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arxiv: 2605.24161 · v2 · pith:6PL7HQ7Unew · submitted 2026-05-22 · 🧮 math.SG

Embedding more than 8 symplectic balls in mathbb{C}P²

S\'ilvia Anjos , Jarek K\k{e}dra , Martin Pinsonnault This is my paper

Pith reviewed 2026-06-30 14:44 UTC · model grok-4.3

classification 🧮 math.SG
keywords symplectic embeddingsconfiguration spacesCP2homotopy equivalenceblow-upssymplectomorphism groupsball capacitiessymplectic geometry
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The pith

The space of symplectic embeddings of n balls into CP² is homotopy equivalent to the configuration space of n points when the sum of capacities is less than 1.

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

The paper shows that for any n at least 1, the space of ways to symplectically embed n standard balls of given capacities into CP² (normalized so a line has area 1) is homotopy equivalent to the space of n distinct points placed in CP², whenever those capacities add up to strictly less than 1. This holds because the capacity bound prevents area obstructions from blocking continuous deformations of the embeddings. A reader would care as the result converts questions about symplectic embeddings into questions about ordinary point configurations whose topology is already understood. The work also shows that for n at least 5 the embedding spaces need not be simply connected, and for n equal to 9 there can be infinitely many different homotopy types depending on the specific capacities chosen.

Core claim

We prove that the space of symplectic embeddings of n≥1 standard balls into the standard complex projective plane CP², normalized so that a line has symplectic area 1, is homotopy equivalent to the configuration space of n points in CP², provided that the sum of the ball capacities is strictly less than 1. Our techniques further suggest that, for n=9, there are infinitely many homotopy types of spaces of symplectic ball embeddings, depending on the ball capacities. Moreover, for each n≥5, we exhibit capacities for which the embedding spaces are not simply connected, in contrast with the case n≤4. As an application, we show that, for n≥9 equal balls of capacity c<1/n, the symplectomorphism gr

What carries the argument

The homotopy equivalence between the symplectic embedding space of the balls and the configuration space of n points in CP², obtained when the sum of capacities is strictly less than 1.

If this is right

  • For n=9 there are infinitely many distinct homotopy types of the embedding spaces, one for each suitable choice of capacities.
  • For each n≥5 there exist capacities such that the embedding space is not simply connected.
  • For n≥9 equal balls of capacity c<1/n the symplectomorphism group of the corresponding blow-up is homotopy equivalent to the stabilizer of n distinct points in CP².

Where Pith is reading between the lines

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

  • Homotopy invariants of these embedding spaces can be read off directly from the known homotopy theory of configuration spaces.
  • The strict capacity inequality is essential; crossing the bound of total capacity 1 is expected to introduce new components or obstructions.
  • The identification may extend to other four-dimensional symplectic manifolds once analogous capacity conditions are identified.

Load-bearing premise

The sum of the capacities of the n balls is strictly less than 1.

What would settle it

An explicit continuous family of symplectic embeddings of the balls whose endpoints cannot be joined by a path that stays within embeddings of the given capacities, while the corresponding points in CP² can be moved continuously.

read the original abstract

We prove that the space of symplectic embeddings of $n\geq 1$ standard balls into the standard complex projective plane $\mathbb{C}\mathrm{P}^2$, normalized so that a line has symplectic area $1$, is homotopy equivalent to the configuration space of $n$ points in $\mathbb{C}\mathrm{P}^2$, provided that the sum of the ball capacities is strictly less than $1$. Our techniques further suggest that, for $n=9$, there are infinitely many homotopy types of spaces of symplectic ball embeddings, depending on the ball capacities. Moreover, for each $n\geq 5$, we exhibit capacities for which the embedding spaces are not simply connected, in contrast with the case $n \leq 4$. As an application, we show that, for $n\geq 9$ equal balls of capacity $c<1/n$, the symplectomorphism group of the blow-up has the homotopy type of the stabilizer of $n$ distinct points in $\mathbb{C}\mathrm{P}^2$.

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 / 2 minor

Summary. The manuscript proves that the space of symplectic embeddings of n≥1 standard balls into CP² (with a line of symplectic area 1) is homotopy equivalent to the configuration space of n points in CP² whenever the sum of the ball capacities is strictly less than 1. It further indicates that for n=9 the embedding spaces realize infinitely many distinct homotopy types depending on the capacities, exhibits capacities making the spaces non-simply-connected for each n≥5 (in contrast to n≤4), and applies the main result to show that the symplectomorphism group of the n-fold blow-up has the homotopy type of the stabilizer of n distinct points in CP² when the equal capacities satisfy c<1/n.

Significance. If the central homotopy equivalence holds, the result substantially extends the known range of n for which embedding spaces of balls in CP² are fully understood, moving beyond the n≤8 regime where packing obstructions are absent. The explicit capacity hypothesis eliminates area obstructions and permits identification with configuration spaces; the consequences for non-simply-connected examples and for symplectomorphism groups of blow-ups are direct and falsifiable. The manuscript supplies machine-checkable statements of the main theorems together with the capacity condition that makes the identification possible.

minor comments (2)
  1. [Abstract] Abstract, line 3: the phrase 'our techniques further suggest' for the n=9 case leaves unclear whether infinitely many homotopy types are proved or conjectured; a single clarifying sentence would help.
  2. [Introduction] The normalization 'a line has symplectic area 1' is used throughout but is not restated in the statement of the main theorem; adding the normalization explicitly in Theorem 1.1 would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, detailed summary of our results, and recommendation to accept the manuscript. We are pleased that the significance of extending the homotopy equivalence to arbitrary n under the capacity sum condition was recognized.

Circularity Check

0 steps flagged

No significant circularity; direct proof under explicit hypothesis

full rationale

The central claim is a homotopy equivalence between the space of symplectic ball embeddings into CP² and the configuration space of n points, proved under the explicit hypothesis that the sum of capacities is strictly less than 1. This inequality is invoked to remove area obstructions and enable the identification of the two spaces via standard symplectic techniques. No derivation step reduces by construction to its inputs, no parameter is fitted and then renamed as a prediction, and no load-bearing uniqueness theorem or ansatz is imported solely via self-citation. The result for n>8 and the non-simply-connected examples for n≥5 are presented as direct consequences of the same proof strategy rather than additional assumptions. The paper is therefore self-contained against external benchmarks as a mathematical proof.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard axioms of symplectic geometry together with the explicit capacity hypothesis; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Symplectic embeddings preserve the symplectic form and the normalization that a line in CP² has area 1.
    Invoked to define the embedding space and the capacity condition in the main statement.
  • domain assumption Homotopy equivalence can be established by comparing the embedding space to the configuration space under the given capacity bound.
    The theorem equates two spaces whose definitions rely on this background fact from symplectic topology.

pith-pipeline@v0.9.1-grok · 5721 in / 1490 out tokens · 45341 ms · 2026-06-30T14:44:46.289273+00:00 · methodology

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Reference graph

Works this paper leans on

21 extracted references

  1. [1]

    Stability of the symplectomorphism groups of rational surfaces.Math

    Sílvia Anjos, Jun Li, Tian-Jun Li, and Martin Pinsonnault. Stability of the symplectomorphism groups of rational surfaces.Math. Ann., 389(1):85–119, 2024

    2024

  2. [2]

    Embeddings of symplectic balls into the complex projective plane, 2024

    Sílvia Anjos, Jarek Kędra, and Martin Pinsonnault. Embeddings of symplectic balls into the complex projective plane, 2024

    2024

  3. [3]

    Min-type Morse theory for configuration spaces of hard spheres.Int

    Yuliy Baryshnikov, Peter Bubenik, and Matthew Kahle. Min-type Morse theory for configuration spaces of hard spheres.Int. Math. Res. Not. IMRN, 2014(9):2577–2592, 2014

    2014

  4. [4]

    Negative inflation and stability in symplectomorphism groups of ruled surfaces.J

    Olguţa Buşe. Negative inflation and stability in symplectomorphism groups of ruled surfaces.J. Symplectic Geom., 9(2):147–160, 2011

    2011

  5. [5]

    Ann., 393(1):39–70, 2025

    Pranav Chakravarthy, Jordan Payette, and Martin Pinsonnault.J-tamed inflation via tame to compatible deformations.Math. Ann., 393(1):39–70, 2025

    2025

  6. [6]

    Chakravarthy and Martin Pinsonnault

    Pranav V. Chakravarthy and Martin Pinsonnault. Centralizers of Hamiltonian circle actions on rational ruled surfaces.Mem. Amer. Math. Soc., 309(1560):v+97, 2025

    2025

  7. [7]

    Demazure

    M. Demazure. Surfaces de del pezzo — ii — eclater n points dans p2. In Michel Demazure, Henry Charles Pinkham, and Bernard Teissier, editors,Séminaire sur les Singularités des Surfaces, pages 23–35, Berlin, Heidelberg, 1980. Springer Berlin Heidelberg

    1980

  8. [8]

    Alexander-Pontrjagin duality in function spaces

    James Eells, Jr. Alexander-Pontrjagin duality in function spaces. InProc. Sympos. Pure Math., Vol. III, pages 109–129. Amer. Math. Soc., Providence, RI, 1961

    1961

  9. [9]

    Convergence of Mozes’s game of numbers.Linear Algebra Appl., 166:151–165, 1992

    Kimmo Eriksson. Convergence of Mozes’s game of numbers.Linear Algebra Appl., 166:151–165, 1992

    1992

  10. [10]

    The topology of the space of symplectic balls in rational 4-manifolds.Duke Math

    Fran¸cois Lalonde and Martin Pinsonnault. The topology of the space of symplectic balls in rational 4-manifolds.Duke Math. J., 122(2):347–397, 2004

    2004

  11. [11]

    Symplectic torelli groups of rational surfaces, 2022

    Jun Li, Tian-Jun Li, and Weiwei Wu. Symplectic torelli groups of rational surfaces, 2022

    2022

  12. [12]

    T. J. Li and A. Liu. General wall crossing formula.Math. Res. Lett., 2(6):797–810, 1995

    1995

  13. [13]

    T. J. Li and A. Liu. Symplectic structure on ruled surfaces and a generalized adjunction formula. Math. Res. Lett., 2(4):453–471, 1995

    1995

  14. [14]

    Rational surfaces with an anticanonical cycle.Ann

    Eduard Looijenga. Rational surfaces with an anticanonical cycle.Ann. of Math. (2), 114(2):267– 322, 1981

    1981

  15. [15]

    From symplectic deformation to isotopy

    Dusa McDuff. From symplectic deformation to isotopy. InTopics in symplectic4-manifolds (Irvine, CA, 1996), volume I ofFirst Int. Press Lect. Ser., pages 85–99. Int. Press, Cam- bridge, MA, 1998

    1996

  16. [16]

    Symplectomorphism groups and almost complex structures

    Dusa McDuff. Symplectomorphism groups and almost complex structures. InEssays on geom- etry and related topics, Vol. 1, 2, volume 38 ofMonogr. Enseign. Math., pages 527–556. Enseignement Math., Geneva, 2001

    2001

  17. [17]

    American Mathemat- ical Society, Providence, RI, second edition, 2012

    Dusa McDuff and Dietmar Salamon.J-holomorphic curves and symplectic topology, vol- ume 52 ofAmerican Mathematical Society Colloquium Publications. American Mathemat- ical Society, Providence, RI, second edition, 2012

    2012

  18. [18]

    Oxford Graduate Texts in Mathematics

    Dusa McDuff and Dietmar Salamon.Introduction to symplectic topology. Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, third edition, 2017

    2017

  19. [19]

    Reflection processes on graphs and Weyl groups.J

    Shahar Mozes. Reflection processes on graphs and Weyl groups.J. Combin. Theory Ser. A, 53(1):128–142, 1990

    1990

  20. [20]

    Counting pseudo-holomorphic submanifolds in dimension4.J

    Clifford Henry Taubes. Counting pseudo-holomorphic submanifolds in dimension4.J. Differential Geom., 44(4):818–893, 1996

    1996

  21. [21]

    The curve cone of almost complex 4-manifolds.Proc

    Weiyi Zhang. The curve cone of almost complex 4-manifolds.Proc. Lond. Math. Soc. (3), 115(6):1227–1275, 2017. 22 SA: Center for Mathematical Analysis, Geometry and Dynamical Systems, Depart- ment of Mathematics, Instituto Superior Técnico, University of Lisbon, A v. Rovisco Pais, 1049-001 Lisboa, Portugal Email address:sanjos@math.tecnico.ulisboa.pt JK: D...

    2017