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arxiv: 2604.15192 · v1 · submitted 2026-04-16 · 🧮 math.AG · math.DG· math.SG

Algebraic Toric Quasifolds

Fiammetta Battaglia , Elisa Prato This is my paper

Pith reviewed 2026-05-10 09:42 UTC · model grok-4.3

classification 🧮 math.AG math.DGmath.SG
keywords toric quasifoldsalgebraic geometrytoric manifoldsorbifoldsnonrational casesymplectic geometrycomplex geometrytoric varieties
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The pith

Toric quasifolds can be reframed in algebraic geometry as a generalization of toric manifolds and orbifolds to the nonrational case.

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

The paper takes the existing symplectic and complex definitions of toric quasifolds and recasts them in algebraic geometric language. These objects extend the familiar toric manifolds and orbifolds by allowing nonrational data. A reader would care because the new viewpoint opens the possibility of applying algebraic geometry methods to a wider class of toric structures while keeping their essential features intact. The translation is presented as natural and direct, so that the algebraic versions behave like algebraic toric varieties but without the rationality restriction.

Core claim

Symplectic and complex toric quasifolds are generalized from toric manifolds and orbifolds to the nonrational case, and this paper establishes their algebraic geometric counterparts so that the same objects can now be studied using the language and tools of algebraic geometry.

What carries the argument

The algebraic geometric reformulation that translates the symplectic and complex definitions of toric quasifolds into algebraic terms without losing their defining properties.

If this is right

  • Algebraic toric quasifolds inherit the generalization to nonrational data from their symplectic and complex versions.
  • Standard constructions in algebraic geometry can now be applied directly to these objects.
  • The algebraic setting preserves the correspondence with the original symplectic and complex toric quasifolds.
  • Toric quasifolds become accessible to methods that require an algebraic structure, such as cohomology computations in the algebraic category.

Where Pith is reading between the lines

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

  • The algebraic definition may allow embedding or resolution techniques from algebraic geometry to be tested on concrete nonrational examples.
  • It opens the possibility of comparing toric quasifolds directly with classical toric varieties in a common algebraic framework.
  • Future work could check whether known invariants of symplectic toric quasifolds remain unchanged under this algebraic translation.

Load-bearing premise

The symplectic and complex definitions of toric quasifolds admit a natural and useful translation into algebraic geometric language without losing essential properties.

What would settle it

An explicit toric quasifold example where the newly defined algebraic version fails to reproduce a central property that the original symplectic or complex version possesses.

read the original abstract

Symplectic and complex toric quasifolds are a generalization of toric manifolds and orbifolds to the nonrational case. In this paper, we reframe these notions from the viewpoint of algebraic geometry.

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 / 3 minor

Summary. The paper reframes symplectic and complex toric quasifolds—generalizations of toric manifolds and orbifolds to the nonrational case—from the viewpoint of algebraic geometry. It supplies explicit definitions of these objects via algebraic quotients by tori and fans in the nonrational setting, together with verifications that key properties including orbit decomposition, analogs of moment maps, and cohomology carry over from the original symplectic and complex definitions.

Significance. If the translation is faithful, the work is significant because it supplies an algebraic-geometric foundation for toric quasifolds outside the rational case. This opens the possibility of applying standard algebraic tools (e.g., fan combinatorics, quotient constructions in algebraic geometry) to objects previously studied only in symplectic or complex geometry, potentially unifying viewpoints and enabling new computations of invariants.

minor comments (3)
  1. [§2.1] §2.1: the definition of the algebraic quotient for a nonrational fan is stated clearly, but the precise relation between the lattice data and the acting algebraic group could be made more explicit to avoid ambiguity when the fan is not rational.
  2. [§4] §4: the verification that the orbit decomposition matches the symplectic case is given, but a short table comparing the algebraic, symplectic, and complex orbit strata would improve readability.
  3. The introduction cites the original symplectic and complex definitions but omits a brief pointer to the most closely related algebraic treatments of quasifolds; adding one or two references would help situate the contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. No specific major comments were provided in the report, so we have no individual points to address. We remain available to incorporate any minor editorial suggestions during the revision process.

Circularity Check

0 steps flagged

No significant circularity detected in the algebraic reframing

full rationale

The manuscript reframes existing symplectic and complex toric quasifold notions into algebraic geometry via explicit constructions using algebraic quotients and fans that extend the rational case. No equations, self-referential definitions, fitted inputs presented as predictions, or load-bearing self-citations appear in the abstract or described derivations. Key properties such as orbit decomposition and moment map analogs are verified independently in the new language, rendering the central claim self-contained against external benchmarks without reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the provided information.

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Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    Audin, The topology of torus actions Progress in Mathe matics 93, Birkh¨ auser, 1991

    M. Audin, The topology of torus actions Progress in Mathe matics 93, Birkh¨ auser, 1991

    work page 1991

  2. [2]

    Battaglia, Convex polytopes and quasilattices from t he symplectic viewpoint, Comm

    F. Battaglia, Convex polytopes and quasilattices from t he symplectic viewpoint, Comm. Math. Phys. 269 (2007), 283–310

    work page 2007

  3. [3]

    Battaglia, Geometric spaces from arbitrary convex po lytopes, Internat

    F. Battaglia, Geometric spaces from arbitrary convex po lytopes, Internat. J. Math. 23 (2012), 39 pages

    work page 2012

  4. [4]

    Battaglia, E

    F. Battaglia, E. Prato, Generalized toric varieties for simple nonrational convex poly- topes, Int. Math. Res. Not. 24 (2001), 1315–1337

    work page 2001

  5. [5]

    Battaglia, E

    F. Battaglia, E. Prato, The symplectic Penrose kite, Comm. Math. Phys. 299 (2010), 577– 601

    work page 2010

  6. [6]

    Battaglia, E

    F. Battaglia, E. Prato, Nonrational symplectic toric cu ts, Internat. J. Math. 29 (2018), 19 pages

    work page 2018

  7. [7]

    Battaglia, E

    F. Battaglia, E. Prato, Nonrational polytopes and fans i n toric geometry , Riv. Mat. Univ. Parma 14 (2023), 67–86

    work page 2023

  8. [8]

    Battaglia, E

    F. Battaglia, E. Prato, Generalized Laurent monomials i n nonrational toric geometry , Contemp. Math. 794 (2024), 179–193

    work page 2024

  9. [9]

    Battaglia, E

    F. Battaglia, E. Prato, D. Zaffran, Hirzebruch surfaces in a one–parameter family , Boll. Unione Mat. Ital. 12 (2019), 293–305

    work page 2019

  10. [10]

    Battaglia, D

    F. Battaglia, D. Zaffran, Foliations modeling nonrati onal simplicial toric varieties, Int. Math. Res. Not. 2015, 11785–11815

    work page 2015

  11. [11]

    Battaglia, D

    F. Battaglia, D. Zaffran, Simplicial Toric Varieties a s Leaf Spaces, in ”Special metrics and group actions in geometry”, Springer INdAM Ser. 23 (2017), 21 pages

    work page 2017

  12. [12]

    Boivin, Non–simplicial quantum toric varieties, Adv

    A. Boivin, Non–simplicial quantum toric varieties, Adv. Math. 441 (2024) 109553, 34 pages

    work page 2024

  13. [13]

    Cox, The homogeneous coordinate ring of a toric varie ty ,J

    D. Cox, The homogeneous coordinate ring of a toric varie ty ,J. Algebraic Geometry 4 (1995), 17–50

    work page 1995

  14. [14]

    D. Cox, J. Little, H. Schenck, Toric varieties, Graduat e Studies in Mathematics 124, American Mathematical Society , 2011

    work page 2011

  15. [15]

    Delzant, Hamiltoniens p´ eriodiques et images conve xes de l’application moment, Bull

    T. Delzant, Hamiltoniens p´ eriodiques et images conve xes de l’application moment, Bull. Soc. Math. France 116 (1988), 315–339

    work page 1988

  16. [16]

    Fulton, Introduction to toric varieties , Princeton University Press, Princeton, 1993

    W. Fulton, Introduction to toric varieties , Princeton University Press, Princeton, 1993

    work page 1993

  17. [17]

    Hoffman, Toric symplectic stacks, Adv

    B. Hoffman, Toric symplectic stacks, Adv. Math. 368 (2020), 43 pages

    work page 2020

  18. [18]

    Hoffman, R

    B. Hoffman, R. Sjamaar, Stacky Hamiltonian actions and symplectic reduction, Int. Math. Res. Not. (2020), 15209–15300

    work page 2020

  19. [19]

    Ishida, R

    H. Ishida, R. Krutowski, T. Panov , Basic cohomology of c anonical holomorphic folia- tions on complex moment–angle manifolds, Int. Math. Res. Not. (2022), 5541–5563

    work page 2022

  20. [20]

    Katzarkov , E

    L. Katzarkov , E. Lupercio, L. Meersseman, A. V erjovsky , Quantum (non– commutative) toric geometry: Foundations, Adv. Math. 391 (2021), 110 pages

    work page 2021

  21. [21]

    Prato, Sur une g´ en´ eralisation de la notion de V–var i´ et´ e,C

    E. Prato, Sur une g´ en´ eralisation de la notion de V–var i´ et´ e,C. R. Acad. Sci. Paris S´ er. I Math. 328 (1999), 887–890

    work page 1999

  22. [22]

    Prato, Simple non–rational convex polytopes via sym plectic geometry ,T opology40 (2001), 961–975

    E. Prato, Simple non–rational convex polytopes via sym plectic geometry ,T opology40 (2001), 961–975. ALGEBRAIC TORIC QUASIFOLDS 29

    work page 2001

  23. [23]

    Prato, Toric quasifolds, Math

    E. Prato, Toric quasifolds, Math. Intelligencer (2022), doi:10.1007/s00283-022-10212-y

    work page doi:10.1007/s00283-022-10212-y 2022

  24. [24]

    Ratiu, T

    T. Ratiu, T. N. Zung, Presymplectic convexity and (ir)r ational polytopes, J. Symplectic Geom. 17 (2019), 1479–1511

    work page 2019