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This paper proves that the Floer homotopy type of an ample smooth divisor complement splits into a wedge of Spanier-Whitehead duals of Thom spectra whenever a single spectral Gromov-Witten class vanishes, and it computes the splitting in ma

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load-bearing objection Real new machinery for spectral symplectic cohomology, but the main splitting theorem currently rests on a gap: Proposition 4.11 proves only an integral-homology isomorphism, yet invokes an HZ-module Whitehead theorem without checking the HZ-module hypothesis. the 3 major comments →

arxiv 2601.08506 v2 pith:FJNMQZ3I submitted 2026-01-13 math.SG math.AGmath.AT

Ample divisor complements, Floer spectra, and relative Gromov-Witten theory

Kenneth Blakey This is my paper
classification math.SG math.AGmath.AT MSC 53D4014N3555P42
keywords Floer homotopysymplectic cohomologydivisor complementsrelative Gromov-Witten theorylog PSS morphismweight filtrationThom spectradel Pezzo surfaces
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.

This paper gives a computation of the Floer homotopy type — the spectrum whose homology is symplectic cohomology — of the complement of an ample smooth divisor in a complex projective manifold. The author builds a spectral lift of the low-energy log PSS morphism, the map that in ordinary cohomology computes the weight filtration of symplectic cohomology from the topology of the pair, and proves that on the level of spectra it computes the associated graded of the weight filtration. The single piece of data not determined by the associated graded is a stable homotopy class, called the spectral Gromov-Witten obstruction, which is defined from genus-0 relative Gromov-Witten moduli spaces; when this class vanishes, the Floer homotopy type splits as a wedge of Spanier-Whitehead duals of Thom spectra indexed by winding number. The paper establishes the splitting in a range of concrete cases, including the affine part of every smooth projective hypersurface of degree at least two, several del Pezzo surfaces, and the quadric complement that models the cotangent bundle of the sphere. The interest is that Floer homotopy types are rarely computable, and here the computation is reduced to relative Gromov-Witten data plus one checkable stable-homotopy class.

Core claim

Under an assumption that gives X a stable real polarization — a real vector bundle Λ whose complexification is stably isomorphic to the tangent bundle — the paper proves Theorem 1.3. The weight filtration of the Floer homotopy type F^Λ has associated graded pieces equal to the Spanier-Whitehead duals of explicit Thom spectra: one for X twisted by a virtual bundle V_0, and one for each winding number k≥1 twisted by V_{w(k)} on the circle bundle of the divisor. A spectral low-energy log PSS morphism, constructed from marked thimbles, is shown to be a homotopy equivalence onto these graded pieces. The failure of the filtration to split is then captured by a single stable homotopy class GW built

What carries the argument

The central object is the stable R-polarization Λ, a real vector bundle whose complexification is stably isomorphic to T X; Assumption 1(3) produces it, and without it no Floer homotopy type exists. The argument is carried by marked thimbles: genus-0 curves with fixed marked points, one negative cylindrical end asymptotic to a chosen 1-periodic orbit, and transverse intersections with the divisor at the marked points. The paper also uses enhanced spheres — genus-0 relative holomorphic spheres with a quotient only by R-translations — and partially-incident thimbles, which allow marked points to avoid the divisor, in order to build the compactified moduli spaces needed for the obstruction. Twi

Load-bearing premise

One must be able to split the tangent bundle, after adding trivial complex factors and powers of the divisor's line bundle, as a direct sum of complexified real vector bundles twisted by powers of that same line bundle, with the real factors oriented and spin on the complement; this is what produces the stable real polarization needed to define the Floer homotopy type. The paper notes that some natural del Pezzo degree-2 cases do not satisfy it, so for those cases the main th

What would settle it

Take the quadric pair M = {z_0^2 = z_1^2 + ... + z_{n+1}^2} ⊂ CP^{n+1}, D = {z_0 = 0}, where the paper proves a splitting for T^*S^n. The proof kills the obstruction by exhibiting a fixed-point-free involution on the moduli space of enhanced lines through a point. Check that this involution has no fixed points on the full compactified moduli space, including enhanced sphere bubbles, and that it preserves the twisted stable framing on every boundary stratum. A single fixed point or framing mismatch would produce a nonzero GW and break the claimed splitting.

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

If this is right

  • If GW=0, F^Λ is explicitly a wedge of Spanier-Whitehead duals of Thom spectra, with the twists V_k determined by the polarization data, the divisor line bundle, and the normal bundle of D.
  • The associated graded of the weight filtration is computed by a spectral low-energy log PSS morphism that is a homotopy equivalence, upgrading the cohomological computation to spectrum level.
  • The affine part of every smooth projective hypersurface of degree at least two has a split Floer homotopy type; this answers the spectral form of a question that was open at cohomology level.
  • In the quadric case, the splitting recovers the classical stable splitting of the free loop space of the sphere via the spectral cotangent-bundle isomorphism.
  • The alternative splitting criterion using exact Lagrangians allows one to prove splitting without computing GW, as done for the affine hypersurfaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The paper's noted failure of Assumption 1(3) for the natural degree-2 del Pezzo models indicates that a multi-divisor generalization, sketched in Section 6.1.6, is the natural next step; if it works, the same machinery should cover del Pezzo degrees 1, 5, and 7.
  • The paper's Question 1 exposes a sharp distinction between cohomological and homotopical splitting: one could search for a manifold where F^Λ ∧ HZ splits but F^Λ does not, using the Lagrangian-injectivity criterion as a constraint on possible counterexamples.
  • The pattern that classical stable splittings of free loop spaces match vanishing of GW suggests a transferable principle: framed bordism of relative Gromov-Witten moduli spaces controls filtration splittings in Floer homotopy, not just in this divisor-complement setting.
  • The spectral obstructions are defined from genus-0 relative Gromov-Witten moduli spaces, so any new method to compute framed bordism classes of those moduli spaces — for example through algebro-geometric models for T^*CP^n or T^*HP^n — would immediately yield new splittings.

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

3 major / 4 minor

Summary. The paper proposes a spectral lift of Ganatra–Pomerleano's low-energy log PSS morphism for ample smooth divisor complements, defines a 'spectral Gromov–Witten obstruction' in stable homotopy, and proves (Theorem 1.3) that vanishing of this obstruction implies a wedge splitting of the Floer homotopy type into Spanier–Whitehead duals of Thom spectra. A secondary criterion (Theorem 1.6) is given in terms of exact Lagrangian spheres. The paper also contains many computations, including cotangent bundles of spheres, del Pezzo surfaces, and affine parts of smooth projective hypersurfaces. The overall architecture is coherent and follows the Ganatra–Pomerleano cohomological template together with Abouzaid–Blumberg flow-category foundations, but the proof of the key spectral equivalence in Proposition 4.11 contains a serious gap.

Significance. If the main results are correct, this is a substantial contribution: it gives a spectral refinement of a well-studied cohomological computation, formulates an explicit stable-homotopy obstruction, and computes many nontrivial examples. The paper is also commendably explicit about the scope of its key Assumption 1(3), noting in §6.1.6 that natural del Pezzo degree-2 candidates fail it. However, the central proof currently rests on an invalid Whitehead-type inference, and several auxiliary constructions are deferred by 'standard' or 'appropriately modify' statements. The result is therefore significant but not yet established at the level claimed.

major comments (3)
  1. [§4.5, proof of Proposition 4.11, specifically (4.71)] The proof identifies the induced map on integral homology with Ganatra–Pomerleano's isomorphism and then concludes that LePSS^k_log is a homotopy equivalence 'since … are bounded below spectra, this proposition follows after using a spectral Whitehead theorem for HZ-module spectra.' This inference is invalid as stated: bounded-below spectra are not automatically HZ-module spectra, and an integral-homology isomorphism does not imply a stable equivalence for bounded-below spectra (e.g. Σ^∞CP^2 and S^0∨S^2∨S^4 have isomorphic integral homology but are not stably equivalent). Neither D(S_D^M)^{-V_w(k)} nor F^Λ_{k,k-1} is shown to be an HZ-module; indeed the Thom spectra in the examples generally have nontrivial Steenrod operations. Since Theorem 1.3 uses the homotopy equivalence in diagram (5.4) to transfer vanishing of GW to null-homotopy of the Puppe maps, the main splitting theorem is uns
  2. [§5.7.5, Lemma 5.21 and framing compatibility] The construction of the framed flow 2-simplex W(a,x) is the heart of the reduction of the Puppe connecting map to the spectral Gromov–Witten obstruction. The proof identifies the boundary strata and states that the twisted stable framing 'follows by direct inspection.' This is not a formality: the framing must be shown to restrict to the prescribed framings on both ends, compatible with all higher corner strata. Since the entire conclusion of Theorem 1.3 depends on this homotopy, the framing compatibility deserves a complete proof rather than an assertion.
  3. [§5.2–§5.3, Lemmas 5.3 and 5.7 and related deferrals] Several index theorems that underwrite the twisted stable framings are disposed of by 'Appropriately modify the proof of Lemma 4.4.' These are not cosmetic substitutions: Lemma 5.3 concerns line-bundle Cauchy–Riemann operators on CP^1 with incidence at +∞, and Lemma 5.7 concerns partially-incident thimbles with new boundary strata involving enhanced spheres. The index computations are load-bearing for the framed flow bimodules used in Theorem 1.3. Please provide full proofs or precise references that cover the modified settings.
minor comments (4)
  1. [§1.2 and §6.1.7] The abstract says 'all smooth projective hypersurfaces of degree at least 2,' but Proposition 6.15 is stated for a smooth hyperplane section D of a smooth hypersurface in CP^{n+1}. Please make the scope of the claim uniform.
  2. [§6.1.4, proof of Proposition 6.12] The notation 'GW^{S^1}(a,b)≠0' appears; it should be '≠∅' since it refers to a moduli space, not a numerical invariant.
  3. [References] The page range for [Mil85] is listed as '411–410', which is evidently a typo. Please correct.
  4. [Remark 1.8] The remark already warns that splitting of SH^*(X;Z) need not imply splitting of F^Λ∧HZ. This makes the gap in Proposition 4.11 more conspicuous: the proof of Proposition 4.11 appears to assume exactly the kind of HZ-module Whitehead statement that the remark cautions against.

Circularity Check

0 steps flagged

No circular derivation; central splitting result is built from moduli spaces and external GP input, with minor non-load-bearing self-citations.

full rationale

The claimed derivation chain does not reduce to its inputs. Assumption 1(3) is an explicit existence hypothesis (vector-bundle splitting (1.2)) and is not fitted to the desired splitting (1.11). The spectral Gromov-Witten obstruction GW is defined in Definition 5.20 from relative GW moduli spaces (auxiliary hybrid enhanced spheres), not assumed equal to the target splitting. Proposition 4.11 identifies the homology effect of the spectral low-energy log PSS map with Ganatra-Pomerleano's independently proved ring isomorphism (equations (4.69)-(4.71)); this is an external benchmark, not a self-referential definition. The later claim that an integral-homology isomorphism for bounded-below spectra upgrades via an HZ-module Whitehead theorem (Section 4.5) is a genuine correctness gap—bounded-below spectra need not be HZ-modules and integral homology isomorphisms do not generally imply homotopy equivalence—but that is a missing proof/invalid inference, not circularity. Self-citations such as [Blaa] for the spectral Viterbo isomorphism (Theorem 6.10) and [Bla24]/[Blab]/[BB25] for flow-category framings and related foundations are present, but they are used for technical support or sanity checks; the main theorems additionally rely on external references (Lar21, PS24b, PS25c, GP20/GP21). No parameter is fitted and renamed as a prediction, and no uniqueness theorem from the authors is invoked to force the conclusion. Hence score 2 reflects only a minor self-citation burden, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 1 invented entities

The paper does not fit numerical parameters to data. Its main cost is the highly restrictive Assumption 1(3), which postulates a specific vector bundle splitting that defines the stable R-polarization. The other axioms are standard background results or deferrals to the cited foundations. The only introduced entity, the spectral Gromov–Witten obstruction, is constructed from moduli spaces rather than postulated arbitrarily, but it lacks independent evidence outside the paper.

axioms (5)
  • standard math Ganatra–Pomerleano's low-energy log PSS morphism is an isomorphism of rings (GP20 Theorem 1.1).
    Used in §4.5 to prove the spectral log PSS maps are homotopy equivalences by reducing to the cohomological ring isomorphism.
  • standard math Framed flow categories model spectra, with gluing/compactness/stable framing analysis from Large, Porcelli–Smith, and Abouzaid–Blumberg.
    The paper repeatedly defers the construction of moduli compactifications and twisted stable framings to [Lar21], [PS24b], [PS25c], and [AB24].
  • ad hoc to paper Assumption 1(3): existence of d, m, n_ν and oriented real bundles F_ν with T M ⊕ C^d ⊕ L^m ≅ ⊕_ν (F_ν ⊗_R C) ⊗_C L^{n_ν}.
    This is the central polarization hypothesis. Without it, the Floer homotopy type F^Λ is not defined. The paper concedes in §6.1.6 that this excludes some natural examples.
  • standard math Spectral Whitehead theorem for bounded-below HZ-module spectra.
    Used at the end of §4.5 to upgrade an integral homology isomorphism to a homotopy equivalence of spectra.
  • domain assumption Milnor fibration theorem and existence of exact Lagrangian vanishing cycles for hypersurface complements.
    Used in Proposition 6.15 to realize the homotopy spheres in the bouquet as embedded exact Lagrangian spheres in the affine hypersurface.
invented entities (1)
  • Spectral Gromov–Witten obstruction GW no independent evidence
    purpose: Stable homotopy class encoding the obstruction to the Floer homotopy type splitting into its associated graded.
    Defined in Definition 5.20 using genus-0 relative Gromov–Witten moduli spaces. Its vanishing is checked in examples inside the paper, but there is no external falsifiable handle beyond the paper's own computations.

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read the original abstract

We spectrally lift Ganatra-Pomerleano's low-energy log PSS morphism to compute the associated graded of Floer homotopy types of ample smooth divisor complements. Moreover, we show the obstruction to splitting into the associated graded is encoded in a stable homotopy class defined via (higher-dimensional) genus 0 relative Gromov-Witten moduli spaces. We compute numerous examples of splittings, including the affine part of all smooth projective hypersurfaces of degree at least 2.

Figures

Figures reproduced from arXiv: 2601.08506 by Kenneth Blakey.

Figure 1
Figure 1. Figure 1: This is a k-marked thimble, k ≥ 1. The arrow at z1 indicates our auxiliary choice of real tangent ray made in order to define the enhanced evaluation map; in future figures, we will omit this choice from the picture. For any x ∈ χ(X; Hℓ ), we consider the moduli space RHℓ ,Jℓ k (x) of k-marked Hℓ -thimbles, i.e., maps u : Σk → M satisfying    [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: This is a broken (low-energy) k-marked thimble. Note, each yj has winding number equal to x’s winding number. We now move on to investigating the smooth structure of R Hℓ ,Jℓ w(x) (x). Proposition 4.7. R Hℓ ,Jℓ w(x) (x) can be given the structure of a compact smooth manifold with corners whose codimension 1 boundary is enumerated by gluing maps of the form R Hℓ ,Jℓ w(y) (y) × F Hℓ ,Jℓ (y, x) → R Hℓ ,Jℓ w(x… view at source ↗
Figure 3
Figure 3. Figure 3: This is a (low-energy) hybrid 0-marked thimble; here, x has winding number 0. Second, for any x ∈ χk(X; Hℓ ) with k ≥ 1 and a ∈ Crit(fS), we consider the moduli space R Hℓ ,Jℓ w(x) (a, x) of hybrid w(x)-marked Hℓ -thimbles, i.e., the standard Gromov-compactification of EnEval−1 z1 [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: This is a (low-energy) hybrid k-marked thimble, k ≥ 1; here, x has winding number k. Technically, the curve going into a should lie in SDM. 4.5. Computing the associated graded. This subsection is dedicated to proving the following result. Proposition 4.11. LePSSk log is a homotopy equivalence. Remark 4.12. Before the proof, perhaps some remarks are in order. (1) At this point, the reader may be wondering … view at source ↗
Figure 5
Figure 5. Figure 5: This is a schematic picture of the proof of Proposition 4.11. In particular, the degenerations of the middle picture, with respect to the parameter r measuring the length of the shaded re￾gion, to the top and bottom pictures depicts the commutativity of the diagram (4.69); moreover, the “equality” in the bottom picture depicts the equality (4.70), i.e., the fact that GP’s low-energy log PSS morphism is an … view at source ↗
Figure 6
Figure 6. Figure 6: This is an enhanced sphere. The arrow at z1 indicates our auxiliary choice of real tangent ray made in order to define the enhanced evaluation map (of course, we can not make such a choice if we had quotiented by the S 1 -action); in future figures, we will omit this choice from the picture. We define GWg ≡ a A∈HS 2 (M;Z) A·D=1 GWg(A); (5.14) we analogously define GWS 1 and GW. The evaluation map at 0 resp… view at source ↗
Figure 7
Figure 7. Figure 7: This is a k-marked {2, . . . , k}-PI thimble, k ≥ 2. We will fix (if 1 ∈ J c ) an element in Sz1Σk in order to define an enhanced evaluation map at z1: EnEvalz1 : R Hℓ ,Jℓ k J (x) → SDM. (5.25) Let u ∈ R Hℓ ,Jℓ k J (x); we will require a single family of Fredholm operators. Con￾sider a Cauchy Riemann operator DLm,J,u : W1,2 (Σk; u ∗Lm) → L 2 (Σk; u ∗Lm) (5.26) with asymptotic operator ∂t + τ · Id at the ne… view at source ↗
Figure 8
Figure 8. Figure 8: This picture is multi-purpose. In the context of the naive compactification, the sphere bubble at z1 is well-defined up to the C ∗ -action (as would be other sphere bubbles at other marked points). In the context of the auxiliary compactification, the sphere bubble at z1 is well-defined up to the R-action (as would be other sphere bubbles at other marked points). Finally, in the context of the enhanced com… view at source ↗
Figure 9
Figure 9. Figure 9: This is a hybrid enhanced 1-pointed relative sphere [PITH_FULL_IMAGE:figures/full_fig_p042_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: This is a hybrid k-marked {1, . . . , k − 1}-PI thimble, k ≥ 2. This is, for generic data, a compact smooth manifold with corners whose codi￾mension 1 boundary is enumerated by gluing maps of the form R Hℓ ,Jℓ k J,S1 (a, y) × F Hℓ ,Jℓ (y, x) → R Hℓ ,Jℓ k J,S1 (a, x), (5.69) R Hℓ ,Jℓ k J∪{1},S1 (x) ×X GWS 1 (a) → R Hℓ ,Jℓ k J,S1 (a, x), (5.70) DMSDM(a, b) × R Hℓ ,Jℓ k J,S1 (b, x) → R Hℓ ,Jℓ k J,S1 (a, x). … view at source ↗
Figure 11
Figure 11. Figure 11: This is an auxiliary hybrid marked {1}-PI thimble. This is, for generic data, a compact smooth manifold with corners. By splitting the short exact sequence 0 → TR Hℓ ,Jℓ w(x) {1},S1 (b, x) → T R Hℓ ,Jℓ w(x) {1},S1 (x) → TW u (b; fX) → 0, (5.91) we may endow R Hℓ ,Jℓ w(x) {1},S1 (b, x) with a twisted stable framing TR Hℓ ,Jℓ w(x) {1},S1 (b, x) + ind TF,x ∼= R −I(b) + Eval∗ z1 Tew(x)−1 + Eval∗ z1 Tw(x)−1 (5… view at source ↗
Figure 12
Figure 12. Figure 12: This is an auxiliary enhanced sphere. This is, for generic data, a compact smooth manifold with corners. By splitting the short exact sequence 0 → TGWS 1 (a, b) → TGWS 1 (a) → TW s (b; fX) → 0, (5.95) we may endow GWS 1 (a, b) with a twisted stable framing TGWS 1 (a, b) + Eval∗ 0T X + R −I(b) ∼= R −I(a)−1 + EnEval∗ +∞TeGW + Eval∗ +∞T GW. (5.96) By rearranging and stabilizing, we may rewrite this as TGWS 1… view at source ↗
Figure 13
Figure 13. Figure 13: This is a schematic picture of the proof of Lemma 5.21. and Eval0,s : [0, +∞] × GWS 1 (a) → [0, +∞] × X (5.114) [PITH_FULL_IMAGE:figures/full_fig_p049_13.png] view at source ↗

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