{"id":"074e7875-d353-48a5-b64a-770f52b875bf","arxiv_id":"1908.03016","paper_version":3,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"New explicit almost complex structures show that closed anti-invariant 2-forms range from infinite-dimensional to zero-dimensional on R^4, and produce compact examples with maximal or arbitrarily large anti-invariant cohomology.","lead":"This paper constructs explicit almost complex structures, both on Euclidean 4-space and on compact manifolds, showing that the space of closed anti-invariant 2-forms can be infinite-dimensional, one-dimensional, zero-dimensional, or arbitrarily large. It answers two open questions about when such structures must be integrable.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader's identified weak point is the external inequality h^-_J <= b^+ and the value b^+(KT)=2. I checked these against standard self-duality of anti-invariant 2-forms in dimension 4 and computed the Kodaira-Thurston intersection form from the given structure equations; both support the paper. The explicit PDE families in Section 3 also check out: the functions w=(t+is)e^{s x1 + t y1} solve the perturbed Cauchy-Riemann system exactly when s^2+t^2+t=0, and the compact-support argument in Theorem 3.8 is a valid harmonic-function plus unique-continuation argument. The typos I found are real but cosmetic: the (1,0)-coframe displayed before Proposition 4.2 should have its signs adjusted if one wants a genuine coframe, but the forms theta1 and theta2 are verified directly; and the Nijenhuis coefficient in Proposition 5.1 should carry a factor 1/f, but the nonvanishing criterion is unchanged. Thus the verdict should remain unchanged.","tokens_in":12840,"tokens_out":40075,"duration_ms":402017,"concrete_test":"Compute the intersection form of the Kodaira-Thurston manifold in the Nomizu basis {E1∧E3, E1∧E4, E2∧E3, E2∧E4}; confirm it has two positive eigenvalues, so b^+=2. Then verify directly with the structure equations that d(theta1)=0, d(e^lambda theta2)=0, and that theta1 and e^lambda theta2 are pointwise independent. If any of these checks fails, Proposition 4.2's maximality claim would need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After rechecking the PDE computations in Theorems 3.7 and 3.8, the gluing in Corollary 3.9, and the explicit forms in Propositions 4.2 and 5.1, the central existence claims are internally consistent. The sharpness statement h^-_J=2 on the Kodaira-Thurston manifold is the only place where an external quantitative input is needed: the bound h^-_J <= b^+ for compact almost complex 4-manifolds and the value b^+(KT)=2. Both are standard and correct for this nilmanifold, and theta1 and e^lambda theta2 are closed, pointwise independent anti-invariant forms. I find no load-bearing gap. The displayed (1,0)-coframe in Section 4 and the coefficient in the Nijenhuis computation in Section 5 contain minor sign or scaling typos, but the subsequent anti-invariance and nonintegrability conclusions can be verified directly and are unaffected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the space of closed J-anti-invariant 2-forms on almost complex manifolds, with attention to both noncompact and compact settings. The main results are: on R^4, a non-integrable almost complex structure with infinite-dimensional space of closed anti-invariant forms (Theorem 3.7); compactly supported perturbations of the standard complex structure yielding h^-_J = 1 or h^-_J = 0 (Theorem 3.8 and Corollary 3.9); a two-parameter family of non-integrable almost complex structures on the Kodaira-Thurston manifold attaining the maximal value h^-_J = 2 (Proposition 4.2); and, in dimension 6, compact examples with h^-_J >= 2g^2 (Proposition 5.1). The paper also observes that these examples show compactness and dimension 4 are essential hypotheses in the Drăghici-Li-Zhang conjecture that h^-_J >= 3 forces integrability on compact 4-manifolds.","tokens_in":13032,"tokens_out":41633,"duration_ms":350657,"significance":"If the results stand, the paper makes a solid contribution to the study of anti-invariant cohomology. The constructions are explicit and self-contained: closed anti-invariant forms are exhibited directly, and the non-integrability checks are computable. The noncompact examples clarify why compactness is needed in the conjectures, and the Kodaira-Thurston example affirmatively answers Question 3.23 from Drăghici-Li-Zhang. The 6-dimensional examples show that the special behavior in dimension 4 is not shared in higher dimensions, which is a useful counterpoint to existing expectations. The paper is not overly reliant on external machinery; its main PDE computations are verifiable by direct substitution.","major_comments":[{"comment":"The sequence (s_n, t_n) = (sqrt((n-1)/n), -1/n) does not satisfy the defining condition s_n^2 + t_n^2 + t_n = 0. For n > 1 the left-hand side equals (n-1)^2/n^2, which is nonzero, so the forms alpha_n = t_n e^{s_n x_1 + t_n y_1} beta - s_n e^{s_n x_1 + t_n y_1} gamma are not closed as written. The theorem is nevertheless correct: the condition is satisfied by s_n = sqrt(n-1)/n (equivalently s_n^2 = (n-1)/n^2), and with this replacement the exponential factors are distinct and the linear-independence argument goes through. This correction is load-bearing because the infinite-dimensionality claim of Theorem 3.7 depends on the existence of such a sequence.","section":"Section 3, proof of Theorem 3.7"}],"minor_comments":[{"comment":"The displayed inclusion chain 'Z^-_J(M) subset H^+_{g_J} subset H^-_J(M)' is misstated: self-dual harmonic forms are not generally J-anti-invariant, and the space H^-_J(M) of anti-invariant harmonic forms is a subspace of H^+_{g_J}, not a superspace. The intended and correct statement is Z^-_J(M) subset H^-_J(M) subset H^+_{g_J}. Since the inequality h^-_J <= b_+ is used essentially in Proposition 4.2, this passage should be rewritten for accuracy.","section":"Section 2"},{"comment":"The complex operators in the display of the perturbed Cauchy-Riemann system are missing conjugation bars: the system derived from (5) should read partial_{\\bar z_1} w + (i/2) partial_{z_2}(f(w+\\bar w)) = 0 and partial_{\\bar z_2} w = 0. As printed, with unbarred operators, the equations are not equivalent to (5), and the solution in Theorem 3.7 does not satisfy the printed system unless the barred operators are intended.","section":"Section 3, equations (6) and (8)"},{"comment":"The displayed formula for the solution w in terms of z_1 and \\bar z_1 has an incorrect sign: the exponent should be s ((z_1 + \\bar z_1)/2) + t ((z_1 - \\bar z_1)/(2i)), not s ((z_1 - \\bar z_1)/2) + t ((z_1 - \\bar z_1)/(2i)). The subsequent forms alpha_{s,t} use the correct exponent e^{s x_1 + t y_1}, so this is a typographical error.","section":"Section 3, proof of Theorem 3.7"},{"comment":"The Nijenhuis computation gives N_J(E_1,E_3) = -e^{lambda+mu} lambda' E_2, with a factor e^{mu} that is missing in the displayed formula. The conclusion that J is non-integrable for nonconstant lambda is unaffected because e^{lambda+mu} is nonzero.","section":"Section 4, Lemma 4.1"},{"comment":"In the displayed Nijenhuis computation, 'partial/partial t_t' is a typo for 'partial/partial t_1', and the coefficients appear to omit factors of 1/f. The conclusion that J is non-integrable for nonconstant f is correct, as can be verified by a direct computation, but the displayed intermediate expression should be corrected.","section":"Section 5, Proposition 5.1"},{"comment":"There are several minor typographical errors, including 'strcuture' in Lemma 3.4, 'Lapacian' in the Introduction, and 'aswer' in Remark 5.2. These do not affect the mathematics.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The only substantive issue I found is the arithmetic error in the sequence in Theorem 3.7, which is easily corrected. The other issues are presentation problems. The paper's central constructions are sound, and the results are likely correct after the correction. I recommend major revision rather than minor because the proof of a central theorem contains a load-bearing miscomputation, even though the fix is straightforward."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know about this paper because it settles two open questions in the anti-invariant cohomology world with very concrete constructions. The main results: on R^4 there are non-integrable almost complex structures with infinite-dimensional, 1-dimensional, and 0-dimensional spaces of closed anti-invariant forms, and on compact manifolds there are families with h^- = 2 on Kodaira-Thurston (answer to DLZ Question 3.23) and arbitrarily large h^- in dimension 6 (answer to ATZ Question 5.2). The proofs are elementary and explicit: they write down a (1,0)-coframe and solve the PDE system (7) for closed anti-invariant forms. I checked the key computations in Theorems 3.7 and 3.8 and in Propositions 4.2 and 5.1; they are consistent. The exponential solutions in Theorem 3.7 genuinely give an infinite family, and the compact-support argument in Theorem 3.8 correctly forces a=0 and b constant. The global forms theta1 and e^lambda theta2 in Proposition 4.2 are closed and independent, and the bound h^- <= b+ = 2 is the standard DLZ inequality. So the results hold.\n\nThe paper is also honest about what is external: the upper bound h^- <= b+ for compact 4-manifolds is imported, but that is the right tool. The typos are minor — there is a missing factor or sign in the Nijenhuis computation in Section 5, and the displayed coframe in Section 4 has an exponent that should be adjusted, but the anti-invariance and non-integrability conclusions are unaffected. The only genuinely soft spot is Remark 3.10: the claim that the almost complex structures can all be assumed compatible with the standard symplectic form on R^4 is justified by citing Moser and Taubes, but the noncompact nature of R^4 is glossed over. This is irrelevant to the main theorems, so I would not hold it against the paper.\n\nWho this is for: anyone working on almost complex cohomology or the geography of h^-_J. It is a nice, readable note with real answers and no heavy machinery. I would send it to a serious referee; it deserves publication, probably more or less as is after a typo-fix round.\n\nBest regards,\n[Your name]","headline":"Two open questions in anti-invariant cohomology answered by elementary explicit constructions; the paper is correct and worth a serious referee.","tokens_in":13536,"tokens_out":7730,"would_cite":true,"duration_ms":78780,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C55","53C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs almost complex structures whose spaces of closed anti-invariant 2-forms range from infinite-dimensional to zero-dimensional on R^4, and maximal or arbitrarily large on compact manifolds.","keywords":["almost complex structure","anti-invariant form","anti-invariant cohomology","non-integrable almost complex structure","Kodaira-Thurston manifold","closed 2-forms","de Rham cohomology"],"falsifier":"On the Kodaira-Thurston manifold with $J_{\\lambda,\\mu}$, compute the de Rham cohomology classes of $\\theta_1$ and $e^{\\lambda(x_4)}\\theta_2$; if these two classes are linearly dependent, the argument for $h^-_J = 2$ would need a third independent closed anti-invariant form to reach dimension 2, and the asserted maximality would fail without one. Alternatively, for any compactly supported $f$, solve the PDE system (7) and look for a nonzero solution $a$; the proof of Theorem 3.8 asserts $a = 0$, so a single nonzero solution would disprove that theorem.","tokens_in":12684,"feed_emoji":"📐","tokens_out":10137,"duration_ms":87779,"temperature":0.7,"pith_summary":"This paper studies the vector space of closed real 2-forms that are anti-invariant under an almost complex structure $J$, and asks how large that space can be on noncompact and compact manifolds. The authors construct non-integrable almost complex structures on $\\mathbb{R}^4$ whose space of closed $J$-anti-invariant forms is infinite-dimensional, and other structures, differing by compactly supported perturbations, for which the space is exactly one-dimensional or zero-dimensional. In the compact setting they build a two-parameter family on the Kodaira-Thurston manifold with anti-invariant cohomology of maximum possible dimension $h^-_J = 2$, and six-dimensional compact examples with arbitrarily large anti-invariant cohomology. Together these examples show that the finiteness and maximality phenomena known in dimension four depend essentially on compactness and on the dimension being four.","feed_headline":"On R^4 non-integrable structures admit infinitely many closed forms","feed_subtitle":"A single family of almost complex structures shows the count of closed anti-invariant forms jumps from infinite to one to zero.","key_machinery":"The machine that drives the noncompact examples is the family $J_f$ on $\\mathbb{C}^2$, defined by twisting the standard complex structure through an angle controlled by a smooth function $f$; a $J$-anti-invariant form is a real 2-form $\\alpha$ satisfying $\\alpha(JX,JY) = -\\alpha(X,Y)$. In the global coframe attached to $J_f$, every anti-invariant 2-form is $a\\beta + b\\gamma$, and closedness is equivalent to a linear system of PDEs in $a,b$, rewritten as a perturbed Cauchy-Riemann equation for $w = a - ib$. Choosing $f$ switches this PDE from a regime admitting infinitely many independent exponential solutions ($f = x_2$) to a regime where harmonic-function arguments force $a = 0$ and $b$ constant (compactly supported $f$). In the compact arguments the load-bearing identity is the inequality $h^-_J(M) \\leq b_+(M)$ for compact almost complex 4-manifolds, matched with two explicit closed forms $\\theta_1$ and $e^{\\lambda(x_4)}\\theta_2$ on the Kodaira-Thurston manifold, and the product structure on $\\Sigma_g \\times \\Sigma_g \\times \\mathbb{T}^2$ for the large-dimensional examples.","core_discovery":"The central discovery is that the size of the space of closed $J$-anti-invariant 2-forms is governed by the global behavior of a function $f$ used to define the almost complex structure, not merely by integrability. For the explicit family $J_f$ on $\\mathbb{C}^2$, closed anti-invariant forms correspond to solutions of a linear PDE system; choosing $f(x_1,x_2,y_1,y_2) = x_2$ yields infinitely many linearly independent exponential solutions, giving an infinite-dimensional space, while choosing $f$ with compact support forces the coefficient $a$ to vanish and leaves only constant multiples of one form, giving $h^-_{J_f} = 1$; a glueing construction then gives $h^-_{J} = 0$. On compact manifolds the paper proves that the Kodaira-Thurston manifold carries non-integrable almost complex structures with $h^-_J = 2$, the maximum allowed by the bound $h^-_J \\leq b_+$, and that products $\\Sigma_g \\times \\Sigma_g \\times \\mathbb{T}^2$ carry non-integrable structures with $h^-_J \\geq 2g^2$, so the anti-invariant cohomology can be made arbitrarily large in dimension six.","pith_inferences":["An implication the paper leaves implicit is that the exponential solutions for $f = x_2$ are almost Kähler local models; transplanting them into compact symplectic manifolds might force lower bounds on anti-invariant cohomology in wider families, a testable extension.","The contrast between infinite-dimensional and one-dimensional regimes suggests that growth conditions on $f$ control the dimension of $Z^-_J$; testing intermediate polynomial-growth functions could interpolate between the two extremes.","Since the six-dimensional examples can make $h^-_J$ arbitrarily large, a natural question the paper does not address is whether any upper bound in terms of Betti numbers exists for $h^-_J$ in dimensions above four."],"forward_implications":["On $\\mathbb{R}^4$, non-integrable almost complex structures can admit infinitely many linearly independent closed $J$-anti-invariant forms, so the finiteness of $Z^-_J$ on compact manifolds does not extend to the noncompact setting.","A compactly supported perturbation of the standard complex structure on $\\mathbb{C}^2$ can cut the space of closed anti-invariant forms from infinite-dimensional to exactly one-dimensional, supporting the picture that anti-invariant forms typically vanish under non-integrable perturbation.","The Kodaira-Thurston manifold carries non-integrable almost complex structures with $h^-_J = 2$, giving an affirmative answer to the question whether the previously known family of examples with $h^-_J = 2$ is exhaustive.","In dimension six there are compact non-integrable almost complex manifolds with $h^-_J \\geq 2g^2$, so the four-dimensional maximality bound is genuinely special to dimension four.","The constructed structures are almost Kähler, compatible with symplectic forms, so these extremes occur within symplectic geometry rather than in exotic non-symplectic settings."],"supporting_citations":[{"why":"Supplies the inequality $h^-_J(M) \\leq b_+(M)$ for compact almost complex 4-manifolds, which converts the two explicit closed forms on the Kodaira-Thurston manifold into the maximality statement $h^-_J = 2$.","marker":"[5]"},{"why":"States the conjectures and the question that motivate the paper, and provides the earlier family of non-integrable structures with $h^-_J = 2$ to which the Kodaira-Thurston examples are compared.","marker":"[6]"},{"why":"Introduces the $J$-anti-invariant and $J$-invariant cohomology groups and the notion of $C^\\infty$-pure-and-full almost complex structures that give the paper its objects of study.","marker":"[13]"},{"why":"Provides the elliptic operator $E$ and the unique continuation principle used to conclude that a closed anti-invariant form vanishing on an open set must vanish identically.","marker":"[10]"},{"why":"Gives the result that integrable structures agreeing with the standard complex structure outside a compact set are biholomorphic to $\\mathbb{C}^2$, providing the contrast $h^-_J = \\infty$ for the zero-dimensional example.","marker":"[17]"},{"why":"Invoked to show that the compactly supported almost complex structures, being tamed by a symplectic form, are actually compatible with a symplectic form, so the examples are almost Kähler.","marker":"[8]"}],"fun_headline_variants":["Closed anti-invariant forms: from infinite to zero on R^4","Non-integrable almost complex structures yield huge cohomology","Arbitrarily large anti-invariant cohomology in six dimensions","Anti-invariant forms: one family shows infinite, 1, or 0","R^4 structures with anti-invariant cohomology infinite or trivial"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is an external inequality stating that in dimension four the anti-invariant cohomology cannot exceed the number of self-dual harmonic 2-forms; the maximality result for the Kodaira-Thurston manifold collapses to a lower bound if that inequality is false.","fun_headline_variants_meta":{"raw":{"variants":["Closed anti-invariant forms: from infinite to zero on R^4","Non-integrable almost complex structures yield huge cohomology","Arbitrarily large anti-invariant cohomology in six dimensions","Anti-invariant forms: one family shows infinite, 1, or 0","R^4 structures with anti-invariant cohomology infinite or trivial"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00019,"raw_usage":{"total_tokens":1323,"prompt_tokens":912,"completion_tokens":411,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":528,"completion_tokens_details":{"reasoning_tokens":314}},"tokens_in":528,"tokens_out":411,"duration_ms":4695,"temperature":1.0,"reasoning_tokens":314,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:29:33.778249+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the Kodaira-Thurston manifold with $J_{\\lambda,\\mu}$, compute the de Rham cohomology classes of $\\theta_1$ and $e^{\\lambda(x_4)}\\theta_2$; if these two classes are linearly dependent, the argument for $h^-_J = 2$ would need a third independent closed anti-invariant form to reach dimension 2, and the asserted maximality would fail without one. Alternatively, for any compactly supported $f$, solve the PDE system (7) and look for a nonzero solution $a$; the proof of Theorem 3.8 asserts $a = 0$, so a single nonzero solution would disprove that theorem.","supporting_citations":[{"cited_title":"Drˇ aghici, T.-J","cited_arxiv_id":null,"evidence_quote":"Supplies the inequality $h^-_J(M) \\leq b_+(M)$ for compact almost complex 4-manifolds, which converts the two explicit closed forms on the Kodaira-Thurston manifold into the maximality statement $h^-_J = 2$."},{"cited_title":"Drˇ aghici, T.-J","cited_arxiv_id":null,"evidence_quote":"States the conjectures and the question that motivate the paper, and provides the earlier family of non-integrable structures with $h^-_J = 2$ to which the Kodaira-Thurston examples are compared."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the $J$-anti-invariant and $J$-invariant cohomology groups and the notion of $C^\\infty$-pure-and-full almost complex structures that give the paper its objects of study."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the elliptic operator $E$ and the unique continuation principle used to conclude that a closed anti-invariant form vanishing on an open set must vanish identically."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the result that integrable structures agreeing with the standard complex structure outside a compact set are biholomorphic to $\\mathbb{C}^2$, providing the contrast $h^-_J = \\infty$ for the zero-dimensional example."},{"cited_title":"Gromov, Pseudo holomorphic curves in symplectic mani folds, Invent","cited_arxiv_id":null,"evidence_quote":"Invoked to show that the compactly supported almost complex structures, being tamed by a symplectic form, are actually compatible with a symplectic form, so the examples are almost Kähler."}],"review_version":1}