ)=ker(δ^λ_g) and degenerate subspaces D^k_{D,λ}=Ω^k(M)⊗K^k_g(λ), observes that on these subspaces the Spencer differential reduces to the exterior differential, defines a projection from degenerate Spencer cocycles to de Rham cohomology, proves mirror invariance of the degeneration condition, and applies the framework to K3 surfaces with the claimed goal of systematically producing algebraic (1,1)-Hodge classes. The application depends on 'parallelizable K3 surfaces' and on numerical dimensions of the kernels K^k_{su(2)}(λ) that are asserted and deferred to the author's preprints.

Significance. If the advertised program worked, a canonical map from degenerate Spencer cocycles to algebraic Hodge classes on K3 surfaces would be a substantial bridge between constraint geometry and algebraic geometry. However, as it stands, the central simplification (Theorem 3.2) is an immediate restatement of the definition of the Spencer differential; the K3 application is set on an empty domain because no compact parallelizable manifold can be a K3 surface; and the crucial kernel dimensions are not proved in this paper. The mirror-stability argument in Section 6 is formally correct once the identity δ^{-λ}_g=-δ^λ_g is accepted, and the paper is clearly organized, but these strengths do not offset the load-bearing gaps. The result, if reduced to its sound core, is a one-line observation rather than a complete degeneration theory.

major comments (5)
  1. [Assumption 2.1(1), Section 5 (Construction 5.1, Proposition 5.2, Theorem 5.6)] Assumption 2.1(1) fixes the base manifold M to be connected, compact, orientable and parallelizable for the entire paper. Section 5 then applies the construction to 'parallelizable K3 surfaces'. Such a manifold does not exist: a parallelizable compact manifold admits a nowhere-vanishing vector field, so Poincaré-Hopf forces χ(M)=0, whereas every K3 surface has Euler characteristic 24 (c2=24). Consequently the domain of Construction 5.1 is empty, and the dimensions in Proposition 5.2 and the surjectivity claim in Theorem 5.6 are statements about a non-existent object. This is a load-bearing error, not a presentation issue.
  2. [Theorem 3.2, Definition 2.10] Theorem 3.2 is the definition of the Spencer differential restated. From Definition 2.10, the Spencer differential D(omega⊗s) equals d omega⊗s plus (-1)^k omega⊗delta(s); if s lies in the kernel of delta, the second term vanishes identically. The proof is a one-line substitution. The advertised 'bridge' between Spencer theory and de Rham theory therefore contains no new analytical content beyond the definition, and all later structural consequences are formal manipulations of the same identity.
  3. [Example 4.1, Example 3.1, Proposition 5.2, Lemma 2.9] The dimension dim K^1_{su(2)}(lambda)=1 for lambda different from zero is asserted in Example 4.1 with the proof deferred to [Zhe25c, Zhe25a]. This dimension is used in Example 3.1 and Proposition 5.2 to produce the K3 numbers dim Z^1_deg=0 and dim Z^2_deg=22·dim K^2_{su(2)}(lambda). Proposition 5.2 also uses dim K^0_{su(2)}(lambda)=1 without any proof. Moreover, the surjectivity of the composite map in Theorem 5.6 requires K^2_{su(2)}(lambda) to be nonzero, which is never proved. Separately, Lemma 2.9 asserts the nilpotency of delta and the mirror identity; the proof of Theorem 2.11 relies on this nilpotency, but the proof is deferred to self-citations. Thus the numerical output of the K3 computation and the Spencer complex property itself are not independently supported by the present manuscript.
  4. [Theorem 5.6, Definition 5.5, Abstract and Introduction] The claimed algebraicity application is not established. The composite map Phi_{D,lambda} sends Z^2_deg to H^{1,1}(X,C) through Z^2_dR, the quotient to H^2_dR(X,C), and the Hodge projection pi_{1,1}. No condition ensuring that the image lies in the integral lattice H^2(X,Z) is stated or verified anywhere in the paper. Lefschetz's (1,1)-theorem applies to integral (1,1) classes, but the paper never shows that any of the constructed classes are integral. Therefore the abstract's assertion that the framework can 'systematically identify (1,1)-Hodge classes satisfying algebraicity conditions' and the Introduction's claim that such classes 'are indeed algebraic' are unsupported.
  5. [Example 3.1 Eq. (2), Example 5.1 Eq. (3), Proposition 5.2] The dimension formulas are contradictory and conflate cocycles with cohomology classes. Example 3.1, Eq. (2) writes dim Z^2_deg = b_2(M)·dim K^2_{su(2)}(lambda)=22·d, treating Z^2_dR(M) as a finite-dimensional space. Example 5.1, Eq. (3) instead writes dim Z^2_deg ≈ ∞. The space of closed 2-forms on a manifold is infinite-dimensional; it is the de Rham cohomology H^2_dR that is finite-dimensional. This conflation affects the statement of Proposition 5.2, where 'dim H^k_deg = b_k(X)·dim K^k_{su(2)}(lambda)' is only plausible if H^k_deg is interpreted as a cohomology group, not as a space of cocycles.
minor comments (3)
  1. [Theorems 3.2, 5.2, Algorithm 1] Several cross-references are broken: Theorem 3.2's proof cites 'Definition??' instead of Definition 2.10, Example 3.1 and Proposition 5.2 refer to 'Example??' instead of Example 4.1, and Algorithm 1 refers to 'Theorem??' instead of Theorem 5.6. These should be corrected.
  2. [Definition 5.5 and Corollary 5.7] The composition in Definition 5.5 writes Z^2_dR(X) is embedded in H^2_dR(X), but a closed form is not naturally a cohomology class without passing to the quotient; the notation conflates cocycles and cohomology classes and should be replaced by the quotient map. Corollary 5.7 then says the image of Phi_{D,lambda} has dimension at most 20 while also saying it can reach all closed 2-forms, which is confusing for the same reason.
  3. [Theorems 2.17 and 2.19] The isomorphism symbols in Theorems 2.17 and 2.19 are garbled in the text and should be typeset as standard isomorphism arrows rather than appearing as partial glyphs.

Circularity Check

3 steps flagged · score 7.0 of 10

The Spencer–de Rham bridge reduces to the defining kernel condition, and the K3 application depends on the author's own unpublished preprints rather than on proofs in this paper.

  1. self definitional [Definition 3.1 and Theorem 3.2, Section 3.1]
    "Degenerate Kernel Space: ... K^k_g(λ):=ker(δ^λ_g: Sym^k(g)→Sym^{k+1}(g)) ... Theorem 3.2: For any degenerate Spencer element α=ω⊗s∈D^k_{D,λ}, ... D^k_{D,λ}(ω⊗s)=dω⊗s. Since α is a degenerate element, its symmetric tensor part s∈K^k_g(λ), which means δ^λ_g(s)=0."

    The theorem's proof is substitution into the defining formula D^k_{D,λ}(ω⊗s)=dω⊗s+(-1)^k ω⊗δ^λ_g(s). The degenerate subspace D^k is defined as Ω^k(M)⊗ker δ^λ_g, so δ^λ_g(s)=0 holds by construction. Hence the asserted 'degeneration' is exactly the defining formula evaluated on the kernel; there is no independent derivation of a bridge to de Rham cohomology, only a relabeling of the condition δ^λ_g(s)=0.

  2. self citation load bearing [Example 4.1, Section 4.2; used in Example 3.1, Proposition 5.2 and Example 5.1]
    "If λ≠0, the kernel space is precisely the one-dimensional subspace ... dimK^1_{su(2)}(λ)= (3 if λ=0, 1 if λ≠0). The detailed derivation of this result from the constructive definition is provided in [Zhe25c, Zhe25a]. This dimension calculation is fundamental for the applications on K3 surfaces discussed in the subsequent chapters."

    The numerical content of the K3 application—dim H^1_deg = b_1·1 = 0 and the placement of dim H^2_deg = 22·dim K^2_{su(2)}(λ)—depends on the asserted dimension dim K^1_{su(2)}(λ)=1 and on K^2 being nonzero. That assertion is not proved here; it is deferred to two unpublished preprints by the same author. The same self-citation chain supplies the nilpotency of δ (Lemma 2.9) and the Spencer-Hodge decomposition (Theorem 2.16), so the paper's central application is forced by inputs borrowed from the author's own overlapping preprints rather than by a self-contained argument.

1 more flagged steps
  1. renaming known result [Theorem 5.6, Remark 5.1, Section 5.3; cf. abstract and Introduction]
    "Φ_{D,λ} is a surjection, whose image space is the entire H^{1,1}(X,C). Therefore ... h^{1,1}(X)=20. ... we prove that (1,1)-Hodge classes constructed by Spencer theory that satisfy mirror stability and other algebraic constraints are indeed algebraic, which is consistent with the results of Lefschetz's (1,1)-theorem."

    Surjectivity onto H^{1,1}(X,C) is assembled from the definitional isomorphism Z^2_deg≅Z^2_dR⊗K^2 (Proposition 3.5), the quotient map to H^2_dR, and the standard Hodge projection. No integrality condition is verified, so the claimed 'algebraicity' cannot come from Spencer theory; it is imported from the classical Lefschetz (1,1)-theorem. The new framework therefore repackages standard Hodge projection under the name 'degenerate Spencer cocycles' rather than deriving a new algebraicity criterion.

full rationale

Two circular steps are load-bearing. First, Theorem 3.2, advertised as the key mechanism connecting Spencer and de Rham theory, is the defining identity D(ω⊗s)=dω⊗s+(-1)^kω⊗δ(s) evaluated on the kernel of δ, so the bridge is definitional. Second, the K3 application's dimension inputs come from the author's own unpublished preprints [Zhe25c, Zhe25a] rather than from proofs in this paper; Lemma 2.9, Theorem 2.16, and Example 4.1 all defer to this self-citation chain. The claimed algebraic detection is additionally a renaming of the standard Hodge projection plus Lefschetz (1,1), since integrality is never checked. Separately, the application is also vacuous as a geometric statement: the global assumption that M is parallelizable cannot hold for a K3 surface, which has Euler characteristic 24, while compact parallelizable manifolds have zero Euler characteristic. This is not a case of a self-contained paper with minor self-citation; the central claims reduce to definitions and to the author's own unproved inputs, so a score of 7 is warranted.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The paper's central claim rests on a definitional identity plus several unproved, self-cited assumptions. The most damaging is the false parallelizability of K3 surfaces, on which the entire application section depends.

free parameters (2)
  • dual constraint function λ = nonzero element of g* (for K3, su(2)*)
    The kernel K^k_g(λ)=ker δ^λ_g depends on λ; the paper uses only λ≠0 and leaves the actual value unspecified.
  • dim K^2_{su(2)}(λ) = d = unspecified, assumed ≥ 1
    The K3 degenerate cohomology dimension is 22·d, but d is never computed in the paper (Example 5.1).
assumptions (4)
  • ad hoc to paper The Spencer prolongation operator δ^λ_g is nilpotent of order two.
    Lemma 2.9: proof deferred to [Zhe25c, Zhe25a]; without it the Spencer complex property (D^2=0) fails.
  • ad hoc to paper For su(2) and λ≠0, dim K^1_{su(2)}(λ)=1.
    Example 4.1: the derivation is not given, only a citation to [Zhe25c, Zhe25a].
  • domain assumption The base manifold M is parallelizable.
    Assumption 2.1(1); this is false for K3 surfaces, which have Euler characteristic 24.
  • domain assumption A nonzero λ satisfying the modified Cartan equation exists on the SU(2)-bundle over the K3 surface.
    Construction 5.1 asserts existence without proof; needed for the compatible pair and the K3 application.
invented entities (1)
  • Degenerate Spencer cohomology H^k_deg(D,λ)
    purpose: Cohomology of Spencer cocycles whose symmetric tensor part lies in the kernel of δ^λ_g
    By Proposition 3.5 and Theorem 3.8 it is isomorphic to H^k_dR(M)⊗K^k_g(λ), so it carries no information beyond ordinary de Rham cohomology; no external falsifiable prediction is made.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Spencer Differential Degeneration Theory and Its Applications in Algebraic Geometry." pith.science (2026). https://pith.science/paper/CUDFXH3R

@misc{pith2026250607410,
  author       = {Pith},
  title        = {Pith review of: Spencer Differential Degeneration Theory and Its Applications in Algebraic Geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CUDFXH3R}},
  note         = {Machine review of arXiv:2506.07410}
}
abstract

Based on the compatible pair theory of principal bundle constraint systems, this paper discovers and establishes a complete Spencer differential degeneration theory. We prove that when symmetric tensors satisfy a $\lambda$-dependent kernel condition $\delta_{\mathfrak{g}}^{\lambda}(s)=0$, the Spencer differential degenerates to the standard exterior differential, thus establishing a precise bridge between the complex Spencer theory and the classical de Rham theory. One of the advances in this paper is the rigorous proof that this degeneration condition remains stable under mirror transformations, revealing the profound symmetry origins of this phenomenon. Based on these rigorous mathematical results, we construct a canonical mapping from degenerate Spencer cocycles to de Rham cohomology and elucidate its geometric meaning. Finally, we demonstrate the application potential of this theory in algebraic geometry, particularly on K3 surfaces, where we preliminarily verify that this framework can systematically identify (1,1)-Hodge classes satisfying algebraicity conditions. This work provides new perspectives and technical approaches for studying algebraic invariants using tools from constraint geometry.

Discussion (0). Sign in to comment.

Forward citations

Cited by 2 Pith papers

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

  1. The Rigidity of Constraint: A Spencer-Hodge Theoretic Approach to the Hodge Conjecture

    math.GM 2025-06 reject novelty 4.0 of 10

    The paper reduces the Hodge conjecture to premises that already assert the desired equality, and its K3 example rests on an invalid elliptic fibration claim.

  2. Extension Research of Principal Bundle Constraint System Theory on Ricci-flat K\"ahler Manifolds

    math.GM 2025-06 reject novelty 3.0 of 10

    The paper asserts that compatible-pair constraint theory is curvature-independent, but the proofs it gives contain a sign error and an unjustified Spencer differential construction.

Reference graph

Works this paper leans on

25 extracted references · 16 canonical work pages · cited by 2 Pith papers

  1. [1]

    Vladimir I. Arnold. Mathematical Methods of Classical Mechanics . Springer-Verlag, 1978

    1978

  2. [2]

    Atiyah and Isadore M

    Michael F. Atiyah and Isadore M. Singer. The index of elliptic operators on compact manifolds. Bulletin of the American Mathematical Society , 69(3):422--433, 1963

    1963

  3. [3]

    Michael F. Atiyah. Complex analytic connections in fibre bundles. Transactions of the American Mathematical Society , 85(1):181--207, 1957

    work page 1957

  4. [4]

    Géométrie des surfaces K3: modules et périodes , volume 126

    Arnaud Beauville. Géométrie des surfaces K3: modules et périodes , volume 126. 1985

    work page 1985

  5. [5]

    Barth, Klaus Hulek, Chris A

    Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven. Compact Complex Surfaces . Springer-Verlag, 2004

    work page 2004

  6. [6]

    Le théorème de riemann-roch

    Armand Borel and Jean-Pierre Serre. Le théorème de riemann-roch. Bulletin de la Société Mathématique de France , 86:97--136, 1958

    work page 1958

  7. [7]

    Characteristic classes of hermitian manifolds

    Shiing-Shen Chern and Andr \'e Weil. Characteristic classes of hermitian manifolds. American Journal of Mathematics , 81(4):725--757, 1959

    work page 1959

  8. [8]

    Sur l'analysis situs des variétés à n dimensions , volume 10

    Georges de Rham. Sur l'analysis situs des variétés à n dimensions , volume 10. 1931

    work page 1931

Show all 25 references
  1. [9]

    Variétés différentiables

    Georges de Rham. Variétés différentiables . Hermann, 1955

    1955

  2. [10]

    Principles of Algebraic Geometry

    Phillip Griffiths and Joseph Harris. Principles of Algebraic Geometry . Wiley, 1978

    1978

  3. [11]

    Éléments de géométrie algébrique ii

    Alexander Grothendieck. Éléments de géométrie algébrique ii. Publications Mathématiques de l'IHÉS , 8:5--222, 1961

    1961

  4. [12]

    William V. D. Hodge. The topological invariants of algebraic varieties. Proceedings of the International Congress of Mathematicians , 1:182--192, 1950

    1950

  5. [13]

    Foundations of Differential Geometry , volume 1

    Shoshichi Kobayashi and Katsumi Nomizu. Foundations of Differential Geometry , volume 1. Interscience Publishers, 1963

    1963

  6. [14]

    L'analysis situs et la géométrie algébrique

    Solomon Lefschetz. L'analysis situs et la géométrie algébrique. Gauthier-Villars , 1924

    1924

  7. [15]

    Daniel G. Quillen. Formal schemes and formal groups. Séminaire Bourbaki , 6:191--220, 1963

    1963

  8. [16]

    Theorie der Abel'schen Funktionen

    Bernhard Riemann. Theorie der Abel'schen Funktionen . Borntraeger, 1857

  9. [17]

    Géométrie algébrique et géométrie analytique

    Jean-Pierre Serre. Géométrie algébrique et géométrie analytique. Annales de l'institut Fourier , 6:1--42, 1956

    1956

  10. [18]

    Donald C. Spencer. Deformation of structures on manifolds defined by transitive, continuous pseudogroups. Annals of Mathematics , 76(2):306--445, 1962

    1962

  11. [19]

    Chen-Ning Yang and Robert L. Mills. Conservation of isotopic spin and isotopic gauge invariance. Physical Review , 96(1):191--195, 1954

    1954

  12. [20]

    Constructing two metrics for spencer cohomology: Hodge decomposition of constrained bundles

    Dongzhe Zheng. Constructing two metrics for spencer cohomology: Hodge decomposition of constrained bundles. arXiv preprint arXiv:2506.00752 , 2025

    2025 arXiv

  13. [21]

    Dynamical geometric theory of principal bundle constrained systems: Strong transversality conditions and variational framework for gauge field coupling

    Dongzhe Zheng. Dynamical geometric theory of principal bundle constrained systems: Strong transversality conditions and variational framework for gauge field coupling. arXiv preprint arXiv:2505.16766 , 2025

    2025 arXiv

  14. [22]

    Geometric duality between constraints and gauge fields: Mirror symmetry and spencer isomorphisms of compatible pairs on principal bundles

    Dongzhe Zheng. Geometric duality between constraints and gauge fields: Mirror symmetry and spencer isomorphisms of compatible pairs on principal bundles. arXiv preprint arXiv:2506.00728 , 2025

    2025 arXiv

  15. [23]

    Mirror symmetry in geometric constraints: Analytic and riemann-roch perspectives

    Dongzhe Zheng. Mirror symmetry in geometric constraints: Analytic and riemann-roch perspectives. arXiv preprint arXiv:2506.06610 , 2025

    2025

  16. [24]

    Mirror symmetry of spencer-hodge decompositions in constrained geometric systems

    Dongzhe Zheng. Mirror symmetry of spencer-hodge decompositions in constrained geometric systems. arXiv preprint arXiv:2506.05816 , 2025

    2025 arXiv

  17. [25]

    Spencer-riemann-roch theory: Mirror symmetry of hodge decompositions and characteristic classes in constrained geometry

    Dongzhe Zheng. Spencer-riemann-roch theory: Mirror symmetry of hodge decompositions and characteristic classes in constrained geometry. arXiv preprint arXiv:2506.05915 , 2025

    2025 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.