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arxiv: 2606.17370 · v1 · pith:52M32XDZnew · submitted 2026-06-15 · 🧮 math.SG

Relative symplectic cohomology in complex projective spaces

Adi Dickstein , Yaniv Ganor This is my paper

Pith reviewed 2026-06-27 01:35 UTC · model grok-4.3

classification 🧮 math.SG
keywords relative symplectic cohomologyNovikov ringcomplex projective spacedisplacement energyFloer complexesMorse-Bott cascadessymplectic invariants
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The pith

Relative symplectic cohomology over the Novikov ring is computed for balls and their complements in CP^n, yielding new estimates for stable displacement energy of their boundaries.

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

The paper establishes a computation of relative symplectic cohomology, an invariant for compact subsets of closed symplectic manifolds, but now over the Novikov ring instead of the more common Novikov field. Most prior examples stayed over the field because ring-level work demands explicit control of Floer complexes for Hamiltonians that are not small Morse functions. The authors supply those complexes for J-shaped Hamiltonians on CP^n in the Morse-Bott setting with cascades, obtain the groups for balls and complements, and convert the result into bounds on the stable displacement energy of ball boundaries.

Core claim

We present a computation of relative symplectic cohomology over the Novikov ring for balls and their complements in CP^n. Our computation relies on explicit descriptions of Floer complexes, in the Morse-Bott setting with cascades, for J-shaped Hamiltonians on CP^n. This allows us to deduce new estimates for the stable displacement energy of the boundaries of balls in CP^n.

What carries the argument

Relative symplectic cohomology over the Novikov ring, obtained from the Floer complexes of J-shaped Hamiltonians in the Morse-Bott setting with cascades.

If this is right

  • The stable displacement energy of the boundary of any ball in CP^n admits a new upper or lower bound derived from the computed groups.
  • The relative symplectic cohomology groups of a ball and of its complement are now known explicitly as modules over the Novikov ring.
  • The same method supplies the invariant for both a compact set and its complement inside the same manifold.
  • The collection of ring-level computations is enlarged by one family of examples.

Where Pith is reading between the lines

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

  • The same style of explicit Floer-complex calculation could be attempted on other toric manifolds whose Hamiltonians admit similar J-shaped profiles.
  • The new energy bounds can be compared directly with displacement-energy estimates coming from other symplectic capacities.
  • If the groups turn out to be sensitive to the radius of the ball, they may distinguish ball sizes that are not distinguished by the field-level invariant.

Load-bearing premise

The explicit descriptions of the Floer complexes for J-shaped Hamiltonians on CP^n in the Morse-Bott setting with cascades correctly determine the relative symplectic cohomology groups over the Novikov ring.

What would settle it

An independent calculation of the same relative symplectic cohomology groups for a ball in CP^2 that produces a different answer over the Novikov ring would falsify the result.

Figures

Figures reproduced from arXiv: 2606.17370 by Adi Dickstein, Yaniv Ganor.

Figure 2
Figure 2. Figure 2: The complex of Hℓ on CP 2 . The different dash styles and shades of gray are used as a visual cue to separate the subcomplexes The complex thus splits as a direct sum of n + 1 subcomplexes. Since all of these subcomplexes have the same form, it suffices to analyze a single summand; the analysis for the remaining summands is identical. xˇi xˇi+n+1 xˇi+2(n+1) xˆi+1 xˆi+n+2 xˆi+2n+3 [PITH_FULL_IMAGE:figures/… view at source ↗
Figure 3
Figure 3. Figure 3: A subcomplex in the direct sum decomposition. These subcomplexes are [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The i-th zigzag, together with the information obtained from C n . By a similar combination of index formulas for the trivializations τB and τT , and the positivity of intersections, we deduce the vanishing of most matrix elements of the continuation map φ: CF(Hℓ ;Z) → CF(Hℓ ′; Z) for ℓ ′ ≥ ℓ. In fact, all matrix elements vanish except for ⟨φxˇi , xˇi⟩ and ⟨φxˆi , xˆi⟩. See Theorem 5.20. Now, H0 provides e… view at source ↗
Figure 5
Figure 5. Figure 5: The continuation φ: CF(H0;Z) → CF(Hℓ ; Z) in CP 2 . Note the leftmost generator is the generator of CF(H0; Z) corresponding to the global minimum, and the pair on the right are the generators corresponding to the global maximum at the CP 1 at infinity. Since CP n is a closed symplectic manifold, the continuation map φ: CF(H0; Z) → CF(Hℓ ; Z) is a quasi-isomorphism, in particular, for all 0 ≤ i ≤ n, φ(xi) i… view at source ↗
Figure 6
Figure 6. Figure 6: Differential of Floer subcomplex. This argument is detailed in Section 6.4. Next, we return to the complexes over the Novikov ring, using Formula (11), relating indices and energy, to compute the topological energy of the solutions. After a change of basis consisting only of multiplications by +1, see Proposition 9.3, we end up with n + 1 copies of the following complex: T n(1−∆) T ∆ T ∆ T ∆ T n(1−∆) T ∆ T… view at source ↗
Figure 7
Figure 7. Figure 7: The i-th subcomplex with Novikov coefficients weighted by energy. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The limit is an infinite zigzag Now, we take the completion of the complex. All the bottom generators, ( ˇxi) ∞ i=0, are closed, and, in fact they generate ker d as a complete module. In order to compute the homology, namely ker d / im d, we use the fact that the notion of a Schauder basis applies verbatim in the setting of torsion-free Novikov modules. We thus find a Schauder basis for ker d, (ei)i∈N, suc… view at source ↗
Figure 9
Figure 9. Figure 9: The complex of Hℓ on CP 2 . The different shades of gray are used as a visual cue to separate the subcomplexes. One of them is labeled with the signs A and B. Employing similar techniques we also compute the continuation maps between the Floer cochain complexes of the acceleration data of two different balls: Let 0 < ∆′ ≤ ∆ < 1. Consider additionally the corresponding acceleration data (H′ ℓ )ℓ≥0 for the b… view at source ↗
Figure 11
Figure 11. Figure 11: The 2-cube and the Hamiltonians at its corners. Let us define the following space of 2-cubes of Hamiltonians, fix ϵ < 1 2 : H (ℓ, ∆, ℓ′ , ∆ ′ ) = n H: [0, 1] × [0, 1] → C ∞(S 1 × CP n , R) | H = H0 in ϵ-balls around the verticeso equipped with the C ∞ topology. Moreover, let us denote by J ∞ 2 (M, ω) (the 2 denoting 2-cubes), the space of all 2-cubes of compatible almost complex structure, such that they … view at source ↗
Figure 12
Figure 12. Figure 12: The recursion step is demonstrated. J ′ m+1 is depicted to be chosen in Nm+1 and the dashed line represents the homotopy constructed. 235 [PITH_FULL_IMAGE:figures/full_fig_p235_12.png] view at source ↗
read the original abstract

Relative symplectic cohomology is an invariant of compact subsets of a closed symplectic manifold, introduced by Varolgunes. There are many examples of computations of this invariant over the Novikov field, but the collection of computed examples over the Novikov ring is still quite limited. One reason for this is that such computations require determining the relevant Floer complexes for Hamiltonians that are not necessarily $C^2$-small Morse functions. In this work, we present a computation of relative symplectic cohomology over the Novikov ring for balls and their complements in $\mathbb{C}P^n$. Our computation relies on explicit descriptions of Floer complexes, in the Morse--Bott setting with cascades, for J-shaped Hamiltonians on $\mathbb{C}P^n$. This allows us to deduce new estimates for the stable displacement energy of the boundaries of balls in $\mathbb{C}P^n$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper computes relative symplectic cohomology over the Novikov ring for balls and their complements in CP^n. The computation relies on explicit descriptions of the Floer complexes, in the Morse-Bott setting with cascades, for J-shaped Hamiltonians on CP^n. From these groups the authors deduce new estimates for the stable displacement energy of the boundaries of balls in CP^n.

Significance. If the computation is correct, the work enlarges the small set of explicit examples of relative symplectic cohomology over the Novikov ring (as opposed to the Novikov field) and supplies concrete new bounds on stable displacement energy in CP^n. Both contributions are of interest to researchers working on symplectic invariants and capacities.

minor comments (1)
  1. The abstract refers to 'J-shaped Hamiltonians' without a definition or citation; a short explanation or pointer to the relevant section would improve accessibility.

Simulated Author's Rebuttal

0 responses · 1 unresolved

We thank the referee for their review and for acknowledging the potential interest of the explicit computations over the Novikov ring and the resulting stable displacement energy bounds, conditional on correctness. No specific major comments appear in the report.

standing simulated objections not resolved
  • The source of the 'uncertain' recommendation is not articulated, and the referee notes uncertainty about whether the computation is correct without identifying particular issues in the Floer complex descriptions or cascade arguments that would allow a targeted response.

Circularity Check

0 steps flagged

No significant circularity; direct Floer computation

full rationale

The paper's central claim is a computation of relative symplectic cohomology groups over the Novikov ring, obtained from explicit descriptions of Floer complexes for J-shaped Hamiltonians on CP^n in the Morse-Bott cascade setting. The abstract and strongest claim present this as arising from standard Floer data without any reduction of outputs to fitted parameters, self-definitions, or self-citation chains. No load-bearing step equates a derived quantity to its input by construction. The derivation is self-contained against external benchmarks of Floer theory, consistent with the reader's score of 2.0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The computation rests on standard Floer-theoretic constructions whose details are not supplied here.

axioms (1)
  • standard math Standard properties of Floer homology, Morse-Bott cascades, and the Novikov ring in symplectic geometry hold for the chosen Hamiltonians on CP^n.
    The abstract invokes these background structures to define the complexes.

pith-pipeline@v0.9.1-grok · 5668 in / 1156 out tokens · 32784 ms · 2026-06-27T01:35:00.493104+00:00 · methodology

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