{"id":"9f5998ec-8eb1-42d1-a523-99897e276ca3","arxiv_id":"2607.05575","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Virtual integrals over GIT quotients X//G equal Jeffrey–Kirwan residues of virtual integrals over T-fixed loci, for perfect obstruction theories and oriented (−2)-shifted symplectic structures, in cohomology and K-theory.","lead":"The paper proves Jeffrey–Kirwan localisation formulas for virtual fundamental classes of GIT quotients, in both cohomology and K-theory. This gives a practical way to compute virtual integrals on moduli spaces by reducing them to fixed-locus contributions.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim is a virtual extension of a classical localisation theorem whose geometric skeleton (algebraic cut + ABBV + residue vanishing + abelianisation) is already well-tested. The only step without a smooth counterpart is the control of virtual normal-bundle weights on the new fixed loci; that control is supplied by explicit, elementary calculations (Propositions 6.3 and 11.5) that do not rely on any unproved black box. The remainder of the argument (descent of obstruction theories, projectivity under weak η-semiprojectivity, toric JK residues, Martin–Maddock abelianisation) is standard and carefully documented. Consequently the reader’s ACCEPT verdict with moderate confidence is appropriate; no adjustment is warranted. The concrete test above is a minimal sanity check that would catch a miscalculation in the weight description if one existed.","tokens_in":62745,"tokens_out":630,"duration_ms":6404,"concrete_test":"Take the standard C* action on P^1 with the trivial perfect obstruction theory (so virtual = ordinary). Form the algebraic cut by a 1-dimensional cone Σ = Cone(ψ) with ψ > 0, apply the localisation formula (40), and verify by hand that the residue of the new-fixed-locus contribution at μ = 0 vanishes while the old fixed-point contributions reproduce the classical JK formula for P^1//C*. If the numerical identity fails, the weight-control claim is false; if it holds, the load-bearing step is confirmed in the simplest non-trivial case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly isolates the most delicate step: vanishing of residues from new fixed loci of the algebraic cut, which rests on the virtual-weight control of Propositions 6.3 and 11.5. Those propositions follow directly from the explicit description of the induced obstruction theory on the cut (diagrams (33) and (80) together with Lemmas 4.5 and 6.5 / Proposition 11.3). The weights of the first summand lie in span(σ) because the stratum X_σ is fixed by the complementary torus; the remaining weights are the projected generators of the edges of Σ ∩ τ_p by the same calculation that holds classically. Once those poles are known, the η-wide condition (Lemma 8.3) places p+ε outside the positive cone of the normalised weights, so Proposition 7.3 kills the residue. No internal contradiction or hidden circularity appears; the argument is the natural virtual extension of the Jeffrey–Kogan / Edidin–Graham strategy and is written with matching care in both the perfect and (−2)-shifted settings.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves virtual Jeffrey–Kirwan localisation formulae that express integrals (and K-theoretic Euler characteristics) over virtual cycles of GIT quotients X//G in terms of residues of integrals over virtual cycles of T-fixed loci X^T. The results cover both perfect obstruction theories (Behrend–Fantechi/Li–Tian) and oriented (−2)-shifted symplectic structures (Borisov–Joyce/Oh–Thomas), in cohomology and K-theory, for reductive G acting on polarised projective-over-affine Deligne–Mumford stacks with compact quotient (under weak η-semiprojectivity). The method constructs an algebraic cut X_Σ by a simplicial cone, induces the corresponding virtual structure on the cut, applies virtual ABBV localisation, and uses η-wide cones together with control of virtual normal-bundle weights to kill contributions of “new” fixed loci; nonabelian formulae are then deduced by a Martin–Maddock abelianisation.","tokens_in":62973,"tokens_out":753,"duration_ms":6508,"significance":"Virtual JK localisation is a natural and useful tool for enumerative geometry of moduli spaces that arise as GIT quotients (or have such presentations). Completing the square with virtual ABBV for both perfect and CY4-type virtual cycles, and supplying matching K-theoretic formulae, is a substantial contribution. The algebraic-cut construction for higher-rank tori on (possibly reducible, non-reduced, noncompact) DM stacks, the descent of obstruction theories and orientations, and the careful residue-vanishing argument for new fixed loci are technically nontrivial and carefully written. The results sit squarely inside the convex hull of classical JK and virtual ABBV, but the virtual and stacky extensions are new and should be immediately usable.","major_comments":[],"minor_comments":[{"comment":"The introduction’s “complete the square” framing is clear, but a short roadmap paragraph at the end of §1 listing which sections treat perfect vs (−2)-shifted cases (and cohomological vs K-theoretic) would help the reader navigate the parallel arguments.","section":null},{"comment":"Notation (2) on maximalist scheme structure for irreducible components is important for reducible X; a brief forward pointer when it is first used in the cut construction (§3) would reduce the chance of misreading.","section":null},{"comment":"In §7 the JK residue is defined via Brion–Vergne; a one-line comparison with the more common Jeffrey–Kirwan residue operator (or a pointer to the dictionary already in Prop. 7.2) would help readers coming from the symplectic literature.","section":null},{"comment":"Figures 3, 6 and 8 are helpful; ensuring that the colour coding of “old / new / special” fixed loci is consistent across all three would improve readability.","section":null},{"comment":"A few minor typos (e.g. “F antechi” in the §4 heading, occasional missing spaces around //) should be cleaned in copy-editing.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is long and technical but the proofs appear complete and the weakest step (vanishing for new fixed loci) is handled carefully via Propositions 6.3/11.5 and Lemma 8.3. I see no load-bearing gap. Fit for a top algebraic-geometry journal is good; the paper is closer to Inventiones/JAMS level in technical ambition than to a short note, so length is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper delivers the virtual Jeffrey–Kirwan formulae that people in DT and moduli theory have been using informally or deriving case-by-case. The main theorems (abelian and nonabelian, perfect and (−2)-shifted, cohomology and K-theory, with noncompact X allowed when the quotient is compact) are new as a package; the ingredients (virtual ABBV, algebraic cuts, Martin–Maddock abelianisation, Brion–Vergne residues) were already there, but combining them cleanly for virtual cycles on DM stacks is the contribution.\n\nWhat they do well is keep the geometry visible. They define an algebraic cut by a simplicial cone, descend both perfect obstruction theories and oriented (−2)-shifted structures to the cut, apply virtual localisation, then kill the “new” fixed loci by an η-wide cone plus explicit control of the virtual normal weights (Props 6.3 and 11.5). The weight statements follow directly from the induced obstruction theory on the cut (diagrams (33), (80) and the comparison lemmas), so the residue vanishing is not hand-waved. The K-theoretic side is cleaner still, as expected once one has formal characters and toric residues. Nonabelian reduction via Maddock’s algebraic version of Martin is cleanly adapted. Citations are honest; self-citations to OT are for the virtual cycles being used, not circular.\n\nThe soft spot is exactly the one the reader flagged: if the virtual weights ever escaped the claimed span(σ) ∪ edges of Σ ∩ τ_p, the residue argument would fail. On the page that control is proved from the cut construction, and the stress-test confirms there is no hidden circularity. It is the most delicate step, but it is written with matching care in both settings and looks correct under the stated hypotheses (weak η-semiprojectivity, T-semistable = T-stable, etc.). No free parameters, no load-bearing fitting.\n\nThis is for people who actually compute virtual integrals on GIT quotients or need wall-crossing formulae in a concrete form. It deserves a serious referee; the manuscript is long but the strategy is transparent and the technical core is fully written. I would accept it for peer review and would cite the statements when I next need a virtual JK formula.","headline":"Solid virtual JK package that fills a real computational gap; the delicate weight-control step is handled carefully and the proofs look complete.","tokens_in":63645,"tokens_out":561,"would_cite":true,"duration_ms":8372,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N35","14L24","53D20","19E08"],"pacs":[],"model":"grok-4.5","headline":"Virtual integrals over GIT quotients reduce to Jeffrey–Kirwan residues of integrals over torus-fixed virtual cycles.","keywords":["Jeffrey–Kirwan localisation","virtual cycles","GIT quotients","perfect obstruction theories","shifted symplectic structures","K-theoretic localisation","algebraic cut"],"falsifier":"Compute both sides of the formula for a concrete non-smooth GIT quotient with known virtual class (for example a moduli space of sheaves on a Calabi–Yau fourfold with a torus action) and check whether the JK residue of the fixed-locus contributions recovers the virtual integral over the quotient; a mismatch on any such example falsifies the claim.","tokens_in":63601,"feed_emoji":"📐","tokens_out":736,"duration_ms":7737,"temperature":0.7,"pith_summary":"The paper shows that when a reductive group acts on a polarised Deligne–Mumford stack carrying either a perfect obstruction theory or an oriented (−2)-shifted symplectic structure, the virtual fundamental class of the GIT quotient can be recovered from the virtual classes of the torus-fixed loci by taking Jeffrey–Kirwan residues. The same reduction works in K-theory with virtual structure sheaves, and it continues to hold when the original stack is noncompact provided the quotient is compact. The construction is algebraic: an algebraic cut replaces the classical symplectic cut, virtual ABBV localisation is applied to the cut, and residues of the resulting contributions yield the virtual JK formula. A sympathetic reader cares because many moduli spaces arise as GIT quotients of stacks with virtual cycles; the formula converts hard integrals on those quotients into residues of simpler integrals on fixed loci, often reducing to a single fixed-point contribution.","feed_headline":"Virtual GIT integrals become residues on fixed loci","feed_subtitle":"Algebraic cuts turn virtual ABBV into Jeffrey–Kirwan formulae for obstruction theories and CY4 cycles","key_machinery":"The algebraic cut of a polarised stack by a simplicial cone Σ (Definition 3.1), equipped with the induced perfect obstruction theory or (−2)-shifted symplectic structure; virtual ABBV localisation on that cut, followed by Jeffrey–Kirwan residues that kill the contributions of newly created fixed loci when Σ is η-wide.","core_discovery":"Under the stated hypotheses (T-semistable points stable, weak η-semiprojectivity, and either a perfect obstruction theory or an oriented (−2)-shifted symplectic structure), the virtual integral of a class α0 over [X//G]vir equals (1/|W|) times a sum, over those T-fixed components F with positive moment pairing against η, of the Jeffrey–Kirwan residue of the equivariant Euler class of the adjoint representation times the integral of αT over the virtual class of F divided by the virtual Euler class (or its square root in the CY4 case) of the virtual normal bundle. Parallel identities hold for virtual structure sheaves in K-theory.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Virtual GIT integrals equal JK residues on fixed loci","Virtual cycles of X//G via Jeffrey–Kirwan residues on XT","JK localisation of virtual integrals for GIT quotients","From virtual [X//G] to residues of virtual fixed classes","Virtual ABBV yields JK formulae on T-fixed components"],"cache_read_input_tokens":49280,"weakest_assumption_plain":"The residues of the new fixed loci created by the algebraic cut vanish only if the weights of the virtual normal bundles stay inside the span of the face and the edges of the cone; if virtual weights escape that control the residue argument fails.","fun_headline_variants_meta":{"raw":{"variants":["Virtual GIT integrals equal JK residues on fixed loci","Virtual cycles of X//G via Jeffrey–Kirwan residues on XT","JK localisation of virtual integrals for GIT quotients","From virtual [X//G] to residues of virtual fixed classes","Virtual ABBV yields JK formulae on T-fixed components"]},"model":"grok-4.5","effort":"low","cost_usd":0.005922,"raw_usage":{"total_tokens":1489,"prompt_tokens":657,"num_sources_used":0,"completion_tokens":84,"cost_in_usd_ticks":59220000,"prompt_tokens_details":{"text_tokens":657,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":748,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":657,"tokens_out":84,"duration_ms":5890,"temperature":1.0,"reasoning_tokens":748,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T05:33:16.484727+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute both sides of the formula for a concrete non-smooth GIT quotient with known virtual class (for example a moduli space of sheaves on a Calabi–Yau fourfold with a torus action) and check whether the JK residue of the fixed-locus contributions recovers the virtual integral over the quotient; a mismatch on any such example falsifies the claim.","supporting_citations":[],"review_version":1}