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Virtual integrals over GIT quotients reduce to Jeffrey–Kirwan residues of integrals over torus-fixed virtual cycles.

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2026-07-11 05:33 UTC pith:MZFHW3SQ

load-bearing objection Solid virtual JK package that fills a real computational gap; the delicate weight-control step is handled carefully and the proofs look complete.

arxiv 2607.05575 v1 pith:MZFHW3SQ submitted 2026-07-06 math.AG math.SG

Virtual Jeffrey--Kirwan localisation

Riccardo Ontani , Richard P. Thomas This is my paper
classification math.AG math.SG MSC 14N3514L2453D2019E08
keywords Jeffrey–Kirwan localisationvirtual cyclesGIT quotientsperfect obstruction theoriesshifted symplectic structuresK-theoretic localisationalgebraic cut
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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.

What carries the argument

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.

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

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.

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.

minor comments (5)
  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. A few minor typos (e.g. “F antechi” in the §4 heading, occasional missing spaces around //) should be cleaned in copy-editing.

Circularity Check

1 steps flagged

No significant circularity: virtual JK formulas are derived from independent virtual ABBV (GP/OT) applied to algebraic cuts plus residue vanishing, without self-definitional reduction or fitted inputs.

specific steps
  1. self citation load bearing [Sec 1 (Virtual ABBV); Sec 6 (40); Sec 10–11 (88, Thm 11.1); Sec 12 (virtual structure sheaves)]
    "There is a similar formula [OT] for the virtual cycles of [BJ] when E is symmetric 3-term and oriented [CGJ]... For the virtual cycles of [BJ, OT] we require that X’s symmetric obstruction theory comes from a G-equivariant oriented (−2)-shifted symplectic derived structure. Then (JK2)..."

    The CY4 virtual ABBV formula and twisted virtual structure sheaves are taken from the authors’ prior [OT] and used as the starting point for residue extraction. This is load-bearing for the (−2)-shifted and K-theoretic statements, but [OT] is an independent construction of virtual cycles (not a restatement of JK), so the circularity is only mild self-citation of a black-box tool rather than a definitional loop.

full rationale

The paper constructs algebraic cuts (Thm 3.3, Prop 3.5), induces obstruction theories/virtual cycles on them (Cor 4.4, Lem 4.5, Sec 10, Prop 10.5), applies existing virtual ABBV localization ((40) from GP; (88) from OT), extracts JK residues (Sec 7–8, Prop 7.3–7.4), and kills new-fixed-locus contributions via weight control (Prop 6.3/11.5 from diagrams (33)/(80) and Lem 6.5/Prop 11.3) plus η-wide cones (Lem 8.3). Nonabelian and K-theoretic cases reduce to abelian via Martin–Maddock abelianization (Thm 9.3, 12.12) and toric residues (Def 12.14). Self-citations to OT supply the virtual cycles and ABBV formula used as black boxes (exactly as classical JK uses ABBV); they are not redefined here, nor do they encode the target JK statement. No parameters are fitted, no uniqueness is imported to force the answer, and no claim equals its input by construction. The derivation is the natural virtual extension of Lerman/Jeffrey–Kogan/Edidin–Graham and is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 2 invented entities

The paper works entirely inside standard algebraic geometry, GIT, and derived symplectic geometry. No numerical free parameters appear. The axioms are the usual foundations plus the existence of virtual cycles for perfect obstruction theories and oriented (−2)-shifted symplectic structures. Invented entities are definitional constructions (algebraic cut, η-wide cone) rather than new physical objects.

axioms (7)
  • domain assumption Existence and basic properties of Behrend–Fantechi virtual fundamental classes for perfect obstruction theories on DM stacks.
    Used throughout §§4–9; cited as [BF, LT, Kr1].
  • domain assumption Existence of virtual cycles and square-root Euler classes for oriented (−2)-shifted symplectic derived DM stacks (Oh–Thomas / Kiem–Park).
    Used in §§10–11; cited as [OT, BJ, KP].
  • domain assumption Stability equals semistability for the linearised actions under consideration, so GIT quotients are DM stacks.
    Standing assumption from Notation (5) onward; needed for finite stabilisers.
  • domain assumption Weak η-semiprojectivity (or η-semiprojectivity) of (X,L) so that X//T and the algebraic cut are projective.
    Definitions 5.1, 5.4; used to guarantee properness of integrals and finite-dimensionality of invariants.
  • standard math Brion–Vergne consistency of the Jeffrey–Kirwan residue operator JK_η^ξ on the ring of rational functions with poles on a hyperplane arrangement.
    Proposition 7.2; cited [BrV].
  • standard math De Concini–Procesi relation between toric JK residues and constant-term functionals.
    Proposition 12.16; used to pass from n≫0 to n=0 in K-theory.
  • domain assumption Martin–Maddock abelianisation relating integrals (or Euler characteristics) on X//G to those on X//T.
    Theorem 9.3 / 12.12; extended from projective varieties to projective-over-affine DM stacks.
invented entities (2)
  • Algebraic cut X_Σ of a polarised DM stack by a simplicial cone Σ no independent evidence
    purpose: Provides a compact T-space on which virtual ABBV can be applied, with X//T appearing as a fixed component.
    Definition 3.1 generalises Edidin–Graham / Jeffrey–Kogan cuts to DM stacks; purely definitional, no independent physical content.
  • η-wide cone and weak η-semiprojectivity no independent evidence
    purpose: Guarantee that new fixed loci contribute zero after JK residue and that quotients remain projective.
    Definitions 5.1, 5.4 and Lemma 8.3; technical hypotheses, not new geometric objects.

pith-pipeline@v1.1.0-grok45 · 66652 in / 3473 out tokens · 34450 ms · 2026-07-11T05:33:16.484727+00:00 · methodology

0 comments
read the original abstract

We express integrals over virtual cycles of GIT quotients $X/\!\!/G$ in terms of integrals over virtual cycles of fixed loci $X^T$. The results hold for both perfect obstruction theories and $(-2)$-shifted symplectic structures, in cohomology and in $K$-theory, and for noncompact Deligne-Mumford stacks acted on by a reductive group with compact quotient.

Figures

Figures reproduced from arXiv: 2607.05575 by Riccardo Ontani, Richard P. Thomas.

Figure 1
Figure 1. Figure 1: Stratification of a 2-dimensional moment polytope of an irre￾ducible Xs. Example 2.4. Let t ∈ C ∗ act on P 1 × P 1 , O(1, 1) by (tx0, x1), (y0, t−1y1). The moment polyhedron ∆ = [−1, 1] has zero-dimensional stratum ∂0∆ = {−1, 0, 1} with the interior point 0 = µ(0 × ∞) = µ(∞ × 0) being the moment polyhedron of two distinct fixed points. Thus not all pairwise intersections of the polyhedra comprising ∂d∆ li… view at source ↗
Figure 2
Figure 2. Figure 2: Stratification of the cut XΣ seen from ∆ ∩ Σ. 3.1. Definition of algebraic cut. Fix an algebraic torus T of rank r acting on a polarised projective-over-affine Deligne-Mumford stack (X, L), with moment polyhedron ∆ such that L-semistable points are L-stable. Let Σ := Cone(ψ1, . . . , ψk) ⊂ t ∨ Q be a simplicial cone of dimension k := dim(Σ) ≤ r spanned by primitive integral elements ψi ∈ t ∨ Z . We denote … view at source ↗
Figure 3
Figure 3. Figure 3: The classification of the components of XT Σ into special, old and new fixed loci, together with the corresponding face σ from (24). Remark 3.6. We illustrate Proposition 3.5 for a full dimensional Σ ⊂ t ∨ Q in [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two examples of cones Σ not transversal to ∆. On the left (1) fails because p ∈ ∂0∆; on the right (2) fails because p ∈ ∂1∆ \ ∂ sm 1 ∆. 8We already know this intersection lies in one of the ∂r−dim σ ∆(Xi) so this last condition is that it should not lie in an intersection of two of them unless they coincide locally [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: An example of Σ transversal to ∆ when X = Xa∪Xb is reducible. Although p belongs to σ ∩(∂2∆b \ ∂ sm 2 ∆b), this does not break Definition 3.8 (2) because that only applies to ∂d with d ≤ dim ∆b = 1. Note that we could have asked for (1) and (2) to hold for all d ≤ dim ∆ for any X. But by dealing with its irreducible components separately, thus letting d vary, we allow for transversality in examples like th… view at source ↗
Figure 6
Figure 6. Figure 6: Minus the weights of Nvir Fp/XΣ = Nvir Fp/Xσ ⊕ NXσ/XΣ described in terms of Σ and τp. On the right we picture the tangent space to t ∨ Q at p. The T-weights of Nvir Fp/Xσ (in green) belong to span(σ), while minus the weights of NXσ/XΣ |Fp (in blue) are the generators of the simplicial cone Σ ∩ τp. Remark 6.4. If X is irreducible, minus the weights of the T action on the classical normal bundle NFp/XΣ lie a… view at source ↗
Figure 7
Figure 7. Figure 7: An η ∈ tQ such that {η ≥ 0} ∩ ∆ is bounded and 0 ∈/ η(µ(XT)). The η-wide cone Σ approximates the half-space {η > 0} as far as µ(XT) is concerned. We can recover Theorem 8.1 from this by replacing α by αec T 1 (L) . The rest of this Section will be devoted to proving Theorem 8.1. Then Theorem 8.2 will follow by some modifications and perturbations. 8.1. The η-wide cone. We begin by forming an algebraic cut … view at source ↗
Figure 8
Figure 8. Figure 8: This picture is in the affine subspace τp ⊆ t ∨ Q . The intersection Σ ∩µ(Y )∩τp lies in {η ≥ 0}, as does at least one of its generating directions at p (but not, in this example, the other). But p (the red point in [PITH_FULL_IMAGE:figures/full_fig_p031_8.png] view at source ↗

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Reference graph

Works this paper leans on

53 extracted references · 18 canonical work pages · 14 internal anchors

  1. [1]

    Aganagic, E

    M. Aganagic, E. Frenkel, A. Okounkov, Quantum q-Langlands correspondence, Transactions of the Moscow Mathematical Society 79 (2018), 1--83. arXiv:1701.03146

    Pith/arXiv arXiv 2018

  2. [2]

    Structure results for torus fixed loci

    J. Alper and F. Janda, Structure results for torus fixed loci, arXiv:2402.10823

    work page internal anchor Pith review Pith/arXiv arXiv

  3. [3]

    Atiyah and R

    M. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1--28

    1984

  4. [4]

    Behrend and B

    K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), 45--88. alg-geom/9601010 https://arxiv.org/abs/alg-geom/9601010

    Pith/arXiv arXiv 1997

  5. [5]

    Benini, R

    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N= 2 gauge theories with rank-one gauge groups, Lett. Math. Phys. 104 (2014), 465--493. arXiv:1305.0533

    Pith/arXiv arXiv 2014

  6. [6]

    Berline and M

    N. Berline and M. Vergne, Classes caract \'e ristiques \'e quivariantes. Formule de localisation en cohomologie \'e quivariante , CR Acad. Sci. Paris 295 (1982), 539--541

    1982

  7. [7]

    Wall-crossing for Calabi-Yau fourfolds: framework, tools, and applications

    A. Bojko, Wall-crossing for Calabi-Yau fourfolds: framework, tools, and applications, arXiv:2507.05922

    work page internal anchor Pith review Pith/arXiv arXiv

  8. [8]

    Borisov and D

    D. Borisov and D. Joyce, Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds, Geom. & Topol. 21 (2017), 3231--3311. arXiv:1504.00690

    Pith/arXiv arXiv 2017

  9. [9]

    Brion, On the general faces of the moment polytope, Int

    M. Brion, On the general faces of the moment polytope, Int. Math. Res. Notices 1999 (1999), 185--201

    1999

  10. [10]

    Invariant Hilbert schemes

    M. Brion, Invariant Hilbert schemes, arXiv:1102.0198

    work page internal anchor Pith review Pith/arXiv arXiv

  11. [11]

    Brion and M

    M. Brion and M. Vergne, Arrangement of hyperplanes. I. Rational functions and Jeffrey-Kirwan residue, Ann. Sci. \'Ec. Norm. Sup\'er. 32 (1999), 715--741. 9903178

    1999

  12. [12]

    Lagrangian structures on mapping stacks and semi-classical TFTs

    D. Calaque, Lagrangian structures on mapping stacks and semi-classical TFTs, Contemporary Math. 643 (2015), 1--23. arXiv:1306.3235

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  13. [13]

    Y. Cao, J. Gross and D. Joyce, Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi--Yau 4-folds, Adv. Math. 368 (2020), 107134. arXiv:1811.09658

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  14. [14]

    Conrad, Reductive group schemes, In Autour des sch \'e mas en groupes, Vol 1 (2014), 93--444

    B. Conrad, Reductive group schemes, In Autour des sch \'e mas en groupes, Vol 1 (2014), 93--444

    2014

  15. [15]

    De Concini, C

    C. De Concini, C. Procesi, On the geometry of toric arrangements, Transformation Groups 10 (2005), 387--422. 0505351

    2005

  16. [16]

    Algebraic Cuts

    D. Edidin and W. Graham, Algebraic cuts, Proc. Amer. Math. Soc. 126 (1998), 677--685. alg-geom/9608028 https://arxiv.org/abs/alg-geom/9608028

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  17. [17]

    Riemann-Roch theorems and elliptic genus for virtually smooth Schemes

    B. Fantechi and L. G \"o ttsche, Riemann-Roch theorems and elliptic genus for virtually smooth schemes, Geom. & Topol. 14 (2010), 83--115. arXiv:0706.0988

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  18. [18]

    Guillemin and J

    V. Guillemin and J. Kalkman, The Jeffrey-Kirwan localization theorem and residue operations in equivariant cohomology, Journal f\"ur die reine und angewandte Mathematik 470 (1996), 123--142

    1996

  19. [19]

    M. G. Gulbrandsen, L. H. Halle and K. Hulek, A relative Hilbert--Mumford criterion, Manuscr. Math. 148 (2015), 283--301. arXiv:1404.7267

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  20. [20]

    Localization of virtual classes

    T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1997), 487--518. alg-geom/9708001 https://arxiv.org/abs/alg-geom/9708001

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  21. [21]

    Halpern-Leistner, A categorical perspective on non-abelian localization, arXiv:2509.24009

    D. Halpern-Leistner, A categorical perspective on non-abelian localization, arXiv:2509.24009

    work page arXiv

  22. [22]

    Hausel and F

    T. Hausel and F. Rodriguez Villegas, Cohomology of large semiprojective hyperk\"ahler varieties, Ast \'erisque 370 (2015), 113--156. arXiv:1309.4914

    Pith/arXiv arXiv 2015

  23. [23]

    Jeffrey and M

    L. Jeffrey and M. Kogan, Localization theorems by symplectic cuts, Festschrift in Honor of Alan Weinstein (2005), 303--326. 0310222

    2005

  24. [24]

    Jeffrey and F

    L. Jeffrey and F. Kirwan, Localization for nonabelian group actions, Topology 34 (1995), 291--327. alg-geom/9307001 https://arxiv.org/abs/alg-geom/9307001

    Pith/arXiv arXiv 1995

  25. [25]

    Jeffrey and F

    L. Jeffrey and F. Kirwan, Localization and the quantization conjecture, Topology 36 (1997), 647--693

    1997

  26. [26]

    Joyce, Enumerative invariants and wall-crossing formulae in abelian categories, arXiv:2111.04694

    D. Joyce, Enumerative invariants and wall-crossing formulae in abelian categories, arXiv:2111.04694

    Pith/arXiv arXiv

  27. [27]

    Karpov and M

    I. Karpov and M. Moreira, Generalized K -theoretic invariants and wall-crossing via non-abelian localization, arXiv:2512.22360

    Pith/arXiv arXiv

  28. [28]

    Kiem and H

    Y. Kiem and H. Park, Localizing virtual cycles for Donaldson-Thomas invariants of Calabi-Yau 4-folds, Jour. Alg. Geom. 32 (2023), 585--639. arXiv:2012.13167

    Pith/arXiv arXiv 2023

  29. [29]

    Kinjo, H

    T. Kinjo, H. Park and P. Safronov, Cohomological Hall algebras for 3-Calabi-Yau categories, arXiv:2406.12838

    arXiv

  30. [30]

    Kresch, Cycle groups for Artin stacks, Invent

    A. Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), 495--536. 9810166

    1999

  31. [31]

    Kresch, On the geometry of Deligne-Mumford stacks , Algebraic geometry--Seattle 2005

    A. Kresch, On the geometry of Deligne-Mumford stacks , Algebraic geometry--Seattle 2005. Part 1. Providence, RI: Amer. Math. Soc. (2009), 259--271

    2005

  32. [32]

    Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton University Press 31 (1984)

    F. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton University Press 31 (1984)

    1984

  33. [33]

    N. Kuhn, H. Liu and F. Thimm, Wall-crossing for invariants of equivariant 3CY categories, arXiv:2512.23012

    arXiv

  34. [34]

    Lerman, Symplectic cuts, Math

    E. Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), 247--258

    1995

  35. [35]

    Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

    J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, Jour. AMS 11 (1998), 119--174. alg-geom/9602007 https://arxiv.org/abs/alg-geom/9602007

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  36. [36]

    A ratio of integration between quotients in geometric invariant theory

    Z. Maddock, A ratio of integration between quotients in geometric invariant theory, Transform. Groups 19 (2014), 131--158. arXiv:1210.4253

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  37. [37]

    Martin, Symplectic quotients by a nonabelian group and by its maximal torus, 0001002

    S. Martin, Symplectic quotients by a nonabelian group and by its maximal torus, 0001002

  38. [38]

    Metzler, Cohomological localization for manifolds with boundary, IMRN 2002 (2002), 1239--1274

    D. Metzler, Cohomological localization for manifolds with boundary, IMRN 2002 (2002), 1239--1274. math.DG/9908002 http://arxiv.org/abs/math/9908002

    work page arXiv 2002

  39. [39]

    Mukai, An introduction to invariants and moduli, Cambridge University Press 81 (2003)

    S. Mukai, An introduction to invariants and moduli, Cambridge University Press 81 (2003)

    2003

  40. [40]

    Oh and R

    J. Oh and R. Thomas, Counting sheaves on Calabi-Yau 4-folds, I, Duke Math. Jour. 172 (2023), 1333--1409. arXiv:2009.05542

    Pith/arXiv arXiv 2023

  41. [41]

    Ontani, The sign rule for ^n ^4 by Jeffrey--Kirwan localisation, preprint

    R. Ontani, The sign rule for ^n ^4 by Jeffrey--Kirwan localisation, preprint

  42. [42]

    Pantev, B

    T. Pantev, B. To \"e n, M. Vaqui \'e and G. Vezzosi, Shifted symplectic structures, Publ. Math. IH \'E S 117 (2013), 271--328. arXiv:1111.3209

    Pith/arXiv arXiv 2013

  43. [43]

    Ressayre, HKKN-stratifications in a non-compact framework, arXiv: \!\!2505.03259 http://arxiv.org/abs/2505.03259

    P.-E Paradan and N. Ressayre, HKKN-stratifications in a non-compact framework, arXiv: \!\!2505.03259 http://arxiv.org/abs/2505.03259

    work page arXiv

  44. [44]

    Park, Shifted symplectic pushforwards, arXiv:2406.19192

    H. Park, Shifted symplectic pushforwards, arXiv:2406.19192

    Pith/arXiv arXiv

  45. [45]

    Park, unpublished notes, 2026

    H. Park, unpublished notes, 2026

    2026

  46. [46]

    Park, Virtual pullbacks in Donaldson-Thomas theory of Calabi-Yau 4-folds

    H. Park, Virtual pullbacks in Donaldson-Thomas theory of Calabi-Yau 4-folds. arXiv:2110.03631

    Pith/arXiv arXiv

  47. [47]

    Shifted symplectic rigidification

    H. Park and J. You, Shifted symplectic rigidification, arXiv:2604.04008

    work page internal anchor Pith review Pith/arXiv arXiv

  48. [48]

    Virtual pullbacks in $K$-theory

    F. Qu, Virtual pullbacks in K -theory, Ann. l'Inst. Fourier 68 (2018), 1609--1641. arXiv:1608.02524

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  49. [49]

    Quasi-Hamiltonian reduction via classical Chern-Simons theory

    P. Safronov, Quasi-Hamiltonian reduction via classical Chern--Simons theory, Adv. Math. 287 (2016), 733--773. arXiv:1311.6429

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  50. [50]

    Szenes and M

    A. Szenes and M. Vergne, Toric reduction and a conjecture of Batyrev and Materov, Invent. Math. 158 (2004), 453--495. math.AT/0306311 http://arxiv.org/abs/math/0306311

    work page arXiv 2004

  51. [51]

    Teleman, The Quantization Conjecture Revisited, Ann

    C. Teleman, The Quantization Conjecture Revisited, Ann. of Math. 152 (2000), 1--43

    2000

  52. [52]

    To \"e n, Derived algebraic geometry, EMS Surv

    B. To \"e n, Derived algebraic geometry, EMS Surv. Math. Sci. 1 (2014), 153--240. arXiv:1401.1044

    Pith/arXiv arXiv 2014

  53. [53]

    To \"e n and G

    B. To \"e n and G. Vezzosi, Homotopical Algebraic Geometry II: Geometric Stacks and Applications, Mem. AMS 193 (2008). 0404373

    2008