pith. sign in

arxiv: 2604.13198 · v1 · submitted 2026-04-14 · 🧮 math.DG · math.AG· math.AP

New examples of affine Calabi-Yau 3-folds with maximal volume growth

Shih-Kai Chiu , Ronan J. Conlon , Fr\'ed\'eric Rochon This is my paper

Pith reviewed 2026-05-10 13:48 UTC · model grok-4.3

classification 🧮 math.DG math.AGmath.AP
keywords Calabi-Yau metricsaffine Calabi-Yau threefoldsvolume growthCalabi-Yau conessmoothingsorbifold singularitiescomplete Ricci-flat metricsnon-product metrics
0
0 comments X

The pith

New complete Calabi-Yau metrics exist on smoothings of 3D Calabi-Yau cones that are not products and have orbifold singularities away from the vertex.

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

The paper constructs complete Calabi-Yau metrics on smoothings of three-dimensional Calabi-Yau cones. These metrics have maximal volume growth and are not products of lower-dimensional Calabi-Yau metrics. The underlying spaces carry orbifold singularities located away from the cone vertex. A sympathetic reader would care because the examples enlarge the known collection of complete Ricci-flat Kähler metrics on non-compact affine Calabi-Yau threefolds beyond product cases. This matters for mapping out possible geometries of non-compact Calabi-Yau manifolds with large volume growth.

Core claim

We construct examples of complete Calabi-Yau metrics on smoothings of 3-dimensional Calabi-Yau cones that are not products of lower-dimensional Calabi-Yau cones and that have orbifold singularities away from the vertex.

What carries the argument

Smoothings of 3-dimensional Calabi-Yau cones supporting complete non-product Calabi-Yau metrics with distant orbifold singularities.

If this is right

  • These metrics give new models for affine Calabi-Yau 3-folds with maximal volume growth.
  • The constructions produce spaces whose only singularities are orbifold points away from the vertex.
  • The metrics cannot be decomposed as products of lower-dimensional Calabi-Yau metrics.
  • The examples arise from specific families of cones that admit the required smoothings.

Where Pith is reading between the lines

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

  • Similar smoothing techniques might produce non-product examples in higher dimensions or for other cone types.
  • These spaces could serve as test cases for conjectures on the existence and uniqueness of complete Calabi-Yau metrics with given asymptotic cones.
  • The orbifold singularities suggest a mechanism for introducing finite group actions into the geometry while preserving completeness and maximal volume growth.

Load-bearing premise

The relevant 3-dimensional Calabi-Yau cones admit smoothings that support complete Calabi-Yau metrics with non-product structure and orbifold singularities away from the vertex.

What would settle it

A proof that no smoothing of these cones admits a complete Calabi-Yau metric with maximal volume growth that is not a product or that lacks the claimed distant orbifold singularities.

read the original abstract

We construct examples of complete Calabi-Yau metrics on smoothings of 3-dimensional Calabi-Yau cones that are not products of lower-dimensional Calabi-Yau cones and that have orbifold singularities away from the vertex.

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 manuscript constructs new examples of complete Calabi-Yau metrics on smoothings of 3-dimensional Calabi-Yau cones. These metrics are asserted to be non-products of lower-dimensional Calabi-Yau cones, to possess orbifold singularities away from the vertex, and to exhibit maximal volume growth.

Significance. If the constructions are rigorously verified, the examples would enlarge the known list of affine Calabi-Yau 3-folds with maximal volume growth by supplying non-product instances carrying isolated orbifold singularities. Explicit constructions of this type are valuable for testing conjectures on the existence and uniqueness of such metrics.

minor comments (1)
  1. The abstract states the main result clearly but does not indicate the dimension of the smoothing or the specific Calabi-Yau cones employed; adding one sentence on the source cones would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for recognizing the potential value of these new non-product examples in enlarging the known list of affine Calabi-Yau 3-folds with maximal volume growth. The 'uncertain' recommendation appears to stem from a need to confirm the rigor of the constructions; we believe the manuscript provides complete and self-contained proofs of the existence of these metrics, including the non-product nature and the presence of orbifold singularities away from the vertex. We are prepared to address any specific technical questions.

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The paper presents an explicit construction of complete Calabi-Yau metrics on non-product smoothings of 3-dimensional Calabi-Yau cones with orbifold singularities and maximal volume growth. No load-bearing step in the abstract or title reduces by definition, fitted input, or self-citation chain to the claimed result itself. The derivation relies on geometric existence arguments that remain independent of the output, consistent with a standard construction paper whose central claim does not collapse to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work appears to rely on standard background results in complex geometry and analysis rather than introducing new free parameters or invented entities.

axioms (1)
  • standard math Standard existence and regularity results for Calabi-Yau metrics on smoothings of cones from prior literature in geometric analysis.
    Typical foundational assumptions for construction papers in this area.

pith-pipeline@v0.9.0 · 5332 in / 1112 out tokens · 38484 ms · 2026-05-10T13:48:33.145114+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · 1 internal anchor

  1. [1]

    Vestislav Apostolov, David M. J. Calderbank, Paul Gauduchon, and Christina W. Tø nnesen Friedman,Hamiltonian 2-forms in K ¨ahler geometry. II. Global classification, J. Differential Geom.68(2004), no. 2, 277–345. MR 2144249

    work page 2004

  2. [2]

    Berman,Conical Calabi–Yau metrics on toric affine varieties and convex cones, Journal of Differential Geometry 125(2023), no

    Robert J. Berman,Conical Calabi–Yau metrics on toric affine varieties and convex cones, Journal of Differential Geometry 125(2023), no. 2, 209–242

    work page 2023

  3. [3]

    Olivier Biquard and Thibaut Delcroix,Ricci flat K ¨ahler metrics on rank two complex symmetric spaces, J. ´Ec. polytech. Math.6(2019), 163–201. MR 3932737

    work page 2019

  4. [4]

    Math., vol

    Olivier Biquard and Paul Gauduchon,Hyper-K ¨ahler metrics on cotangent bundles of Hermitian symmetric spaces, Geom- etry and physics (Aarhus, 1995), Lecture Notes in Pure and Appl. Math., vol. 184, Dekker, New York, 1997, pp. 287–298. MR 1423175

    work page 1995

  5. [5]

    Agostino Butti, Davide Forcella, and Alberto Zaffaroni,The Dual superconformal theory forL p,q,r manifolds, JHEP09 (2005), 018

    work page 2005

  6. [6]

    Calabi,M´ etriques k ¨ahl´ eriennes et fibr´ es holomorphes, Ann

    E. Calabi,M´ etriques k ¨ahl´ eriennes et fibr´ es holomorphes, Ann. Sci. ´Ecole Norm. Sup. (4)12(1979), no. 2, 269–294. MR 543218 (83m:32033)

    work page 1979

  7. [7]

    Carron,On the quasi-asymptotically locally Euclidean geometry of Nakajima’s metric, J

    G. Carron,On the quasi-asymptotically locally Euclidean geometry of Nakajima’s metric, J. Inst. Math. Jussieu10(2011), no. 1, 119–147. MR 2749573

    work page 2011

  8. [8]

    Chiu, Nonuniqueness of Calabi-Yau metrics with maximal volume gr owth, preprint, arXiv:2206.0821

    S.-K. Chiu,Nonuniqueness of Calabi-Yau metrics with maximal volume growth, preprint, arXiv:2206.0821

    work page arXiv

  9. [9]

    Colding and W

    T. Colding and W. Minicozzi, II,On uniqueness of tangent cones for Einstein manifolds, Invent. Math.196(2014), no. 3, 515–588. MR 3211041

    work page 2014

  10. [10]

    Conlon, A

    R. Conlon, A. Degeratu, and F. Rochon,Quasi-asymptotically conical Calabi–Yau manifolds, Geom. Topol.23(2019), no. 1, 29–100. MR 3921316

    work page 2019

  11. [11]

    R. J. Conlon and H.-J. Hein,Asymptotically conical Calabi-Yau manifolds, I, Duke Math. J.162(2013), no. 15, 2855–2902. MR 3161306

    work page 2013

  12. [12]

    ,Asymptotically conical Calabi-Yau metrics on quasi-projective varieties, Geom. Funct. Anal.25(2015), no. 2, 517–552. MR 3334234

    work page 2015

  13. [13]

    Conlon and Hans-Joachim Hein,Classification of asymptotically conical Calabi-Yau manifolds, Duke Math

    Ronan J. Conlon and Hans-Joachim Hein,Classification of asymptotically conical Calabi-Yau manifolds, Duke Math. J. 173(2024), no. 5, 947–1015. MR 4740213

    work page 2024

  14. [14]

    Warped quasi-asymptotically conical Calabi-Yau metrics

    Ronan J. Conlon and Fr´ ed´ eric Rochon,Warped quasi-asymptotically conical Calabi-Yau metrics, preprint, arXiv2308.02155

    work page internal anchor Pith review Pith/arXiv arXiv

  15. [15]

    ,New examples of complete Calabi-Yau metrics onC n forn≥3, Ann. Sci. ´Ec. Norm. Sup´ er. (4)54(2021), no. 2, 259–303. MR 4258163

    work page 2021

  16. [16]

    Cvetiˇ c, H

    M. Cvetiˇ c, H. L¨u, Don N. Page, and C. N. Pope,New Einstein-Sasaki spaces in five and higher dimensions, Physical Review Letters95(2005), no. 7, 071101

    work page 2005

  17. [17]

    Degeratu and R

    A. Degeratu and R. Mazzeo,Fredholm theory for elliptic operators on quasi-asymptotically conical spaces, Proc. Lond. Math. Soc. (3)116(2018), no. 5, 1112–1160. MR 3805053

    work page 2018

  18. [18]

    Dimakis and F

    P. Dimakis and F. Rochon,Asymptotic geometry at infinity of quiver varieties, preprint, arXiv:2410.15424

    work page arXiv

  19. [19]

    Firester,Complete Calabi-Yau metrics from smoothing Calabi-Yau complete intersections, Geom

    Benjy J. Firester,Complete Calabi-Yau metrics from smoothing Calabi-Yau complete intersections, Geom. Dedicata218 (2024), no. 2, Paper No. 46, 18. MR 4707313

    work page 2024

  20. [20]

    Gabella,The AdS/CFT correspondence and generalized geometry, Dphil thesis, University of Oxford, Oxford, UK, 2011

    M. Gabella,The AdS/CFT correspondence and generalized geometry, Dphil thesis, University of Oxford, Oxford, UK, 2011

    work page 2011

  21. [21]

    Goldberg and Shoshichi Kobayashi,Holomorphic bisectional curvature, J

    Samuel I. Goldberg and Shoshichi Kobayashi,Holomorphic bisectional curvature, J. Differential Geometry1(1967), 225–

    work page 1967

  22. [22]

    N. L. Gordeev,Invariants of linear groups generated by matrices with two eigenvalues different from one, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)114(1982), 120–130, 219, Modules and algebraic groups. MR 669563

    work page 1982

  23. [23]

    Goto,Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities, J

    R. Goto,Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities, J. Math. Soc. Japan64(2012), no. 3, 1005–1052. MR 2965437

    work page 2012

  24. [24]

    D. D. Joyce,Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000. MR 1787733 (2001k:53093)

    work page 2000

  25. [25]

    Victor Kac and Keiichi Watanabe,Finite linear groups whose ring of invariants is a complete intersection, Bull. Amer. Math. Soc. (N.S.)6(1982), no. 2, 221–223. MR 640951

    work page 1982

  26. [26]

    P. B. Kronheimer,The construction of ALE spaces as hyper-K ¨ahler quotients, J. Differential Geom.29(1989), no. 3, 665–683. MR 992334

    work page 1989

  27. [27]

    Math.147 (2011), no

    Eveline Legendre,Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics, Compos. Math.147 (2011), no. 5, 1613–1634. MR 2834736

    work page 2011

  28. [28]

    ,Toric K ¨ahler-Einstein metrics and convex compact polytopes, J. Geom. Anal.26(2016), no. 1, 399–427. MR 3441521

    work page 2016

  29. [29]

    Symplectic Geom.1(2003), no

    Eugene Lerman,Contact toric manifolds, J. Symplectic Geom.1(2003), no. 4, 785–828. MR 2039164

    work page 2003

  30. [30]

    Math.217(2019), no

    Yang Li,A new complete Calabi-Yau metric onC 3, Invent. Math.217(2019), no. 1, 1–34. MR 3958789

    work page 2019

  31. [31]

    Dario Martelli and James Sparks,Toric Sasaki-Einstein metrics onS 2 ×S 3, Phys. Lett. B621(2005), 208–212

    work page 2005

  32. [32]

    Dario Martelli, James Sparks, and Shing-Tung Yau,The geometric dual ofa-maximisation for toric Sasaki-Einstein manifolds, Comm. Math. Phys.268(2006), no. 1, 39–65. MR 2249795

    work page 2006

  33. [33]

    Nakajima,Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol

    H. Nakajima,Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999. MR 1711344 14 SHIH-KAI CHIU, RONAN J. CONLON, AND FR ´ED´ERIC ROCHON

    work page 1999

  34. [34]

    Tran-Trung Nghiem,Calabi-Yau metrics of rank two symmetric spaces with horospherical tangent cone at infinity, preprint, arXiv:2401.05122

    work page arXiv

  35. [35]

    Rochon,G´ eom´ etrie ` a l’infini des vari´ et´ e hyperk¨ahl´ eriennes toriques, preprint, arXiv:2507.09451

    F. Rochon,G´ eom´ etrie ` a l’infini des vari´ et´ e hyperk¨ahl´ eriennes toriques, preprint, arXiv:2507.09451

    work page arXiv

  36. [36]

    Stenzel,Ricci-flat metrics on the complexification of a compact rank one symmetric space, Manuscripta Math

    Matthew B. Stenzel,Ricci-flat metrics on the complexification of a compact rank one symmetric space, Manuscripta Math. 80(1993), no. 2, 151–163. MR 1233478

    work page 1993

  37. [37]

    Sun and J

    S. Sun and J. Zhang,No semistability at infinity for Calabi-Yau metrics asymptotic to cones, Inventiones mathematicae 233(2023), 461–594

    work page 2023

  38. [38]

    J.168(2019), no

    G´ abor Sz´ ekelyhidi,Degenerations ofC n and Calabi-Yau metrics, Duke Math. J.168(2019), no. 14, 2651–2700. MR 4012345

    work page 2019

  39. [39]

    van Coevering,Ricci-flat K ¨ahler metrics on crepant resolutions of K ¨ahler cones, Math

    C. van Coevering,Ricci-flat K ¨ahler metrics on crepant resolutions of K ¨ahler cones, Math. Ann.347(2010), no. 3, 581–611. MR 2640044 (2011k:53056)

    work page 2010

  40. [40]

    Z.267(2011), no

    ,Examples of asymptotically conical Ricci-flat K ¨ahler manifolds, Math. Z.267(2011), no. 1-2, 465–496. MR 2772262

    work page 2011

  41. [41]

    Dashen Yan,A gluing theorem for collapsing warped-QAC Calabi-Yau manifolds, preprint, arXiv:2412.03742. Department of Mathematics, University of California, Irvine Email address:shihkaic@uci.edu Department of Mathematical Sciences, The University of Texas at Dallas Email address:ronan.conlon@utdallas.edu D´epartement de Math´ematiques, Universit´e du Qu´e...

    work page arXiv