{"id":"2fdbef34-14f6-4c8e-8dc6-6f36a86feb24","arxiv_id":"1908.03599","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For chi-plurisubharmonic solutions of complex Hessian equations with gradient-dependent right-hand sides on compact Hermitian manifolds, the second covariant derivative of the solution is uniformly bounded in terms of lower-order bounds and the data.","lead":"This paper proves a uniform bound on the size of second derivatives of solutions to complex Hessian equations on Hermitian manifolds, allowing the right-hand side to depend on the gradient of the solution. It is the key regularity ingredient needed to solve these fully nonlinear equations on non-Kähler spaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The maximum principle does not close as written: after (3.22) the paper discards the E_i terms using an inequality imported from [26] that is neither proved nor adapted to Hermitian torsion, and the exponent m on which the inequality may depend is never fixed.","rationale":"I agree with the reader that the most load-bearing point is the unproved inequality after (3.22). The paper contains substantial, mostly checkable computations in Sections 2 and 3, and the overall maximum-principle architecture follows [26], so the main risk is not the strategy but the single closing inequality that lets the E_i terms be discarded. The reader identified this point precisely; my stress-test sharpens it by noting that the proof also never fixes the exponent m in P_m, and that the Hermitian torsion enters the definitions just before the assertion. A conditional verdict remains appropriate: the result is credible and the gap is probably fillable, but the claim is not fully established until the inequality is either proved in the Hermitian setting or shown to be a purely algebraic pointwise estimate with explicit dependence on m and K. No further objection to the main theorem is apparent, and the paper should receive credit for the detailed torsion modifications and the extension to gradient-dependent right-hand sides.","tokens_in":14286,"tokens_out":21516,"duration_ms":217947,"concrete_test":"Take the inequality A_i+B_i+C_i+D_i-E_i >= 0 in the form used after (3.22). Extract from [26] the exact lemma or computation proving the corresponding statement, then transcribe it to the Hermitian covariant derivatives and torsion identities used between (3.20) and (3.22), recording every place where commutation of D_i and D_{\\bar j}, or torsion-freeness, is invoked. If the transcription uses no such commutation and the resulting quadratic-form inequality is verified for all complex vectors v_p = D_i g_{p\\bar p} with explicit hypotheses on m and K, the concern is resolved; if a commutation identity or torsion-freeness is used, Theorem 1.1 is not established as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive step is the assertion immediately after (3.22): \"By the arguments as in [26], we may assume without loss of generality that A_i+B_i+C_i+D_i-E_i >= 0, for every i = 1,...,n.\" The negative E_i terms are otherwise multiplied by a coefficient close to 1, and without this assertion the chain leading to (3.23) and the bound on lambda_1 does not go through. In this paper A_i,...,E_i are built from Hermitian covariant derivatives of g_{p\\bar q}, and the preceding estimate (3.20) has just used a torsion-involving identity of the form D_q u_{j\\bar j} = D_j g_{q\\bar j} - T^a_{jq} u_{a\\bar j} + ..., so the pointwise third-order quantities are not literally the same as those in [26]. No proof is supplied that the inequality survives the torsion, and no restriction on m is stated, although the inequality is sensitive to m in simple cases. Theorem 1.3 inherits the same unproved step after (4.7). The gap may be fillable if the inequality is a purely algebraic quadratic-form estimate, but as written the paper does not give the required verification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a priori second-order estimates for χ-plurisubharmonic solutions of the complex Hessian equation (χ+√−1∂∂u)^k ∧ ω^{n−k} = ψ(z,Du,u)ω^n on compact Hermitian manifolds, under χ ≥ εω. The proof follows the maximum-principle method of Phong–Picard–Zhang [26], with a test function log P_m + φ(|Du|^2)+φ(u) and a modified auxiliary function to handle torsion terms. Theorem 1.1 is the main result; Theorem 1.3 is an outlined analogue when the form χ is replaced by χ+√−1(a⊗∂u−\\bar a⊗\\bar ∂u).","tokens_in":14593,"tokens_out":24333,"duration_ms":220462,"significance":"If the proof can be completed, the result would be a natural and useful extension of [26] from Kähler to Hermitian manifolds, providing the missing second-order estimate needed in the continuity method for these equations. The manuscript contains a detailed and mostly careful maximum-principle calculation; the treatment of the torsion terms via the modified test function is a plausible and interesting idea. The main obstacle is a single but load-bearing algebraic inequality that is imported from [26] without verification in the Hermitian setting, so the current manuscript is not yet a complete proof of the stated theorems.","major_comments":[{"comment":"The assertion 'By the arguments as in [26], we may assume without loss of generality that A_i+B_i+C_i+D_i-E_i ≥ 0, for every i=1,...,n' is load-bearing and unproved. Since 1−δ ≥ 1−φ''/(2φ'^2), the negative E_i terms can be neglected only if the stated pointwise inequality holds for each i. The tensors A_i,...,E_i are formed from Hermitian covariant derivatives D_i g_{p\\bar p}, and the preceding estimates (e.g., (3.15) and (3.20)) use torsion commutation identities, so it is not automatic that the Kähler proof in [26] carries over verbatim. The authors should state the precise algebraic lemma from [26], prove it (or show that it is a purely algebraic inequality valid for arbitrary v_p = D_i g_{p\\bar p}), and specify the range of m for which it holds; m is never fixed in the proof and the inequality may depend on it.","section":"Section 3, after (3.22)"},{"comment":"Theorem 1.3 is only outlined, and its final step says 'the proof is the same as that of Theorem 1.1.' Because the corresponding step in Theorem 1.1 depends on the unproved inequality in the previous comment, Theorem 1.3 inherits the same gap. Please either provide the full verification for the tensors defined with g̃ = χ+√−1∂∂u+√−1(a⊗∂u−\\bar a⊗\\bar ∂u), or explicitly say which parts are identical and prove the nontrivial ones.","section":"Section 4, after (4.7)"}],"minor_comments":[{"comment":"The definition of Γ_k(M) contains the typo 'A^{1,1}(M, R^n)'; it should be 'A^{1,1}(M, R)' since σ_k(h) is a real-valued function.","section":"Equation (2.3)"},{"comment":"Reference [11] appears corrupted as 'Dinew-Ko/suppress lodziej'; it should be 'Dinew-Kołodziej'. Reference [20] contains the typo 'Kähler manfold'.","section":"References"},{"comment":"The constants in the line '- C/β - C/τ' are not tracked precisely: with β=τ=1/(6e^{M(-u+L)}), one obtains -C/β - C/τ = -12C e^{M(-u+L)}, not simply '-Cφ'. The conclusion is unaffected, but the displayed formula should be corrected.","section":"Equation (3.23)"},{"comment":"Theorem 1.3 is labeled an 'Outline of proof'. If this is intended to be a full proof of a main theorem, the paper should either upgrade the outline to a complete argument or clearly state that the details are analogous and available upon request.","section":"Section 4"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the unproved inequality after (3.22). If the authors supply a self-contained proof of the algebraic lemma or a precise statement from [26] that clearly applies to the Hermitian covariant derivatives, the paper may be acceptable after revision. The outlined proof of Theorem 1.3 is also thinner than usual for a main theorem. I would encourage the editor to ask for these clarifications before further consideration."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The new content is the extension of Phong-Picard-Zhang's second-order estimate for chi-plurisubharmonic solutions of complex Hessian equations from Kähler to compact Hermitian manifolds, with right-hand sides depending on Du and u. That is a real, nontrivial technical step: torsion creates additional third-order terms, and the authors modify the auxiliary function to generate enough good terms to absorb them. The main estimate, Theorem 1.1, is the missing ingredient for a continuity-method existence and regularity theory in this generality, and with the Fu-Yau equation as an application it is a useful paper for people working in non-Kähler complex PDEs.\n\nThe proof is detailed and mostly careful. The maximum-principle setup, commutation formulas, and the treatment of the torsion terms Hp are plausible. The paper also deserves credit for stating the gradient-dependent case and for not overclaiming: Theorem 1.3 is explicitly labelled an outline.\n\nNow the soft spot, and it is the one the stress-test flags. Immediately after (3.22), the argument says \"By the arguments as in [26], we may assume without loss of generality that A_i+B_i+C_i+D_i-E_i >= 0 for every i\", and then uses that to discard a negative term. The definitions of A_i through E_i are built from covariant derivatives of g, and in the Hermitian setting those derivatives carry torsion contributions, as the paper's own commutation identities (2.8)-(2.9) show. So this is not literally the same inequality as in [26]. The paper does not verify it, and it does not fix the exponent m. Without it the chain from (3.22) to (3.23) does not close. This is a genuine gap in the written proof, not a stylistic complaint. It may well be fillable, if the inequality is an algebraic consequence of the cone condition and the equation, but the authors need to show that.\n\nTheorem 1.3 inherits the same unproved step after (4.7), plus the outline omits many of the same details. A minor issue is that the relation to Feng-Ge-Zheng [12] could be stated more precisely so the reader knows exactly what is new in the gradient-dependent case.\n\nOverall: the central idea is sound and the result is likely true, but as written the proof has a load-bearing \"by the arguments as in [26]\" at the exact place where the maximum principle has to close. That deserves a serious referee, not a desk rejection. My recommendation: send it to review, and ask the referee to demand a proof of the A_i+B_i+C_i+D_i-E_i inequality in the Hermitian setting, or a citation to a version that covers it, before accepting.","headline":"Good technical extension of PPZ's second-order estimate to Hermitian manifolds, with one unproved imported inequality that stops the proof from fully closing.","tokens_in":15063,"tokens_out":4176,"would_cite":true,"duration_ms":42756,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J15","53C55","58J05","35B45"],"pacs":[],"model":"deepseek-v4-flash","headline":"On compact Hermitian manifolds, χ-plurisubharmonic solutions of complex Hessian equations with gradient-dependent right-hand sides have uniformly bounded second derivatives.","keywords":["complex Hessian equations","second order estimates","Hermitian manifolds","χ-plurisubharmonic functions","fully nonlinear elliptic equations","a priori estimates","torsion","maximum principle"],"falsifier":"On a compact Hermitian manifold with non-zero torsion, take any smooth $\\chi$-plurisubharmonic function with $\\chi \\ge \\varepsilon\\omega$ and compute the tensors $A_i, B_i, C_i, D_i, E_i$ defined after (3.22) at a point where the maximum-principle test function $G$ attains its maximum; if $A_i + B_i + C_i + D_i - E_i < 0$ for some $i$, the proof's final step fails, and the eigenvalue bound would need another argument.","tokens_in":14095,"feed_emoji":"🧮","tokens_out":12172,"duration_ms":105965,"temperature":0.7,"pith_summary":"On a compact Hermitian manifold, the paper proves a uniform second-order estimate for $\\chi$-plurisubharmonic solutions of the complex Hessian equation $(\\chi+\\sqrt{-1}\\,\\partial\\bar\\partial u)^k \\wedge \\omega^{n-k} = \\psi(z,Du,u)\\,\\omega^n$, assuming $\\chi \\ge \\varepsilon\\omega$. The bound $|D\\bar D u|_\\omega \\le C$ depends only on known data, not on the particular solution. This is the missing a priori estimate needed to run the continuity method and conclude existence and regularity for these fully nonlinear equations on non-Kähler manifolds, where torsion produces extra third-order terms. A second theorem extends the estimate to equations whose background Hessian itself depends on the gradient of the unknown.","feed_headline":"Hessian equations on Hermitian manifolds get uniform C^2 bound","feed_subtitle":"The bound completes the regularity chain for solving these nonlinear Hessian equations with gradient terms.","key_machinery":"The maximum-principle test function is $G = \\log P_m + \\phi(|Du|^2) + \\varphi(u)$, where $P_m = \\sum_j \\lambda_j^m$ and $\\lambda_j$ are the eigenvalues of $g = \\chi + \\sqrt{-1}\\,\\partial\\bar\\partial u$ with respect to $\\omega$. Differentiating $G$ twice, contracting with $\\sigma_k^{pq}$, and using Hermitian commutation formulas produces a long inequality in which the third-order terms are controlled by a tensor lemma for $\\sigma_k$ (Lemma 3.1) and by choosing $\\phi$ and $\\varphi$ as exponentials with widely separated exponents. The torsion enters through extra terms $T \\ast D^3u$; the auxiliary function is modified so that these become absorbable, and the argument closes by invoking the inequality $A_i + B_i + C_i + D_i - E_i \\ge 0$ from the Kähler case.","core_discovery":"The central claim is Theorem 1.1: every $C^4$ $\\chi$-plurisubharmonic solution of the Hermitian Hessian equation with gradient-dependent right-hand side and $\\chi \\ge \\varepsilon\\omega$ satisfies $|D\\bar D u|_\\omega \\le C$, with $C$ uniform. The proof achieves this by adapting the Kähler-manifold maximum-principle argument to the Hermitian case, modifying the auxiliary function so that torsion terms of the form $T \\ast D^3u$ are absorbed into favorable third-order terms. Theorem 1.3 gives the analogous bound when the Hessian is replaced by $\\chi + \\sqrt{-1}\\, a \\wedge \\partial u - \\sqrt{-1}\\, \\bar a \\wedge \\bar\\partial u$, so the result covers equations whose coefficients depend on the gradient as well.","pith_inferences":["The inequality $A_i + B_i + C_i + D_i - E_i \\ge 0$ is imported from the Kähler proof without a Hermitian verification; testing it on explicit torsion-modified tensors would either close the gap or expose a counterexample.","A natural next step is to relax the $\\chi$-plurisubharmonic hypothesis to the $\\Gamma_k$ cone; the present technique does not obviously survive that relaxation, since positivity of $g$ is used to control the negative third-order terms.","The same maximum-principle format should generalize to almost Hermitian manifolds for all $k$, using non-integrable commutation formulas, though the torsion estimates would need to be reworked."],"forward_implications":["For $k=n$, the result gives the second-order estimate for the complex Monge-Ampère equation on Hermitian manifolds when the right-hand side depends on the gradient.","Together with standard higher-order elliptic estimates, the bound yields $C^\\infty$ regularity and existence for fully nonlinear Hessian equations on Hermitian manifolds under the $\\chi$-plurisubharmonic admissibility condition.","The gradient-dependent $\\chi$ case (Theorem 1.3) puts equations whose coefficients include first derivatives of the unknown into the same a priori estimate framework.","The uniform bound gives compactness of admissible solution families, which is exactly what the continuity method needs to cross from the linearized equation to the nonlinear one."],"supporting_citations":[{"why":"Establishes the second-order estimate for χ-plurisubharmonic solutions on Kähler manifolds and supplies the final inequality the Hermitian proof invokes.","marker":"[26]"},{"why":"Provides Lemma 3.1, the tensor inequality that absorbs the negative third-order terms in the maximum-principle argument.","marker":"[19]"},{"why":"Gives the commutation formulas for covariant derivatives on Hermitian manifolds used throughout the computation.","marker":"[43]"},{"why":"Shows that second-order bounds combine with standard higher-order elliptic regularity to give smoothness, explaining why the estimate implies solvability.","marker":"[40]"},{"why":"Proves the k=2 case on almost Hermitian manifolds, the closest prior result in non-Kähler settings and the benchmark this paper extends to all k.","marker":"[9]"}],"fun_headline_variants":["Uniform C^2 bound for Hessian equations on Hermitian manifolds","Gradient-dependent Hessian equations get uniform C^2 estimates","C^2 bound for Hessian equations with gradient terms on Hermitian manifolds","Second-order estimates for complex Hessian equations on Hermitian manifolds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument leans on an inequality, imported from the Kähler case, that says $A_i + B_i + C_i + D_i - E_i \\ge 0$ for every $i$; the paper does not prove this inequality for the torsion-modified tensors of Section 3, and if it fails for some Hermitian manifold, the maximum-principle closure and hence the bound may collapse.","fun_headline_variants_meta":{"raw":{"variants":["Uniform C^2 bound for Hessian equations on Hermitian manifolds","Gradient-dependent Hessian equations get uniform C^2 estimates","C^2 bound for Hessian equations with gradient terms on Hermitian manifolds","Second-order estimates for complex Hessian equations on Hermitian manifolds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001023,"raw_usage":{"total_tokens":4203,"prompt_tokens":723,"completion_tokens":3480,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":339,"completion_tokens_details":{"reasoning_tokens":3402}},"tokens_in":339,"tokens_out":3480,"duration_ms":25013,"temperature":1.0,"reasoning_tokens":3402,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:08:26.833983+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a compact Hermitian manifold with non-zero torsion, take any smooth $\\chi$-plurisubharmonic function with $\\chi \\ge \\varepsilon\\omega$ and compute the tensors $A_i, B_i, C_i, D_i, E_i$ defined after (3.22) at a point where the maximum-principle test function $G$ attains its maximum; if $A_i + B_i + C_i + D_i - E_i < 0$ for some $i$, the proof's final step fails, and the eigenvalue bound would need another argument.","supporting_citations":[{"cited_title":"Phong, S","cited_arxiv_id":null,"evidence_quote":"Establishes the second-order estimate for χ-plurisubharmonic solutions on Kähler manifolds and supplies the final inequality the Hermitian proof invokes."},{"cited_title":"Guan, C.Y","cited_arxiv_id":null,"evidence_quote":"Provides Lemma 3.1, the tensor inequality that absorbs the negative third-order terms in the maximum-principle argument."},{"cited_title":"Tosatti, Y","cited_arxiv_id":null,"evidence_quote":"Shows that second-order bounds combine with standard higher-order elliptic regularity to give smoothness, explaining why the estimate implies solvability."},{"cited_title":"The $2$-nd Hessian type equation on almost Hermitian manifolds","cited_arxiv_id":"1707.04072","evidence_quote":"Proves the k=2 case on almost Hermitian manifolds, the closest prior result in non-Kähler settings and the benchmark this paper extends to all k."}],"review_version":1}