{"id":"68a77e11-4a33-453a-a4fb-2a6a29098bfd","arxiv_id":"2411.17978","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The inverse Monge-Ampere flow converges under bounds such as bounded sup or alpha-invariant above n/(n+1), and diverging trajectories generate proper Nadel multiplier ideal sheaves.","lead":"This paper gives new convergence criteria for the inverse Monge-Ampere flow on Kähler manifolds, including the twisted case, and shows that when no Kähler-Einstein metric exists the flow produces Nadel multiplier ideal sheaves. It also establishes a linear growth bound for the potential and constructs asymptotic geodesic rays, extending results known for the Kähler-Ricci flow.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.6 transfers [9]'s long-time existence and a priori estimates to β≥0 by a one-line 'carry over verbatim'; every later convergence, Nadel-sheaf, and geodesic-ray statement depends on this unverified step.","rationale":"I read the paper in good faith: its main results—Theorem 1.1 (boundedness criteria imply convergence), Theorem 1.2 (Nadel multiplier ideal sheaves), Theorem 1.3 (linear C0 bound), and Theorem 1.4 (geodesic rays)—are coherent and, if Theorem 2.6 holds, they follow from the given arguments with some compressed steps. The scalar parabolic equation (2.4) has the same form for β=0 and β≥0, so the carry-over is plausible; nevertheless, Theorem 2.6 is the unique place where the entire paper imports long-time existence and all parabolic regularity from [9] without proof. The reader correctly identifies this as the weakest assumption. I do not see a second, independent objection that is more load-bearing: the semicontinuity step in Theorem 5.4 is standard in the direction used here, the normalization issue for ψ_j is harmless because sup φ_j and the average with respect to ωφ^n differ by a uniformly bounded amount, and the Nadel vanishing applies to the line bundle -(⌊α⌋+1)K_X before twisting by K_X. Thus the central claim should be accepted conditionally on a complete verification of the twisted a priori estimates. Since the reader's verdict is already CONDITIONAL, my stress-test does not change the verdict, and I set verdict_should_be to UNCHANGED.","tokens_in":15973,"tokens_out":37859,"duration_ms":360401,"concrete_test":"Independently derive the maximum-principle and a priori estimates of [9, §4] for the equation (2.4) with Ric(ωφ)=ωφ+β+√−1∂∂ρ, tracking every occurrence of β through the C^0, C^1, and C^2 estimates. In particular, recompute the evolution of Δφ and the quantity log Tr_{ω0}ωφ using (2.3), and check that all extra β-terms have the correct sign or are controlled by the semipositivity of β. If the twisted estimates go through unchanged, Theorem 2.6 is proved and the concern is resolved; if any estimate uses the untwisted identity Ric(ωφ)=ωφ+√−1∂∂ρ in an essential way, the main theorems would need a separate proof for β>0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central risk is in Theorem 2.6 (Section 2), where the twisted inverse MA flow with β≥0 is asserted to exist for all times and to converge when a twisted KE metric exists. The proof is: 'the flow looks the same for both twisted and non-twisted cases, F(φ) is convex, the arguments from [9, Theorem 4.6 and Theorem 4.11] together with [1,4,5] carry over verbatim.' The potential equation (2.4) is formally independent of β, and β≥0 helps the convexity argument in Lemma 2.5, so the statement is plausible. But the estimates in [9] are obtained for the untwisted inverse MA flow on Fano manifolds; they involve not only the scalar equation but also evolution identities for ρ, the metric evolution (2.3) containing β, and maximum-principle/Sobolev arguments. Theorem 2.6 does not show which of those estimates are β-independent, which require only β≥0, and which would need new hypotheses. Since Theorems 1.1–1.4 all begin with a solution of the twisted inverse MA flow, every subsequent claim—the equivalence criteria, the Nadel sheaf construction, the linear L∞ bound, and the geodesic ray—rests on this unverified regularity transfer. This is a proof-gap concern rather than a claim that the theorem is false; the concern would be resolved by a complete verification of the twisted estimates.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the inverse Monge-Ampère flow on a compact Kähler manifold X in the twisted setting where [ω0]+[β]=c1(X) with β≥0. Under the assumption H^0(X,T^{1,0}X)=0, Theorem 1.1 asserts the equivalence of several uniform bounds along the flow (sup φ, average ∫φ, J and d1, I, and L^p integrability of e^{-φ}) and that each of these bounds implies convergence to the twisted Kähler-Einstein metric; it also lists sufficient criteria involving the α-invariant, oscillation, inf φ, and d_p. Theorem 1.2 claims that if no twisted Kähler-Einstein metric exists, then normalized potentials ψ_j admit an L^1 limit ψ_∞ whose multiplier ideal sheaf I(αψ_∞) is proper and satisfies the cohomology vanishing (1.6). Theorem 1.3 claims the linear growth bound ||φ||_{C0}≤M(t+1), and Theorem 1.4 applies this bound to construct weakly asymptotic d_p-geodesic rays. The proofs rely on convexity of F, monotonicity of M, pluripotential compactness, and imported estimates from [9], [2], [20], and [13].","tokens_in":16246,"tokens_out":21002,"duration_ms":178208,"significance":"If correct, the paper would extend the Collins–Hisamoto–Takahashi inverse Monge-Ampère flow to the twisted Kähler-Einstein setting and provide a pluripotential-theoretic analogue of the Kähler-Ricci flow results of Phong–Sesum–Sturm and Rubinshtein, avoiding Perelman estimates and uniform Sobolev inequalities. The paper is explicit about this methodological choice and does not fit parameters or rely on invented entities; the α-invariant, Nadel vanishing, and the Trudinger inequality are external benchmarks. The claims are coherent and plausible, and the recovery of results from [27] and [29] is a genuine application rather than a restatement. However, the verification of the flow-regularity input and of the Nadel-vanishing application is not yet at the standard required for the central theorems.","major_comments":[{"comment":"The long-time existence and convergence for the twisted inverse Monge-Ampère flow is asserted by saying that the arguments from [9, Theorems 4.6 and 4.11] \"carry over verbatim\" because F is convex and the equation for the potential is formally independent of β. This is not a proof of the a priori estimates: the metric evolution (2.3), the evolution of ρ, the maximum-principle and Sobolev steps, and the spectral estimate for L_ρ all involve the twist β through Ric(ω_φ)=ω_φ+β+√-1∂∂ρ. The paper does not identify which estimates are β-independent, which require only β≥0, and which would need new hypotheses. Since every later theorem (1.1–1.4) assumes a solution of this twisted flow, this transfer is load-bearing and must be supplied in detail.","section":"Section 2, Theorem 2.6"},{"comment":"The cohomology vanishing (5.4), H^q(X,-⌊α⌋K_X⊗I(αψ_∞))=0, does not follow from the Nadel vanishing theorem as stated in Theorem 5.2. For α∈(n/(n+1),1), ⌊α⌋=0 and -⌊α⌋K_X is trivial, whereas Nadel vanishing requires a line bundle L with curvature current F_H≥εω, so the proposed vanishing is not a direct consequence of positive curvature. The proof must specify the line bundle L and the singular metric on it for which I(αψ_∞) is the multiplier ideal sheaf, and verify the curvature hypothesis.","section":"Section 5, Theorem 5.4"},{"comment":"The statement of Theorem 1.3 is unconditional, but the Introduction says the linear bound is proved \"under the additional assumption that ω_φ^n is exponentially bounded along the flow\". The proof uses the Trudinger inequality (6.9) to bound log||u||_1 by C/V∫(-ψ)ω_ψ^n+C; this inequality is not valid for arbitrary large exponent B without extra hypotheses or a known exponential integrability bound. In addition, the Moser-iteration step in (6.4)–(6.8) compresses the interpolation argument for ||u||_{1-δ} and the passage from (6.7) to (6.8); the latter requires the elementary bound ||u||_{1-δ}≤||u||_{C0}, which is not stated. If the result is intended to be unconditional, the hypothesis must be removed or the Trudinger input justified; as written, the proof is too terse for a load-bearing claim.","section":"Section 6, Theorem 6.2 (Theorem 1.3)"},{"comment":"In the proof of (2)=>(1), after obtaining L^1 convergence of φ_j, the paper states that the \"effective version of semicontinuity theorem\" guarantees that e^{-pφ_j} converge to e^{-pφ_j} in L^1 (the limit should be e^{-pφ_∞}). This convergence is asserted without a reference or a uniform integrability argument, and it is then used to conclude that c(t) is uniformly bounded and F(φ_j)≥-B. Since this is the step that converts L^1 convergence into the coercivity input for strong convergence, it should be justified explicitly.","section":"Section 4, Proposition 4.1"}],"minor_comments":[{"comment":"\"If X does not admit the KE metric in c_1(X)\" should read \"twisted Kähler-Einstein metric\", since the theorem concerns the twisted equation Ric(ω_φ)=ω_φ+β.","section":"Section 5, proof of Theorem 5.4"},{"comment":"Equation (4.8) uses C0=E(φ) without mentioning that E is normalized to zero or constant along the flow; if E≠0, the constant should be made explicit.","section":"Section 4, Proposition 4.2"},{"comment":"The Introduction states that Theorem 1.3 is proved under an additional exponential volume-form bound, but the theorem statement in Section 6 omits this hypothesis; the inconsistency should be resolved.","section":"Section 1 and Section 6"},{"comment":"There are several typos and formatting issues, including \"Holder continious\" in Section 1, extra closing parentheses in the exponents of (5.5) and (5.6), and the repeated \"converge to e^{-pφ_j}\" in Proposition 4.1, which should be \"e^{-pφ_∞}\".","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the unverified transfer in Theorem 2.6; if the twisted estimates fail, the paper's central claims do not follow as stated. The author should be encouraged either to supply a complete proof of the twisted regularity results or to state them explicitly as assumptions. The remaining gaps in the Nadel-vanishing application and in Theorem 1.3's hypotheses are also load-bearing but appear fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a read if you work on Kähler flows or multiplier ideal sheaves. The paper gives the first convergence criteria for the twisted inverse Monge-Ampere flow, and uses them to produce Nadel obstructions and a linear L∞ bound. The strategy is sound: since no Perelman estimates are available, the author leans on pluripotential theory and the monotonicity of the Mabuchi functional. Theorems 1.1–1.4 are new for this flow, and the recovery of the analogous Kähler-Ricci flow results from Phong-Sesum-Sturm and Rubinshtein is natural.\n\nThe main issue is Theorem 2.6. Long-time existence and a priori estimates for the twisted flow are asserted to follow because the arguments from Collins–Hisamoto–Takahashi 'carry over verbatim.' That is plausible but not proved. The scalar equation (2.4) is formally independent of β, but the metric evolution (2.3) contains β, and the estimates in [9] rely on evolution identities, Sobolev inequalities, and maximum principles where β could matter. Every subsequent theorem assumes a solution to this flow, so the gap is load-bearing. The fix is concrete: either write out the twisted estimates or point to precise statements in Berman–Berndtsson or Berman–Darvas–Lu that cover β≥0.\n\nTwo other spots are compressed. In Theorem 1.3, the Moser iteration has a step bounding e^{-(A+1) inf ψ} by norms of u that is too terse; it can be unpacked, but referees will want that equation. In Theorem 1.2, the passage from unboundedness to diverging L^α integrals is sketched. These are addressable, not signs of a false central claim.\n\nI agree with the conditional verdict. If Theorem 2.6 holds up, this is a solid contribution. The citation pattern is honest, with no self-citation inflation, and the author explicitly flags the absence of Perelman estimates and works around them. I would send this to a knowledgeable referee. The gaps are real but repairable.","headline":"Plausible and likely correct extension of the inverse MA flow to the twisted Fano setting, but the unproved transfer of long-time estimates in Theorem 2.6 is load-bearing.","tokens_in":16824,"tokens_out":2730,"would_cite":true,"duration_ms":24604,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C55","32Q20","32W20","53E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that natural bounds on the potentials of the inverse Monge-Ampere flow force convergence to the twisted Kähler-Einstein metric, and that without such a metric the flow generates a proper multiplier ideal sheaf with…","keywords":["inverse Monge-Ampere flow","twisted Kähler-Einstein metric","multiplier ideal sheaf","alpha-invariant","pluripotential theory","geodesic ray","C0 estimate","Fano manifold"],"falsifier":"Take any compact Kähler manifold with no holomorphic vector fields and a semipositive $\\beta$ with [omega0]+[$\\beta$]=c1(X) for which no twisted Kähler-Einstein metric exists, and solve the twisted inverse Monge-Ampere flow; if sup_X phi or the L^p integral of $e^{{-p phi}}$ stays bounded, or if the flow fails to exist for all time, the paper's main claims fail.","tokens_in":15729,"feed_emoji":"📐","tokens_out":11049,"duration_ms":85917,"temperature":0.7,"pith_summary":"This paper extends the inverse Monge-Ampere flow, a gradient flow originally introduced for untwisted Fano manifolds, to the twisted setting where $[\\omega_0]+[\\beta]=c_1(X)$ with $\\beta$ semipositive. It establishes that, on manifolds without holomorphic vector fields, convergence to the twisted Kähler-Einstein metric is equivalent to any of five natural bounds on the evolving potentials, and that several other familiar bounds, including the $\\alpha$-invariant bound, also force convergence. If no twisted Kähler-Einstein metric exists, the flow cannot satisfy those bounds, and a normalized subsequence converges in $L^1$ to a potential whose multiplier ideal sheaf is proper and satisfies a Nadel-type vanishing theorem. The paper also proves a linear growth bound $\\|\\varphi\\|_{C^0}\\le M(t+1)$ and uses it to produce nontrivial $d_p$-geodesic rays asymptotic to diverging trajectories. These results matter because they remove the assumption that a Kähler-Einstein metric already exists and replace the unavailable gradient estimates of the Ricci-flow setting with pluripotential theory.","feed_headline":"Inverse Monge-Ampere flow converges under any of five bounds","feed_subtitle":"No Kähler-Einstein metric? The flow leaves a proper multiplier ideal sheaf with vanishing cohomology.","key_machinery":"The central mechanism is the inverse Monge-Ampere flow $\\dot{\\varphi}=1-e^{\\rho}$, where $\\rho$ is the Ricci potential of the twisted metric $\\omega_\\varphi$ defined by $\\mathrm{Ric}(\\omega_\\varphi)=\\omega_\\varphi+\\beta+\\sqrt{-1}\\partial\\bar\\partial\\rho$, together with the monotonicity of the twisted Mabuchi energy and the convexity of the functional $F$ along the flow. The crucial identity is the $\\alpha$-invariant inequality $((n+1)\\alpha-n)\\sup_X\\varphi \\le \\log\\left(\\frac{1}{V}\\int_X e^{-\\alpha(\\varphi-\\sup_X\\varphi)}\\,\\omega_0^n\\right)+C$, which converts a bound on the $\\alpha$-invariant into a uniform upper bound on $\\sup_X\\varphi$. For the multiplier ideal sheaf half, the normalized potentials $\\psi_j$ carry the $L^1$ limit, and the effective semicontinuity theorem plus the multiplier ideal sheaf vanishing theorem produce the proper ideal sheaf and the cohomology vanishing. The $L^\\infty$ bound rests on a $C^2$ estimate $\\log\\mathrm{Tr}_{\\omega_0}\\,\\omega_\\varphi \\le C+A(\\varphi-\\inf_X\\varphi)+t-\\inf_X\\varphi$ combined with Sobolev iteration and a sharp $L^1$-energy inequality that bounds the $L^1$ norm of $e^{-B\\psi}$ by the Monge-Ampere energy.","core_discovery":"On a compact Kähler manifold with $H^0(X,T^{1,0}X)=0$ and $[\\omega_0]+[\\beta]=c_1(X)$, $\\beta\\ge 0$ semipositive, the paper claims that along the inverse Monge-Ampere flow the following are equivalent: uniform upper bound of $\\sup_X\\varphi$, uniform upper bound of $\\frac{1}{V}\\int_X\\varphi\\,\\omega_0^n$, uniform bound of $J(\\varphi)$ or $d_1(0,\\varphi)$, uniform bound of $I(\\varphi)$, and uniform bound of $\\frac{1}{V}\\int_X e^{-p\\varphi}\\,\\omega_0^n$ for some $p>1$; each implies the flow converges to the unique twisted Kähler-Einstein metric. It further claims that $\\alpha(X,\\omega_0)>\\frac{n}{n+1}$, bounded oscillation, lower bound on $\\inf_X\\varphi$, or bounded $d_p$-distance for $p>1$ each force the same convergence. If no twisted Kähler-Einstein metric exists, the paper concludes both $\\|\\varphi\\|_{C^0}$ and the average are unbounded along the flow, and for $\\alpha>\\frac{n}{n+1}$ a subsequence of the normalized potentials $\\psi_j=\\varphi_j-\\frac{1}{V}\\int_X\\varphi_j\\,\\omega_{\\varphi_j}^n$ converges in $L^1$ to $\\psi_\\infty$ with $\\mathcal{I}(\\alpha\\psi_\\infty)$ a proper multiplier ideal sheaf and $H^q(X,-\\lfloor\\alpha\\rfloor K_X\\otimes\\mathcal{I}(\\alpha\\psi_\\infty))=0$ for all $q\\ge 1$. Separately, the paper proves $\\|\\varphi\\|_{C^0}\\le M(t+1)$ along the flow, and from this constructs nontrivial $d_p$-geodesic rays weakly asymptotic to diverging trajectories, on which $F$ is convex and decreasing, whose normalized limit $\\varphi_\\infty$ satisfies $\\int_X e^{-\\frac{n}{n+1}\\varphi_\\infty}\\,\\omega_0^n=+\\infty$.","pith_inferences":["The paper does not say this, but the equivalence of the five bounds suggests that global convergence of the inverse Monge-Ampere flow is controlled entirely by finite-energy classes; one could test numerically on toric Fano surfaces whether the $d_1$ bound is the easiest to verify.","An extension the paper leaves implicit is to replace the smooth semipositive form $\\beta$ by a positive current and ask whether the same five-bounds equivalence survives; the pluripotential tools used here are the natural language for that generalization.","If the linear growth estimate is sharp, it would imply that the divergence rate of normalized potentials is at most linear; comparing the constant $M$ in the growth bound with the alpha-invariant could give a quantitative slope for the destabilizing geodesic ray."],"forward_implications":["If any of the five equivalent bounds holds along the flow, the twisted inverse Monge-Ampere flow converges smoothly to the unique twisted Kähler-Einstein metric, so existence of that metric is detected by a single uniform bound on potentials.","On any manifold without a twisted Kähler-Einstein metric, $\\sup_X\\varphi$ and the average potential are necessarily unbounded along the flow, so the flow provides an explicit destabilizing mechanism.","For $\\alpha>\\frac{n}{n+1}$, a diverging inverse Monge-Ampere trajectory produces a proper multiplier ideal sheaf $\\mathcal{I}(\\alpha\\psi_\\infty)$ with vanishing $H^q(X,-\\lfloor\\alpha\\rfloor K_X\\otimes\\mathcal{I}(\\alpha\\psi_\\infty))$ for all $q\\ge 1$, recovering the standard obstruction to Kähler-Einstein metrics from the flow itself.","The linear bound $\\|\\varphi\\|_{C^0}\\le M(t+1)$ gives a concrete qualitative growth estimate for the inverse Monge-Ampere flow, and it allows construction of nontrivial $d_p$-geodesic rays asymptotic to the flow with an integrability failure at exponent $\\frac{n}{n+1}$.","The paper recovers, in the twisted inverse-flow setting, the multiplier ideal sheaf results previously known for the Kähler-Ricci flow."],"supporting_citations":[{"why":"Supplies the inverse Monge-Ampere flow, its long-time regularity, convexity of the functional F, and the C^2 estimate that the twisted case inherits.","marker":"[9]"},{"why":"Supplies the definition of multiplier ideal sheaves, the effective semicontinuity result, and the vanishing theorem used to obtain the cohomology statement.","marker":"[15]"},{"why":"Supplies the functional inequality used to control the L^1 norm of e^{-Bpsi} in the proof of the linear C^0 bound.","marker":"[20]"},{"why":"Supplies the geodesic-ray construction and asymptotic properties that the paper adapts to the inverse flow.","marker":"[13]"},{"why":"Supplies the convexity and uniqueness argument used to identify the limit as the unique twisted Kähler-Einstein metric.","marker":"[5]"},{"why":"Supplies the Kähler-Ricci flow analogue that motivates the five-bounds equivalence and is recovered by the paper's results.","marker":"[27]"},{"why":"Supplies the construction of multiplier ideal sheaves from diverging Ricci-flow trajectories that the paper adapts to the inverse flow.","marker":"[29]"},{"why":"Supplies the original multiplier ideal sheaf existence criterion and vanishing theorem underlying the sheaf obstruction.","marker":"[24]"}],"fun_headline_variants":["Inverse Monge-Ampere flow: five bounds, one convergence","Flow convergence without assuming Kähler-Einstein metric","No Kähler-Einstein? Flow leaves multiplier ideal sheaves","Five equivalent bounds force inverse Monge-Ampere convergence","Inverse Monge-Ampere flow: convergence or Nadel sheaves"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the twisted inverse Monge-Ampere flow with a semipositive beta exists for all time and inherits the standard a priori estimates from the untwisted case, since the paper does not reprove that regularity.","fun_headline_variants_meta":{"raw":{"variants":["Inverse Monge-Ampere flow: five bounds, one convergence","Flow convergence without assuming Kähler-Einstein metric","No Kähler-Einstein? Flow leaves multiplier ideal sheaves","Five equivalent bounds force inverse Monge-Ampere convergence","Inverse Monge-Ampere flow: convergence or Nadel sheaves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000854,"raw_usage":{"total_tokens":3779,"prompt_tokens":1085,"completion_tokens":2694,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":701,"completion_tokens_details":{"reasoning_tokens":2606}},"tokens_in":701,"tokens_out":2694,"duration_ms":19718,"temperature":1.0,"reasoning_tokens":2606,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:39:00.808078+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any compact Kähler manifold with no holomorphic vector fields and a semipositive $\\beta$ with [omega0]+[$\\beta$]=c1(X) for which no twisted Kähler-Einstein metric exists, and solve the twisted inverse Monge-Ampere flow; if sup_X phi or the L^p integral of $e^{{-p phi}}$ stays bounded, or if the flow fails to exist for all time, the paper's main claims fail.","supporting_citations":[{"cited_title":"Differential Geom","cited_arxiv_id":null,"evidence_quote":"Supplies the inverse Monge-Ampere flow, its long-time regularity, convexity of the functional F, and the C^2 estimate that the twisted case inherits."},{"cited_title":"Demailly, J.Kollar.Semi-continuity of complex singularity exponents and K¨ ahler-Einsten metrics on Fano orbifoldsAnn","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of multiplier ideal sheaves, the effective semicontinuity result, and the vanishing theorem used to obtain the cohomology statement."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the functional inequality used to control the L^1 norm of e^{-Bpsi} in the proof of the linear C^0 bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the geodesic-ray construction and asymptotic properties that the paper adapts to the inverse flow."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Kähler-Ricci flow analogue that motivates the five-bounds equivalence and is recovered by the paper's results."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the construction of multiplier ideal sheaves from diverging Ricci-flow trajectories that the paper adapts to the inverse flow."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the original multiplier ideal sheaf existence criterion and vanishing theorem underlying the sheaf obstruction."}],"review_version":1}