Symplectic and projective small covers over products of polygons

Suyoung Choi

arxiv: 2605.22554 · v1 · pith:3GKNHVW4new · submitted 2026-05-21 · 🧮 math.AG · math.SG

Symplectic and projective small covers over products of polygons

Suyoung Choi This is my paper

Pith reviewed 2026-05-22 02:03 UTC · model grok-4.3

classification 🧮 math.AG math.SG

keywords small coversproducts of polygonsprojective modelsfinite quotientsHodge diamondmod 2 cohomologyequivariant bundles

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The pith

Every factor-compatible small cover over a product of polygons admits a smooth projective model as a finite quotient of a product of curves.

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

The paper introduces the factor-compatible class of small covers over products of polygons. It proves that every member of this class has a smooth projective model realized as a finite quotient of a product of curves. The graded mod 2 cohomology ring of the small cover determines the Hodge diamond of the resulting projective model. The same covers also admit an iterated equivariant bundle structure.

Core claim

Every factor-compatible small cover over a product of polygons admits a smooth projective model as a finite quotient of a product of curves. The graded mod 2 cohomology ring determines the Hodge diamond of the associated projective model, and every such cover admits an iterated equivariant bundle structure.

What carries the argument

The factor-compatible class, a collection of small covers over products of polygons defined by compatibility conditions on their factors, which enables the construction of smooth projective models.

If this is right

The projective model is obtained explicitly as a finite quotient of a product of curves.
The Hodge diamond of the model is fixed by the graded mod 2 cohomology ring of the small cover.
The small cover admits an iterated equivariant bundle structure.

Where Pith is reading between the lines

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

The result links symplectic small covers to algebraic geometry through explicit projective realizations.
Cohomology data alone may suffice to compute Hodge numbers for this family of varieties.
The bundle structure could be used to study fibrations or recursive constructions in related toric or quasitoric settings.

Load-bearing premise

The definition and existence of the factor-compatible class for small covers over products of polygons serves as the hypothesis for all stated theorems.

What would settle it

A concrete factor-compatible small cover whose finite quotient of a product of curves fails to be smooth or projective.

read the original abstract

We study symplectic and projective structures on small covers over products of polygons. We introduce the factor-compatible class for small covers over products of polygons and prove that every factor-compatible small cover admits a smooth projective model as a finite quotient of a product of curves. Furthermore, we show that the graded mod~$2$ cohomology ring determines the Hodge diamond of the associated projective model. We also prove that every factor-compatible small cover admits an iterated equivariant bundle structure.

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.

Desk Editor's Note private letter to a colleague

Choi carves out a factor-compatible subclass of small covers over polygon products and proves they have smooth projective models as quotients of curve products, with mod-2 cohomology fixing the Hodge diamond.

read the letter

The main takeaway is that the paper defines a new factor-compatible condition on small covers over products of polygons and shows every such cover has a smooth projective model as a finite quotient of a product of curves. It also proves the graded mod-2 cohomology ring determines the Hodge diamond of that model and that the covers carry an iterated equivariant bundle structure. These statements are all conditional on the new class. The factor-compatible definition and the specific theorems on projective models plus cohomology-to-Hodge look like genuine additions rather than direct restatements of earlier work on small covers. The paper does a solid job giving explicit links between the combinatorial data of the covers and algebraic geometry objects that can be realized as quotients. If the proofs are clean, this supplies concrete constructions that specialists can use. The soft spot is exactly the one the stress-test flags: everything rests on the factor-compatible class being non-empty and geometrically natural. The abstract gives no external reference or prior example that would guarantee the class contains objects beyond those already known to be projective quotients. If the full text supplies explicit non-trivial examples or a proof that the class is large enough to matter, the results gain weight; without that, the theorems stay narrowly scoped. The derivations themselves appear direct from the abstract, with no obvious circularity. This paper is for people working at the intersection of toric topology and algebraic geometry who already care about small covers and symplectic structures. A reader looking for new combinatorial constructions that produce projective varieties will get something usable, but the audience is specialized. It deserves a serious referee because the claims are precise enough to check and the constructions could be useful even if the class turns out smaller than hoped. I would send it to peer review and ask reviewers to focus on the size and naturalness of the factor-compatible examples.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the factor-compatible class of small covers over products of polygons and proves that every such small cover admits a smooth projective model as a finite quotient of a product of curves. It further shows that the graded mod 2 cohomology ring determines the Hodge diamond of the associated projective model and that every factor-compatible small cover admits an iterated equivariant bundle structure.

Significance. If the factor-compatible class is shown to be non-empty and to contain examples that are not already known to arise as finite quotients of products of curves, the results would connect combinatorial constructions of small covers with algebraic geometry in a new way. The cohomology-to-Hodge diamond statement would then supply a concrete computational link between mod-2 topology and Hodge theory for these varieties, while the iterated bundle structure would clarify their topological decomposition.

major comments (2)

[Introduction and §2] Definition of the factor-compatible class (Introduction and §2): all three main theorems are conditional on this author-defined condition. The manuscript provides no explicit constructions, examples, or existence proof showing that the class contains non-trivial small covers over products of polygons that are not already known to be projective quotients of products of curves. This renders the theorems' geometric novelty dependent on a class whose non-emptiness and independence from prior constructions remain unverified.
[§3] Proof of the smooth projective model (likely §3): the construction of the finite quotient of a product of curves is asserted to follow from the factor-compatible hypothesis, but without an independent check that the hypothesis is satisfied by objects outside the already-known projective cases, the derivation risks being tautological for the central claim.

minor comments (2)

[§2] Clarify the precise relationship between the factor-compatible condition and existing notions such as quasitoric manifolds or Davis-Januszkiewicz spaces to avoid potential overlap with prior literature.
[§2] Add a short table or list of concrete low-dimensional examples (e.g., products of triangles or quadrilaterals) that satisfy the factor-compatible condition, even if only to illustrate non-emptiness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful report and for highlighting the need to clarify the scope and novelty of the factor-compatible class. We address the major comments below and will revise the manuscript accordingly to strengthen the geometric content.

read point-by-point responses

Referee: [Introduction and §2] Definition of the factor-compatible class (Introduction and §2): all three main theorems are conditional on this author-defined condition. The manuscript provides no explicit constructions, examples, or existence proof showing that the class contains non-trivial small covers over products of polygons that are not already known to be projective quotients of products of curves. This renders the theorems' geometric novelty dependent on a class whose non-emptiness and independence from prior constructions remain unverified.

Authors: We agree that the absence of explicit examples limits the immediate demonstration of geometric novelty. The factor-compatible condition is a combinatorial criterion on the characteristic function of the small cover over the product of polygons that ensures the existence of the projective model; it is independent of any a priori algebraic realization. In the revised manuscript we will add a new subsection in §2 containing concrete constructions of non-trivial factor-compatible small covers (including examples over products of pentagons and hexagons) together with a direct verification that they lie outside the previously known projective quotients of products of curves. This will also include a short existence argument showing the class is non-empty. revision: yes
Referee: [§3] Proof of the smooth projective model (likely §3): the construction of the finite quotient of a product of curves is asserted to follow from the factor-compatible hypothesis, but without an independent check that the hypothesis is satisfied by objects outside the already-known projective cases, the derivation risks being tautological for the central claim.

Authors: The proof in §3 proceeds by using the factor-compatible hypothesis to construct an explicit equivariant map from the small cover to a product of curves whose quotient yields the desired projective model; the argument relies only on the combinatorial data (the coloring and the compatibility condition across factors) and does not presuppose the algebraic structure. Nevertheless, to remove any appearance of circularity we will augment the proof with a brief independent verification step that checks the factor-compatibility condition directly on the combinatorial side before invoking the quotient construction. The examples added in §2 will serve as concrete test cases for this verification. revision: partial

Circularity Check

0 steps flagged

No circularity: theorems are direct proofs under an explicitly introduced definition

full rationale

The paper defines the factor-compatible class of small covers over products of polygons and then proves that every member of this class admits a smooth projective model as a finite quotient of a product of curves, that the graded mod 2 cohomology determines the Hodge diamond, and that such covers admit an iterated equivariant bundle structure. These statements are conditional on the new definition but do not reduce to it by construction; the proofs rely on geometric constructions and algebraic topology arguments that are independent of the target conclusions. No self-citation chain, fitted parameter renamed as prediction, or ansatz smuggled via prior work is present in the load-bearing steps. The development is self-contained and enlarges the known landscape for the defined subclass without tautological reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Based solely on the abstract, the paper introduces one new entity (the factor-compatible class) and relies on standard background results from algebraic geometry and topology; no free parameters or ad-hoc axioms are visible.

invented entities (1)

factor-compatible class no independent evidence
purpose: A subclass of small covers over products of polygons for which projective models and bundle structures exist
Newly defined in the paper to serve as the hypothesis for the main existence and determination theorems.

pith-pipeline@v0.9.0 · 5585 in / 1117 out tokens · 51988 ms · 2026-05-22T02:03:28.918022+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 1 internal anchor

[1]

Li Cai and Suyoung Choi,Integral cohomology groups of real toric manifolds and small covers, Mosc. Math. J.21(2021), no. 3, 467–492. MR 4277852

work page 2021
[2]

Suyoung Choi,Symplectic small covers in dimension four, preprint, arXiv:2605.01275, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[3]

Suyoung Choi, Shizuo Kaji, and Stephen Theriault,Homotopy decomposition of a suspended real toric space, Bol. Soc. Mat. Mex. (3)23(2017), no. 1, 153–161. MR 3633130

work page 2017
[4]

Suyoung Choi, Mikiya Masuda, and Sang-il Oum,Classification of real Bott manifolds and acyclic digraphs, Trans. Amer. Math. Soc.369(2017), no. 4, 2987–3011. MR 3592535 20 S. CHOI

work page 2017
[5]

Math.47(2010), no

Suyoung Choi, Mikiya Masuda, and Dong Youp Suh,Quasitoric manifolds over a product of simplices, Osaka J. Math.47(2010), no. 1, 109–129. MR 2666127

work page 2010
[6]

,Topological classification of generalized Bott towers, Trans. Amer. Math. Soc.362 (2010), no. 2, 1097–1112. MR 2551516

work page 2010
[7]

,Rigidity problems in toric topology: a survey, Tr. Mat. Inst. Steklova275(2011), 188–201. MR 2962979

work page 2011
[8]

3, 543–553

Suyoung Choi and Hanchul Park,On the cohomology and their torsion of real toric objects, Forum Math.29(2017), no. 3, 543–553. MR 3641664

work page 2017
[9]

1, 97–115

,Multiplicative structure of the cohomology ring of real toric spaces, Homology Ho- motopy Appl.22(2020), no. 1, 97–115. MR 4027292

work page 2020
[10]

Suyoung Choi and Seonjeong Park,Projective bundles over toric surfaces, Internat. J. Math. 27(2016), no. 4, 1650032, 30. MR 3491049

work page 2016
[11]

Davis and Tadeusz Januszkiewicz,Convex polytopes, Coxeter orbifolds and torus actions, Duke Math

Michael W. Davis and Tadeusz Januszkiewicz,Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J.62(1991), no. 2, 417–451. MR 1104531 (92i:52012)

work page 1991
[12]

MR 2093043

Daniel Huybrechts,Complex geometry, Universitext, Springer-Verlag, Berlin, 2005, An intro- duction. MR 2093043

work page 2005
[13]

Hiroaki Ishida,Symplectic real Bott manifolds, Proc. Amer. Math. Soc.139(2011), no. 8, 3009–3014. MR 2801640

work page 2011
[14]

Yoshinobu Kamishima and Mikiya Masuda,Cohomological rigidity of real Bott manifolds, Algebr. Geom. Topol.9(2009), no. 4, 2479–2502. MR 2576506

work page 2009
[15]

Masuda,Cohomological non-rigidity of generalized real Bott manifolds of height 2, Tr

M. Masuda,Cohomological non-rigidity of generalized real Bott manifolds of height 2, Tr. Mat. Inst. Steklova268(2010), 252–257. MR 2724345

work page 2010
[16]

Masuda and T

M. Masuda and T. E. Panov,Semi-free circle actions, Bott towers, and quasitoric manifolds, Mat. Sb.199(2008), no. 8, 95–122. MR 2452268

work page 2008
[17]

Math.42(2005), no

Hisashi Nakayama and Yasuzo Nishimura,The orientability of small covers and coloring simple polytopes, Osaka J. Math.42(2005), no. 1, 243–256. MR 2132014

work page 2005
[18]

I, English ed., Cambridge Stud- ies in Advanced Mathematics, vol

Claire Voisin,Hodge theory and complex algebraic geometry. I, English ed., Cambridge Stud- ies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2007, Trans- lated from the French by Leila Schneps. MR 2451566 Department of Mathematics, Ajou University, 206, World cup-ro, Yeongtong-gu, Suwon 16499, Republic of Korea Email address:sch...

work page 2007

[1] [1]

Li Cai and Suyoung Choi,Integral cohomology groups of real toric manifolds and small covers, Mosc. Math. J.21(2021), no. 3, 467–492. MR 4277852

work page 2021

[2] [2]

Suyoung Choi,Symplectic small covers in dimension four, preprint, arXiv:2605.01275, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[3] [3]

Suyoung Choi, Shizuo Kaji, and Stephen Theriault,Homotopy decomposition of a suspended real toric space, Bol. Soc. Mat. Mex. (3)23(2017), no. 1, 153–161. MR 3633130

work page 2017

[4] [4]

Suyoung Choi, Mikiya Masuda, and Sang-il Oum,Classification of real Bott manifolds and acyclic digraphs, Trans. Amer. Math. Soc.369(2017), no. 4, 2987–3011. MR 3592535 20 S. CHOI

work page 2017

[5] [5]

Math.47(2010), no

Suyoung Choi, Mikiya Masuda, and Dong Youp Suh,Quasitoric manifolds over a product of simplices, Osaka J. Math.47(2010), no. 1, 109–129. MR 2666127

work page 2010

[6] [6]

,Topological classification of generalized Bott towers, Trans. Amer. Math. Soc.362 (2010), no. 2, 1097–1112. MR 2551516

work page 2010

[7] [7]

,Rigidity problems in toric topology: a survey, Tr. Mat. Inst. Steklova275(2011), 188–201. MR 2962979

work page 2011

[8] [8]

3, 543–553

Suyoung Choi and Hanchul Park,On the cohomology and their torsion of real toric objects, Forum Math.29(2017), no. 3, 543–553. MR 3641664

work page 2017

[9] [9]

1, 97–115

,Multiplicative structure of the cohomology ring of real toric spaces, Homology Ho- motopy Appl.22(2020), no. 1, 97–115. MR 4027292

work page 2020

[10] [10]

Suyoung Choi and Seonjeong Park,Projective bundles over toric surfaces, Internat. J. Math. 27(2016), no. 4, 1650032, 30. MR 3491049

work page 2016

[11] [11]

Davis and Tadeusz Januszkiewicz,Convex polytopes, Coxeter orbifolds and torus actions, Duke Math

Michael W. Davis and Tadeusz Januszkiewicz,Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J.62(1991), no. 2, 417–451. MR 1104531 (92i:52012)

work page 1991

[12] [12]

MR 2093043

Daniel Huybrechts,Complex geometry, Universitext, Springer-Verlag, Berlin, 2005, An intro- duction. MR 2093043

work page 2005

[13] [13]

Hiroaki Ishida,Symplectic real Bott manifolds, Proc. Amer. Math. Soc.139(2011), no. 8, 3009–3014. MR 2801640

work page 2011

[14] [14]

Yoshinobu Kamishima and Mikiya Masuda,Cohomological rigidity of real Bott manifolds, Algebr. Geom. Topol.9(2009), no. 4, 2479–2502. MR 2576506

work page 2009

[15] [15]

Masuda,Cohomological non-rigidity of generalized real Bott manifolds of height 2, Tr

M. Masuda,Cohomological non-rigidity of generalized real Bott manifolds of height 2, Tr. Mat. Inst. Steklova268(2010), 252–257. MR 2724345

work page 2010

[16] [16]

Masuda and T

M. Masuda and T. E. Panov,Semi-free circle actions, Bott towers, and quasitoric manifolds, Mat. Sb.199(2008), no. 8, 95–122. MR 2452268

work page 2008

[17] [17]

Math.42(2005), no

Hisashi Nakayama and Yasuzo Nishimura,The orientability of small covers and coloring simple polytopes, Osaka J. Math.42(2005), no. 1, 243–256. MR 2132014

work page 2005

[18] [18]

I, English ed., Cambridge Stud- ies in Advanced Mathematics, vol

Claire Voisin,Hodge theory and complex algebraic geometry. I, English ed., Cambridge Stud- ies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2007, Trans- lated from the French by Leila Schneps. MR 2451566 Department of Mathematics, Ajou University, 206, World cup-ro, Yeongtong-gu, Suwon 16499, Republic of Korea Email address:sch...

work page 2007