{"id":"c67163a4-55ef-4d61-a2ba-1d1bfdda0694","arxiv_id":"2507.01534","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Local existence of solutions to polynomial curvature equations is equivalent to local polystability, with local filtrations and a Kempf-Ness homeomorphism.","lead":"This paper proves a local Kobayashi-Hitchin correspondence for a wide class of curvature PDEs on holomorphic vector bundles: a small deformation admits a solution exactly when it satisfies a local polystability condition. It also builds local Jordan-Holder and Harder-Narasimhan filtrations and a local Kempf-Ness homeomorphism, giving a general toolkit for perturbing special metrics.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the local Kobayashi-Hitchin proof is internally coherent, and the main analytic and GIT steps are supported by explicit statements in the text.","rationale":"I read the full manuscript and focused on the proof of Theorem 4.12. The reader's verdict of CONDITIONAL is reasonable, but the specific weakest assumption named by the reader, openness of the sub-solution locus, is actually proven in Lemma 4.1 by a direct continuity argument: the sub-solution condition is a pointwise positivity condition on P'_zeta(partial), and P'_zeta varies continuously with (zeta,partial) in the chosen C^d or L^p_d topologies. I also checked the more delicate steps that the theorem needs. The deformed slice sigma is constructed by an implicit function theorem at Proposition 4.5; the derivative is the elliptic operator Delta_{zeta_0,nabla_0}, whose kernel is exactly the infinitesimal unitary automorphisms by Lemma 2.8, so the isomorphism needed for sigma is present. The flow argument in Section 6 uses the Lojasiewicz inequality (13) for both the Kaehler moment map mu_{zeta_0} and the deformed moment map tilde_mu_{zeta_0}; the former is standard, and the latter is supported by the analyticity of the implicit function sigma, although the manuscript states this compactly. The GIT degeneration Proposition 5.13 and the polystability argument in Section 6.2 are internally coherent. I found no internal inconsistency or unsupported step that would change the reader's conditional accept. The agreement is marked partial rather than agree because my non-finding does not coincide with the reader's stated weakest assumption, but I do agree that the sub-solution hypothesis itself is load-bearing for the whole framework.","tokens_in":64169,"tokens_out":46769,"duration_ms":578604,"concrete_test":"Verify the two analytic pillars at the base point: check that the map Psi(zeta,b,s)=Pi_{ik_perp} i mu_infty,zeta(e^s . partial_b) in Proposition 4.5 is real analytic in (zeta,b,s) and that its derivative in s at (zeta_0,0,0), namely Delta_{zeta_0,nabla_0} restricted to ik_perp, is an isomorphism C^{d+1}(ik_perp) -> C^{d-1}(ik_perp). If this holds, the analyticity of sigma and tilde_mu are justified, the Lojasiewicz inequality (13) is supported, and no adjustment to the verdict is needed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central equivalence in Theorem 4.12 rests on four pillars: openness of the sub-solution condition, existence and convergence of the finite-dimensional moment-map flows, the deformed Kuranishi slice whose zeroes give genuine P-critical operators, and the GIT degeneration result Proposition 5.13. Each is supplied with a proof or a precise adaptation of a standard result: Lemma 4.1 gives openness in C^d (and L^p_d) topology; Propositions 5.10, 5.11 and Corollary 6.7 give flow existence and convergence; Propositions 4.5 and 5.6 link the zeroes of the deformed finite-dimensional moment map to P-critical operators; Proposition 5.13 plus the polystability argument supplies the orbit-closure half. The reader's identified weakness, openness of sub-solutions, is not a gap: sub-solution positivity is pointwise and P'_zeta(∂) depends continuously on (zeta,∂) in the stated topologies, so Lemma 4.1 is sound. The main compressed point is the real analyticity of the deformed moment map used for the Lojasiewicz inequality (13), but tilde_mu_{zeta_0} is obtained from the analytic implicit-function construction of sigma in Proposition 4.5, so the appeal to the Marle-Guillemin-Sternberg normal form is justified. I therefore do not find a load-bearing gap in the argument for Theorem 4.12.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies local deformations of a holomorphic Hermitian vector bundle E0 whose curvature solves a polynomial equation Pζ0(F) = 0, under a sub-solution positivity condition. It constructs a deformed Kuranishi slice and finite-dimensional moment maps, then proves a local Kobayashi–Hitchin correspondence (Theorem 4.12): for ζ close to ζ0 and b close to 0, the deformed bundle Eb admits a Pζ-critical operator close to ∂0 if and only if Eb is locally Pζ-polystable, with analogous equivalences for the semi-stable and stable versions. The paper also proves uniqueness of solutions modulo unitary gauge (Theorem 6.2), continuity of solutions (Theorem 7.2), a local Kempf–Ness homeomorphism (Theorem 7.4), and local Jordan–Hölder and Harder–Narasimhan filtrations (Theorems 6.3 and 7.1). Section 8 applies the machinery to rank-two bundles on a blow-up of P2 in the context of the J-equation and the deformed Hermitian Yang–Mills equation, giving an explicit numerical existence criterion.","tokens_in":64392,"tokens_out":31677,"duration_ms":483774,"significance":"If the proof is completed, this is a substantial generalization of the local Kobayashi–Hitchin results of Buchdahl–Schumacher and Dervan–McCarthy–Sektnan to a wide class of polynomial curvature equations under the sub-solution assumption. The paper is unusually explicit: the linearization, ellipticity, Kuranishi slice, deformed moment map, and stability equivalences are stated in detail, and the final example gives concrete numerical criteria. The main theorems are not obtained by fitting parameters: the condition is Pζ(F), a Chern-class number, and the proof is a genuine moment-map/GIT construction with explicit flow arguments. No machine-checked proofs are provided, so the strength of the paper lies in its detailed analytic–algebraic bridge and in the breadth of the unified framework.","major_comments":[{"comment":"The Lojasiewicz inequality for tilde{f}_{ζ0} requires real analyticity of tilde{μ}_{ζ0}, or a substitute such as a Lojasiewicz–Simon inequality for this class of functions. Proposition 4.5 constructs σ via the smooth implicit function theorem and states only smoothness of σ and tilde{Φ}; no analyticity of the deformed Kuranishi slice or of tilde{μ} is proved. Since Propositions 5.10 and 5.11, and hence the convergence of the deformed moment-map flow used in the proof of Theorem 4.12, depend on (13), this is a load-bearing point. Please either prove the real analyticity (for example via an analytic Kuranishi slice and the analytic implicit-function theorem) or replace (13) with a justified Lojasiewicz–Simon inequality.","section":"§5.3, Eq. (13)"},{"comment":"In the proof of Lemma 6.6, Corollary 6.5 is applied to the pair b_m and b' = tilde{φ}_{ζ_m}(b_m,t'_m), but Corollary 6.5 as stated applies only when both bundles are locally Pζ-semi-stable. This hypothesis is not verified for an arbitrary sequence b_m → 0, and at this point of the proof Theorem 4.12 or Theorem 4.13 cannot be used to supply it without circularity. Since Lemma 6.6 is the key step proving that the ζ-dependent flows stay in a fixed compact set (Corollary 6.7), the proof needs either a strengthened version of Corollary 6.5 (K-orbit independence of flow limits for arbitrary small points in the same G-orbit) or a different argument in Lemma 6.6.","section":"§6.5, Lemma 6.6"}],"minor_comments":[{"comment":"The definition is written for a sub-bundle F with respect to its admissible sub-bundles, but the statements of Theorems 4.12, 6.2, and related results refer to 'Eb is locally Pζ-(semi/poly)stable'. Please state explicitly that this means the same definition applied to F = E, especially the equality case requiring (G⊥, ∂b) to be holomorphic.","section":"Definition 4.11"},{"comment":"In the displayed formula for d/ds|_{s=0} μ∞,ζ(e^s·∂)v, the term 2π[Im(v), Pζ(∂)] appears to omit the volume-form normalization that appears in Proposition 2.6 and in the definition μ∞,ζ = -2iπ/Vol Pζ. Please check the normalization consistency, since later computations integrate traces of Pζ against the volume form.","section":"§4.1, derivative of μ∞,ζ"},{"comment":"Lemma 5.1 is stated for deformations ∂F + γ of a single holomorphic structure, while Lemma 5.5 uses the two-sided operator ∂_{b,b',α}f = ∂0 f + (Φ(b')+α)f - fΦ(b). Please state the needed two-sided version or explain explicitly why the proof of [5, Proposition 4.5] applies verbatim to this operator.","section":"Lemma 5.1 and Lemma 5.5"},{"comment":"The proof refers to 'Remark 6.1', but the relevant statement appears to be Lemma 6.1; please correct the cross-reference.","section":"Corollary 6.5"},{"comment":"The notation in the proof of the local Kempf–Ness theorem switches between Γ, tilde{Γ}, Γ∞, and Γζ; the text should define all of these maps explicitly and consistently before using them.","section":"Theorem 7.4 and its proof"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is long and has a long chain of dependencies, but the two gaps I identified are local and likely repairable: the missing analyticity argument behind Eq. (13) and the unjustified use of Corollary 6.5 in Lemma 6.6. If the author supplies those missing arguments, I would support publication. The paper fits the journal's scope and the main theorem is substantial."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this paper proves a genuinely local Kobayashi-Hitchin correspondence for the whole class of P-critical equations, and it appears to hold together. I read the full preprint and the stress-test note; I do not find a load-bearing gap.\n\nWhat is new: prior local results covered HYM (Buchdahl-Schumacher) and asymptotic Z-critical equations (Dervan-McCarthy-Sektnan). Delloque treats arbitrary polynomial curvature equations, with simultaneous variation of the bundle and the equation parameters, and adds local Jordan-Holder, Harder-Narasimhan, and Kempf-Ness theorems. The proof is structured and honest: explicit linearization, elliptic sub-solution condition, deformed Kuranishi slice, finite-dimensional moment maps, and GIT flows. The openness of sub-solutions (Lemma 4.1) is sound, and the circularity burden is low—the stability condition is a Chern-class number, not a fitted parameter.\n\nSoft spots: the paper imports a lot of GIT and flow machinery from Georgoulas-Robbin-Salamon, and some adaptations to the non-compact ball are sketched rather than proved in full—particularly Propositions 5.13 and 5.14, and the real analyticity needed for the Lojasiewicz inequality. These are the places I would want a referee to check carefully. The distinction between the simplified and full local stability definitions could also be clearer. None of this amounts to a detected error; it is more a question of exposition and verifying the imported results under the local hypotheses.\n\nWho it is for: anyone working on Hermitian metrics, stability conditions, or deformations of geometric PDEs. The application section on J- and dHYM equations on a blow-up is a nice concrete payoff.\n\nRecommendation: send to a serious referee. The paper deserves referee time, and the referee should be asked to verify the imported GIT/flow propositions in the non-compact local setting, not to desk-reject.","headline":"A genuinely local Kobayashi-Hitchin correspondence for the full class of P-critical equations; the proof looks sound, with the main risk concentrated in imported GIT/flow results.","tokens_in":64985,"tokens_out":1519,"would_cite":true,"duration_ms":19396,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C07","53D20","32G13","14D20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Small perturbations of a holomorphic bundle admit a nearby solution to its curvature polynomial equation exactly when the bundle is locally polystable.","keywords":["P-critical equations","local Kobayashi-Hitchin correspondence","moment maps","geometric invariant theory","holomorphic vector bundles","stability conditions","Kuranishi slice","deformed Hermitian Yang-Mills equation"],"falsifier":"Take E0=L1⊕L2 on X=Bl_p P2 with the J-equation solution from Section 8 and the two non-split extensions E1 and E2. The theorem predicts that after small deformations, E0 has a nearby dHYM solution exactly on the threshold curve A(ε1,ε2)=0, E1 exactly when A<0, and E2 exactly when A>0; numerically solving the dHYM equation in the small C1 neighbourhood and checking these sign boundaries would settle the correspondence.","tokens_in":63913,"feed_emoji":"📐","tokens_out":6528,"duration_ms":76359,"temperature":0.7,"pith_summary":"This paper tries to establish a local Kobayashi–Hitchin correspondence for a broad family of geometric PDEs on holomorphic Hermitian vector bundles: equations in which a polynomial in the reduced curvature form vanishes. Starting from one solution E0 to such a P-critical equation, the author studies small perturbations of both the holomorphic structure and the equation's coefficients. The central claim is that, under a positivity assumption on the initial solution, a nearby perturbed bundle admits a nearby solution if and only if the perturbed bundle is locally polystable, a condition phrased purely in terms of Chern-class weights of admissible subbundles. If true, this converts a hard analytic existence question into a finite algebraic check, covering equations such as the Hermitian Yang–Mills, complex Monge–Ampère, J, and deformed Hermitian Yang–Mills equations in one framework.","feed_headline":"Local polystability decides which bundle perturbations solve their PDE","feed_subtitle":"Near a known solution, the existence of new solutions is decided by admissible subbundles and their Chern-class weights.","key_machinery":"The load-bearing mechanism is the reduction of the infinite-dimensional gauge-theoretic equation to a finite-dimensional moment map problem via a deformed Kuranishi slice. A sub-solution ∂0 makes the linearized operator Δ_{ζ,∇} a nonnegative elliptic operator and gives a moment-map interpretation of the P-critical equation; the deformed Kuranishi slice Φ(ζ,b)=$e^{{σ(ζ,b)}}$·∂b folds small gauge perturbations into a family of moment maps μζ and μ̃ζ on a ball in V, whose zeros correspond exactly to nearby solutions after a unitary gauge change. The finite-dimensional invariant-theory machine—gradient flows of half the squared moment map, the analytic gradient inequality, and one-parameter subgroup degeneration—then matches orbit closure to the algebraic inequalities defining local Pζ-polystability. The key identity that carries the comparison is that along a one-parameter subgroup ξ with eigenspace filtration F_k, the limit pairing ⟨μζ(b∞), iξ⟩ evaluates to 2π rk(E) Pζ(F), relating moment-map zeroes to Chern-class weights.","core_discovery":"Let E0=(E,h,∂0) be Pζ0-critical, meaning Pζ0(E0,∂0,h)=0 and ∂0 is a sub-solution. The author shows there are neighbourhoods Ud of ζ0 and B1 of 0 in the finite-dimensional space V=$H^{{0,1}}$(X,End(E0)) of harmonic (0,1)-forms such that for every ζ∈Ud and b∈B1, the deformed bundle Eb admits a Pζ-critical Dolbeault operator close to ∂0 in its complex gauge orbit if and only if Eb is locally Pζ-polystable. Local Pζ-polystability is an algebraic condition: every admissible subbundle F, meaning one that is holomorphic for ∂b, for ∂0, and whose orthogonal complement is holomorphic for ∂0, must satisfy Pζ(F)/rk(F) ≤ Pζ(E)/rk(E)=0, with equality only if F⊥ is also ∂b-holomorphic. When ζ=ζ0, the same equivalence says that local polystability is exactly closedness of the orbit G·b in V under the automorphism group, matching the finite-dimensional invariant-theory picture. The paper also proves uniqueness of nearby solutions modulo unitary gauge, continuous variation of solution orbits with (ζ,b), Jordan–Hölder and Harder–Narasimhan filtrations by admissible subbundles, and a local Kempf–Ness homeomorphism between the polystable quotient and the solution moduli space.","pith_inferences":["The same machinery should produce algorithmic existence tests in explicit rank-2 examples: since local stability is a finite list of linear inequalities in [ζ−ζ0], one can compute wall-crossing curves, like the threshold curve A(ε1,ε2)=0 in the paper's example, without solving the PDE.","Because the Hermitian Yang–Mills equation has every operator as a sub-solution, this local theorem recovers the known local deformation theory for HYM; extending the sub-solution condition to singular or non-locally-free sheaves would likely extend the correspondence to compactified moduli.","One could test whether openness of the sub-solution condition, which is central here, also holds when the initial operator ∂0 is only a solution in a weak or distributional sense; if not, the theorem would require a different transversality mechanism.","The moment map flow limit provides a canonical graded representative for each semi-stable deformation, suggesting a deformation-theoretic analogue of Harder–Narasimhan stratification that could be used to study jumping phenomena in moduli spaces."],"forward_implications":["Near a polystable base solution, the existence question for all nearby P-critical equations is reduced to checking finitely many inequalities on admissible subbundles, because only finitely many such subbundles appear up to isomorphism.","For the unperturbed equation ζ=ζ0, a small deformation admits a solution exactly when its automorphism-group orbit in the harmonic space V is closed, giving a local invariant-theory criterion for solvability.","When solutions exist they are locally unique modulo unitary gauge and their gauge orbits vary continuously with the equation parameter and the deformation, so canonical metrics constructed this way deform in families.","Every small semi-stable deformation has a Jordan–Hölder filtration by admissible subbundles and a unique graded object, which is polystable and is the limit of the moment map flow; every small deformation has a unique Harder–Narasimhan filtration.","The local Kempf–Ness homeomorphism identifies the moduli of polystable deformed bundles up to complex automorphisms with the moduli of P-critical operators up to unitary gauge, with both sides varying continuously in ζ."],"supporting_citations":[{"why":"Supplies the sub-solution condition, the elliptic linearization Δ_{ζ,∇}, and the moment-map formalism for P-critical and Z-critical equations that the paper adapts.","marker":"[18]"},{"why":"Provides the finite-dimensional invariant-theory toolkit: moment-map flows, the analytic gradient inequality, and orbit-closure and Kempf–Ness theorems used throughout the proofs.","marker":"[24]"},{"why":"Supplies the Kuranishi-slice method and the orbit-closed criterion for local deformations of Hermitian Yang–Mills bundles, which the paper generalizes to all P-critical equations.","marker":"[5]"},{"why":"Introduces the P-critical and Z-critical formalism and conjectures a local stability criterion that Theorem 4.12 confirms.","marker":"[28]"},{"why":"Provides the vector-bundle J-equation and its sub-solution notion, one of the concrete equation families covered by the main theorem.","marker":"[46]"},{"why":"Provides the vector-bundle Monge–Ampère equation, another instance of the P-critical framework treated by the paper's local correspondence.","marker":"[40]"},{"why":"Gives the general moment-map interpretation for geometric PDEs that underlies the infinite-dimensional moment map used here.","marker":"[17]"}],"fun_headline_variants":["Local polystability decides perturbed bundle PDE solutions","Polystability iff perturbed bundle admits solution","Perturbed bundles solve PDE exactly when locally polystable","Local Kobayashi-Hitchin: polystability determines solutions","Polystability: the gatekeeper for perturbed bundle PDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The initial operator ∂0 must be a sub-solution: for every nonzero tangent vector v and endomorphism, a certain pointwise trace expression involving the derivative of the polynomial in the curvature must be strictly positive; if this positivity is not preserved in a neighbourhood, the local equivalence is not known to hold.","fun_headline_variants_meta":{"raw":{"variants":["Local polystability decides perturbed bundle PDE solutions","Polystability iff perturbed bundle admits solution","Perturbed bundles solve PDE exactly when locally polystable","Local Kobayashi-Hitchin: polystability determines solutions","Polystability: the gatekeeper for perturbed bundle PDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001055,"raw_usage":{"total_tokens":4453,"prompt_tokens":996,"completion_tokens":3457,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":612,"completion_tokens_details":{"reasoning_tokens":3389}},"tokens_in":612,"tokens_out":3457,"duration_ms":30817,"temperature":1.0,"reasoning_tokens":3389,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:50:09.631916+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take E0=L1⊕L2 on X=Bl_p P2 with the J-equation solution from Section 8 and the two non-split extensions E1 and E2. The theorem predicts that after small deformations, E0 has a nearby dHYM solution exactly on the threshold curve A(ε1,ε2)=0, E1 exactly when A<0, and E2 exactly when A>0; numerically solving the dHYM equation in the small C1 neighbourhood and checking these sign boundaries would settle the correspondence.","supporting_citations":[{"cited_title":"Robbin, and Dietmar A","cited_arxiv_id":null,"evidence_quote":"Provides the finite-dimensional invariant-theory toolkit: moment-map flows, the analytic gradient inequality, and orbit-closure and Kempf–Ness theorems used throughout the proofs."},{"cited_title":"Polystable Bundles and Representations of their Automorphisms","cited_arxiv_id":null,"evidence_quote":"Supplies the Kuranishi-slice method and the orbit-closed criterion for local deformations of Hermitian Yang–Mills bundles, which the paper generalizes to all P-critical equations."},{"cited_title":"$J$-equation on holomorphic vector bundles","cited_arxiv_id":"2112.00550","evidence_quote":"Provides the vector-bundle J-equation and its sub-solution notion, one of the concrete equation families covered by the main theorem."},{"cited_title":"A vector bundle version of the Monge-Ampere equation","cited_arxiv_id":"1804.03934","evidence_quote":"Provides the vector-bundle Monge–Ampère equation, another instance of the P-critical framework treated by the paper's local correspondence."},{"cited_title":"The Universal Structure of Moment Maps in Complex Geometry, 2024","cited_arxiv_id":null,"evidence_quote":"Gives the general moment-map interpretation for geometric PDEs that underlies the infinite-dimensional moment map used here."}],"review_version":1}