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Calabi-Yau threefolds across quadratic singularities

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Pith's one-line read This survey argues that conifold transitions can carry Calabi-Yau threefolds into non-Kähler complex threefolds that retain the core structures of Calabi-Yau geometry.

desk verdict Solid, honest survey of conifold transitions to non-Kähler threefolds; no new theorems, but a useful map of the area with one overbroad theorem statement worth checking before you cite it. read the letter →

arxiv 2501.19313 v1 pith:QUPF37QC submitted 2025-01-31 math.DG math.AG

Sébastien Picard This is my paper · ORCID
classification math.DGmath.AG MSC 14J3232Q2553C2553C55
keywords Calabi-Yauthreefoldconifoldtransitionordinarydoublepointnon-KählercomplexbalancedmetricHermitian-Yang-MillsequationspecialLagrangiancycle(-11)-curve
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

These lecture notes argue that the right object of study is not a single projective Calabi-Yau threefold but the web of complex threefolds reachable by conifold transitions. A conifold transition contracts disjoint $(-1,-1)$-curves to ordinary double points and then smooths the singular space; topologically it replaces $D^4\times S^2$ neighbourhoods by $S^3\times D^3$, and the resulting threefold need not be Kähler. The notes survey results showing that these non-Kähler limits nevertheless retain the defining features of Calabi-Yau geometry: a holomorphic volume form, vanishing $H^1$, a balanced metric, stable tangent bundle, and special Lagrangian vanishing cycles. The upshot is a proposal to widen the moduli space of Calabi-Yau threefolds to include these analytic threefolds, with the caveat that the existence of the required curve configurations is still open.

What carries the argument

The central object is the conifold transition, a two-step surgery whose local model is the ordinary double point $\{z_1^2+\cdots+z_4^2=0\}\subset\mathbb{C}^4$. One direction contracts the zero section $\mathbb{P}^1$ of $\mathcal{O}(-1)^{\oplus 2}$ (a $(-1,-1)$-curve) to the singular point; the other deforms the singular space to $\{\sum z_i^2=t\}$, diffeomorphic to $T^*S^3$. The global existence of the smoothing is controlled by the linear relation (2.2) among the homology classes of the contracted curves. Across this local model the paper carries metric and gauge-theoretic structures: explicit Ricci-flat Kähler metrics on both sides of the cone, balanced metrics on the global smoothing, solutions of the Hermitian-Yang-Mills equation $F_h\wedge\omega^2=0$, and calibrated special Lagrangian vanishing cycles.

What would settle it

A concrete falsifier would be a smooth projective Calabi-Yau threefold that admits no $(-1,-1)$-curves, or no collection satisfying condition (2.2); such a threefold would be disconnected from the web and refute the connectedness programme surveyed here.

Watch

Extended reading notes

Core claim

The notes' central claim is that passing through a quadratic singularity does not destroy Calabi-Yau structure. Given a projective Calabi-Yau threefold and a collection of disjoint $(-1,-1)$-curves satisfying the smoothing condition (2.2), the smoothed threefolds admit a holomorphic volume form, satisfy $h^{1,0}=h^{0,1}=0$ and the $\partial\bar\partial$-lemma, carry balanced (conformally balanced) metrics, have stable tangent bundle with respect to those metrics, and contain special Lagrangian 3-spheres that replace the contracted $\mathbb{P}^1$s. These properties persist even when the smoothing is non-Kähler, for example when $b_2$ collapses to zero and the threefold is diffeomorphic to a connected sum of $S^3\times S^3$. The paper therefore treats the set of threefolds linked to projective Calabi-Yau threefolds by conifold transitions as a natural extension of the Calabi-Yau moduli space.

Load-bearing premise

The web picture rests on every Calabi-Yau threefold admitting a set of disjoint rigid rational curves whose homology classes satisfy the linear relation (2.2); the notes state that this is an open problem.

Editorial extensions

If this is right

  • The moduli space of Calabi-Yau threefolds, if connected by conifold transitions, must be expanded to include non-Kähler complex threefolds; Kähler metrics cannot exist on them, but balanced metrics and holomorphic volume forms can.
  • A conifold transition swaps holomorphic $\mathbb{P}^1$s for special Lagrangian $S^3$s, and this exchange is geometrically realized even when the smoothing is non-Kähler.
  • Stability of the tangent bundle, previously a consequence of the Calabi-Yau theorem in the Kähler case, survives the transition: the limiting threefolds have stable tangent bundle with respect to the balanced metric, ruling out complex threefolds with destabilizing subsheaves as limits.
  • The $\partial\bar\partial$-lemma and Hodge-Riemann bilinear relations on $H^{2,1}$ hold on the non-Kähler side, so much of the Hodge-theoretic apparatus used for projective Calabi-Yau threefolds remains available.
  • Conifold transitions are continuous in the Gromov-Hausdorff sense when measured with the natural metrics, even though Betti numbers jump discontinuously; the explicit local Ricci-flat models provide the asymptotics near singularities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: the notes leave open the reverse direction of the transition; an immediate test is to solve the balanced or Strominger-type equations on a small resolution obtained by degenerating a projective smoothing, which would create holomorphic $\mathbb{P}^1$s out of vanishing 3-spheres.
  • Because stability of the tangent bundle is the surviving invariant, a complex threefold with an unstable tangent bundle cannot be a conifold limit; scanning known non-Kähler threefolds for stability would give new obstructions to membership in the web.
  • If every Calabi-Yau threefold eventually admits the required curve configurations, the web would form a connected 'moduli space with boundary,' and the boundary members, being diffeomorphic to connected sums of $S^3 \times S^3$ in the known cases, would be a finite list of topological types to classify.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This manuscript is an expository survey of conifold transitions from projective Calabi-Yau threefolds to possibly non-Kahler complex threefolds. Section 1 reviews definitions and basic invariants (Hodge numbers, deformations, rational curves); Section 2 describes the conifold transition, Reid's conjecture, Friedman's condition, the local model, and explicit examples; Section 3 surveys analytic results on balanced metrics, Hermitian-Yang-Mills metrics, special Lagrangian vanishing cycles, and Gromov-Hausdorff convergence across the transition. The paper argues that the 'web' W of threefolds connected to projective Calabi-Yau threefolds by conifold transitions inherits several properties of Kahler geometry even though its members need not be Kahler. No proofs are given; the paper is a lecture-note-style summary with extensive references.

Significance. The survey fills a useful niche: it collects the topological model, the local Ricci-flat geometries, and the recent analytic theorems (Fu-Li-Yau, Collins-Picard-Yau, and the Gromov-Hausdorff continuity results) in one place, and it is careful to flag open problems such as Reid's conjecture and Yau's conjecture. The conditional nature of the web W is stated explicitly, and the reliance on recent preprints, including the author's own, is transparent. Its value depends on accurate reporting of the cited theorems; the main issue found here is an internal inconsistency in a foundational remark, plus insufficient precision in the statement of the central structural theorem.

major comments (2)
  1. [Remark 1.2] Remark 1.2 claims that projectivity is redundant in Definition 1.1 because H^1(X,C)=0 implies h^{0,2}=0 and then projectivity follows from the Kodaira embedding theorem. This is contradicted by the paper's own Example 2.9: the smoothing X_t is a connected sum of copies of S^3 x S^3, so b_1(X_t)=b_2(X_t)=0 and H^1(X_t,C)=0, and Section 3.4.2 states that X_t admits a holomorphic volume form; such an X_t cannot be projective. Even the step from h^{0,2}=0 to projectivity is not justified without a Kahler metric and an integral Kahler class. Please correct or remove this remark and check later passages that describe non-Kahler examples.
  2. [Theorem 3.2 and Section 3.4.2] The central structural statement, Theorem 3.2, is quoted as 'let X-hat -> X_0 => X_t be a conifold transition' and then asserts the existence of (Omega_t,g_t,h_t) solving (3.4)-(3.6) with estimates near the vanishing cycles. Since this theorem supports the bullet list in Section 3.4.2 and the definition of the web W, the survey should state the exact hypotheses under which [41] and [27] prove it, including any conditions on the contracted curves beyond Friedman's condition and any role played by stability of T^{1,0}X-hat. As written the reader cannot verify that the statement is not stronger than the cited results.
minor comments (5)
  1. [Example 2.12] The displayed contradiction in Example 2.12 has an unclear Stokes step: if Z is a 3-chain with boundary C, the equality should read int_C omega = int_{dZ} omega = int_Z d omega = 0; please fix the notation.
  2. [Section 3.1, Eq. (3.1)] The notation for the cone coordinate is not consistent: the ansatz sets r^3 = ||z||^2, but Eq. (3.1) writes the metric as dr^2 + r^2 g_Sigma. If r is the cone radial coordinate, the relation should be r = ||z||; if r = ||z||^{2/3}, an extra conformal factor appears. Please align the notation.
  3. [Theorem 3.11] The statement of Theorem 3.11 says that Hermitian metrics converge in the Gromov-Hausdorff sense but does not define the induced Riemannian distance used for this convergence; the phrase 'unique normalized sequence' should also be explained.
  4. [Section 3.4.2] The sentence about contracting only two curves in Example 2.8 should justify that the chosen pair satisfies Friedman's condition (2.2); not every pair of disjoint (-1,-1)-curves is automatically smoothable.
  5. [Title and Abstract] The title and abstract contain typographical artifacts ('Calabi-Y au', 'Quadra tic'); they should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper is a survey whose claims are conditional on cited published theorems, and the key open problem is explicitly flagged rather than hidden.

full rationale

This paper is an expository survey and lecture notes, not a research derivation. It does not fit parameters to data, rename a known result, or derive a prediction from an input. The central object W, the 'web of threefolds', is defined conditionally as the set of complex threefolds connected to projective Calabi-Yau threefolds by conifold transitions, and every substantive geometric assertion about X_t is presented as a summary of published theorems: the existence of holomorphic volume forms and vanishing H^1 from Friedman [37]; the balanced metric and stability of the tangent bundle from Fu-Li-Yau [41] and Collins-Picard-Yau [27]; the \partial\bar\partial-lemma from Li [74], Friedman [39], and Lee [71]. These are citations to external, peer-reviewed or arXiv-posted mathematical results, not arguments whose validity rests on the present survey. The author's own prior work appears among the citations ([26], [27], [40], [85], [86]), but it is used as a source for stated theorems, not as an unexamined premise that forces a conclusion. In particular, Section 3.4.2's bullet list does not rely on the preprint [40] for its main properties. The one potentially load-bearing background assumption, that every Calabi-Yau threefold admits (-1,-1)-curves, is explicitly labeled as open in Remark 2.3 ('It is an open problem whether every Calabi-Yau threefold X admits (-1,-1)-curves'), and the survey does not claim Reid's conjecture is proved. Likewise, the limitations of the Strominger-system program are acknowledged in Section 3.3.1 and Remark 3.7. There is therefore no step in which the paper's claims reduce by construction or by self-citation to their own inputs. A possible caveat, that the exact hypotheses of Theorem 3.2 should be checked against the original statements in [41] and [27], is a correctness or precision concern, not a circularity concern.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The survey rests on several major background theorems from the existing literature; no new entities, free parameters, or ad hoc assumptions are introduced. The axioms listed are the principal unproved inputs the survey depends on, all standard in the field except where noted as recent results.

assumptions (5)
  • standard math Yau's theorem (Theorem 1.6) asserting existence and uniqueness of solutions to the complex Monge-Ampère equation (1.2), used throughout Section 1.
    Invoked as background for the existence of Ricci-flat Kähler metrics on projective Calabi-Yau threefolds.
  • standard math Bogomolov-Tian-Todorov theorem (Theorem 1.9) that deformations of compact Calabi-Yau manifolds are unobstructed.
    Used in Section 1.3 to describe the parameter space of complex structures.
  • standard math Friedman-Kawamata-Ran-Tian theorem (Theorem 2.4) that a smoothing of the contraction exists under Friedman's condition (2.2).
    Central to the definition of conifold transitions in Section 2.1.2.
  • standard math Existence of balanced metrics and solutions to (3.4)-(3.6) on non-Kähler threefolds from [41,27] (Theorem 3.2).
    Underlies the survey's main message in Section 3.3; the theorem is cited, not proved in the notes.
  • standard math Theorem 3.11 on Gromov-Hausdorff continuity and conical asymptotics of Hermitian-Yang-Mills metrics, from [27,40].
    Cited in Section 3.3.2; one source [40] is a preprint.

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Cite this review

Pith. "Pith review of Calabi-Yau threefolds across quadratic singularities." pith.science (2026). https://pith.science/paper/QUPF37QC

@misc{pith2026250119313,
  author       = {Pith},
  title        = {Pith review of: Calabi-Yau threefolds across quadratic singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QUPF37QC}},
  note         = {Machine review of arXiv:2501.19313}
}
read the original abstract

These are lecture notes on non-K\"ahler complex threefolds presented at the MATRIX program ``The geometry of moduli spaces in string theory''. We review some basics of Calabi-Yau geometry in Section 1, describe topological features of the conifold transition in Section 2, and survey recent developments on the geometrization of conifold transitions in Section 3.

Figures

Figures reproduced from arXiv: 2501.19313 by the authors.

Figure 1
Figure 1. The local model of a conifold transition [20, 60]. Illustration taken from H¨ubsch’s website [link]. 2.2.1. More on small resolutions. We now provide more details on the small resolution Vˆ µ→ V0. We define Vˆ = O(−1)⊕2 → P 1 . The space Vˆ = U ∪ U˜ is covered by two coordinate charts with coordinates (u, v, λ) satisfying the coordinate transformation (2.4) λ˜ = λ −1 , u˜ = λu, v˜ = λv. The coordinate λ is in the P … view at source ↗

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. An introduction to conifold transitions

    math.DG 2025-08 conditional novelty 5.0 of 10

    Survey of conifold transitions between Calabi-Yau threefolds with a sketched differential-geometric proof of the necessity part of Friedman's smoothing criterion.

  2. Beyond Algebraic Superstring Compactification

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    Calabi-Yau compactifications and mirror symmetry are conjecturally extended to non-algebraic toric spaces using Laurent deformations and the 'intrinsic limit' completion.

Reference graph

Works this paper leans on

115 extracted references · 45 canonical work pages · cited by 2 Pith papers

  1. [40]

    Friedman, S

    B. Friedman, S. Picard and C. Suan, Gromov-Hausdorff con- tinuity of non-K¨ ahler Calabi-Yau conifold transitions , preprint. [arXiv:2404.11840]

    arXiv

  2. [41]

    J. Fu, J. Li, and S.-T. Yau, Balanced metrics on non-Kahler Calabi- Yau threefolds , J. Differential Geom. 90 (2012), no. 2, 81-129. [arXiv:0809.4748]

    work page Pith review arXiv 2012

  3. [27]

    T. C. Collins, S. Picard, and S. -T. Yau, Stability of the tangent bundle through conifold transitions, Comm. Pure Appl. Math. 77 (2024), 284-

    2024

  4. [1]

    Embeddings of submanifolds and normal bundles

    M. Abate, F. Bracci and F. Tovena, Embeddings of submanifolds and normal bundles , Advances in Mathematics 220 (2009), no. 2, 620-656. [math/0612449]

    work page Pith review arXiv 2009

  5. [2]

    Alessandrini and G

    L. Alessandrini and G. Bassanelli, Metric properties of manifolds bimeromorphic to compact Kahler spaces , J. Differential Gometry 37 (1993), 95-121

    1993

  6. [3]

    Anderson, C

    L. Anderson, C. Brodie, and J. Gray, Branes and bundles through coni- fold transitions and dualities in heterotic string theory , Phys. Rev. D (2023), no. 103. [ arXiv:2211.05804]

    arXiv 2023

  7. [4]

    Ashmore, R

    A. Ashmore, R. Minasian, and Y. Proto, Geometric flows and super- symmetry, Commun. Math. Phys. (2024), 405:16. [ arXiv:2302.06624]

    arXiv 2024

  8. [5]

    Ashmore, X

    A. Ashmore, X. de la Ossa, R. Minasian, C. Strickland-Constable and E.E. Svanes, Finite deformations from a heterotic superpotential: holo- morphic Chern–Simons and an L infinity algebra , J. High Energ. Phys. 179 (2018). [ arXiv:1806.08367]

    arXiv 2018

Show all 115 references
  1. [6]

    Ashmore, J

    A. Ashmore, J. Ibarra, D. McNutt, C. Strickland-Constable, E.E. Svanes, D. Tennyson, S. Winje, A heterotic Kodaira-Spencer theory at one-loop, preprint. [arXiv:2306.10106]

    arXiv

  2. [7]

    Atiyah, On analytic surfaces with double points , Proceedings of the Royal Society of London A 247 (1958), no

    M.F. Atiyah, On analytic surfaces with double points , Proceedings of the Royal Society of London A 247 (1958), no. 1249, 237-244

    1958

  3. [8]

    Batyrev and M

    V. Batyrev and M. Kreuzer, Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions , Adv. Theor. Math. Phys. 14 (2010), 879-898. [ arXiv:0802.3376]

    2010 arXiv

  4. [9]

    Behrend and B

    K. Behrend and B. Fantechi, The intrinsic normal cone , Invent. Math. 128 (1997), no. 1, 45–88. [ arXiv:alg-geom/9601010]

    1997 arXiv

  5. [10]

    Bergshoeff and M

    E. Bergshoeff and M. de Roo, The quartic effective action of the het- erotic string and supersymmetry , Nuclear Physics B 328 (1989), no. 2, 439–468

    1989

  6. [11]

    Bozhkov, Specific complex geometry of certain complex surfaces and three-folds, PhD Thesis at the University of Warwick, 1992

    Y.D. Bozhkov, Specific complex geometry of certain complex surfaces and three-folds, PhD Thesis at the University of Warwick, 1992

    1992

  7. [12]

    Bryan and R

    J. Bryan and R. Pandharipande, The local Gromov-Witten the- ory of curves , J. Amer. Math. Soc. 21 (2008), no. 1, 101–136. [arXiv:math/0411037]

    2008 arXiv

  8. [13]

    Calabi, On Kahler manifolds with vanishing canonical class , in Al- gebraic geometry and topology

    E. Calabi, On Kahler manifolds with vanishing canonical class , in Al- gebraic geometry and topology. A symposium in honor of S. Lefschetz, pp. 78–89. Princeton University Press, Princeton, N. J., 1957

    1957

  9. [14]

    Candelas and R

    P. Candelas and R. Davies, New Calabi-Yau manifolds with small Hodge numbers , Fortschritte der Physik 58 (2010), 383-466. 28 S. PICARD [arXiv:0809.4681]

    2010 arXiv

  10. [15]

    Candelas and X

    P. Candelas and X. de la Ossa, Comments on conifolds, Nuclear Physics B 342 (1990), no. 1, 246-268

    1990

  11. [16]

    Candelas and X

    P. Candelas and X. de la Ossa, Moduli space of Calabi-Yau manifolds , Nuclear Physics B 355 (1991), 455-481

    1991

  12. [17]

    Candelas, X

    P. Candelas, X. de la Ossa, P.S. Green, L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory , Nuclear Physics B 359 (1991), 21-74

    1991

  13. [18]

    Candelas, X

    P. Candelas, X. de la Ossa, and J. McOrist, A metric for het- erotic moduli , Commun. Math. Phys. 356, no. 2 (2017), 567-612. [arXiv:1605.05256]

    2017 arXiv

  14. [19]

    Candelas, X

    P. Candelas, X. de la Ossa, J. McOrist, and R. Sisca. The uni- versal geometry of heterotic vacua , J. High Energ. Phys. 38 (2019). [arXiv:1810.00879]

    2019 arXiv

  15. [20]

    Candelas, P

    P. Candelas, P. Green, and T. H¨ ubsch, Rolling among Calabi-Yau vacua, Nuclear Phys. B 330 (1990), no. 1, 49–102

    1990

  16. [21]

    Candelas, G

    P. Candelas, G. Horowitz, A. Strominger, and E. Witten, Vacuum con- figurations for superstrings , Nuclear Phys. B 258 (1985), no. 1, 46–74

    1985

  17. [22]

    Chiu and G

    S.K. Chiu and G. Szekelyhidi, Higher regularity for singular Kahler–Einstein metrics , Duke Mathematical Journal 172 (2023), no. 18, 3521–3558. [ arXiv:2202.11083]

    2023 arXiv

  18. [23]

    Chuan, Existence of Hermitian-Yang-Mills metrics under coni- fold transitions , Comm

    M.-T. Chuan, Existence of Hermitian-Yang-Mills metrics under coni- fold transitions , Comm. Anal. Geom. 20 (2012), no. 4, 677–749. [arXiv:1012.3107]

    2012 arXiv

  19. [24]

    Clemens, Homological equivalence modulo algebraic equivalence is not finitely generated , Publ

    H. Clemens, Homological equivalence modulo algebraic equivalence is not finitely generated , Publ. Math. IHES 58 (1983), 19-38

    1983

  20. [25]

    Clemens, Double solids , Advances in Mathematics 47 (1983), 107- 230

    H. Clemens, Double solids , Advances in Mathematics 47 (1983), 107- 230

    1983

  21. [26]

    Collins, S

    T. Collins, S. Gukov, S. Picard, and S.-T. Yau, Special Lagrangian cycles and Calabi-Yau transitions , Commun. Math. Phys. 401 (2023), 769–802. [arXiv:2111.10355]

    2023 arXiv

  22. [28]

    Conlon and H.J

    R.J. Conlon and H.J. Hein, Classification of asymptotically conical Calabi–Yau manifolds , Duke Math. J. 173 (2024), no. 5, 947-1015. [arXiv:2201.00870] 29

    2024 arXiv

  23. [29]

    Demailly, Complex Analytic and Differential Geometry

    J.P. Demailly, Complex Analytic and Differential Geometry. Book avail- able on the author’s website. [ agbook.pdf]

  24. [30]

    de la Ossa, and E

    X. de la Ossa, and E. E. Svanes, Holomorphic bundles and the mod- uli space of N = 1 supersymmetric heterotic compactifications , J. High Energ. Phys. 123 (2014). [ arXiv:1402.1725]

    2014 arXiv

  25. [31]

    Dixon, Some world-sheet properties of superstring compactifications, or orbifolds and otherwise

    L. Dixon, Some world-sheet properties of superstring compactifications, or orbifolds and otherwise . In: Superstrings, Unified Theories and Cos- mology 1987 (Trieste, 1987), ICTP Ser. Theoret. Phys. vol. 4, World Sci., Teaneck, NJ, 1988, 67–126

    1987

  26. [32]

    Donaldson, An application of gauge theory to four dimensional topology, J

    S.K. Donaldson, An application of gauge theory to four dimensional topology, J. Differential Geom. 18 (1983), 279-315

    1983

  27. [33]

    Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles , Proc

    S.K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles , Proc. London Math. Soc. 50 (1985), no. 1, 1-26

    1985

  28. [34]

    Eyssidieux, V

    P. Eyssidieux, V. Guedj, and A. Zeriahi, Singular Kahler-Einstein met- rics, Journal of the American Mathematical Society 22 (2009), no. 3, 607–639. [arXiv:math/0603431]

    2009 arXiv

  29. [35]

    Fischer, Complex Analytic Geometry , Springer Berlin, Heidelberg 1970

    G. Fischer, Complex Analytic Geometry , Springer Berlin, Heidelberg 1970

    1970

  30. [36]

    Friedman, Simultaneous resolution of threefold double points , Math

    R. Friedman, Simultaneous resolution of threefold double points , Math. Ann. 274 (1986), no. 4, 671–689

    1986

  31. [37]

    Friedman, On threefolds with trivial canonical bundle, Proc

    R. Friedman, On threefolds with trivial canonical bundle, Proc. Sympos. Pure Math. 53 (1991), 103-134

    1991

  32. [38]

    Friedman, Algebraic Surfaces and Holomorphic Vector Bundles , Springer-Verlag 1998

    R. Friedman, Algebraic Surfaces and Holomorphic Vector Bundles , Springer-Verlag 1998

    1998

  33. [39]

    Friedman, The ∂ ¯∂-lemma for general Clemens manifolds , Pure and Applied Mathematics Quarterly 15 (2019), 1001–1028

    R. Friedman, The ∂ ¯∂-lemma for general Clemens manifolds , Pure and Applied Mathematics Quarterly 15 (2019), 1001–1028. [arXiv:1708.00828]

    2019 arXiv

  34. [42]

    J. Fu, Z. Wang, and D. Wu, Form-type Calabi-Yau equations , Math. Res. Lett. 17 (2010), no. 5, 887–903. [ arXiv:0908.0577]

    2010 arXiv

  35. [43]

    Garcia-Fernandez, Lectures on the Strominger system , Travaux Mathematiques Vol

    M. Garcia-Fernandez, Lectures on the Strominger system , Travaux Mathematiques Vol. XXIV (2016) 7-61. [ arXiv:1609.02615] 30 S. PICARD

    2016 arXiv

  36. [44]

    Garcia-Fernandez and R

    M. Garcia-Fernandez and R. Gonzalez Molina, Futaki invariants and Yau’s conjecture on the Hull-Strominger system , preprint. [arXiv:2303.05274]

    arXiv

  37. [45]

    Garcia-Fernandez, R

    M. Garcia-Fernandez, R. Gonzalez Molina and J. Streets, Pluriclosed flow and the Hull-Strominger system , preprint. [arXiv:2408.11674]

    arXiv

  38. [46]

    Garcia-Fernandez, R, Rubio, C

    M. Garcia-Fernandez, R, Rubio, C. Tipler, Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry , Math- ematische Annalen 369 (2017) 539-595. [ arXiv:1503.07562]

    2017 arXiv

  39. [47]

    Garcia-Fernandez, R

    M. Garcia-Fernandez, R. Rubio, C. Tipler, Holomorphic string algebroids, Trans. Amer. Math. Soc. 373 (2020) 7347-7382. [arXiv:1807.10329]

    2020 arXiv

  40. [48]

    Garcia-Fernandez, R

    M. Garcia-Fernandez, R. Rubio, C. Tipler, Gauge theory for string algebroids , J. Differential Geom. 128 (2024), no. 1, 77-152. [arXiv:2004.11399]

    2024

  41. [49]

    Giusti and C

    F. Giusti and C. Spotti, A Kummer construction for Chern-Ricci flat balanced manifolds, preprint. [arXiv:2309.12909]

    arXiv

  42. [50]

    Giusti and C

    F. Giusti and C. Spotti, Chern-Ricci flat balanced metrics on small resolutions of Calabi-Yau threefolds, preprint. [arXiv:2301.11636]

    arXiv

  43. [51]

    Grauert, Uber Modifikationen und exzeptionelle analytische Mengen , Math

    H. Grauert, Uber Modifikationen und exzeptionelle analytische Mengen , Math. Ann. 146 (1962), 331–368

    1962

  44. [52]

    Green and J.H

    M.B. Green and J.H. Schwarz, Anomaly cancellations in supersymmet- ric D = 10 gauge theory and superstring theory , Physics Letters B. 149 (1984), 117–122

    1984

  45. [53]

    Green and T

    P.S. Green and T. H¨ ubsch, Connecting Moduli Spaces of Calabi-Yau Threefolds, Commun. Math. Phys. 119 (1988), 431-441

    1988

  46. [54]

    Green and T

    P.S. Green and T. H¨ ubsch,Possible Phase Transitions among Calabi- Yau Compactifications, Phys. Rev. Lett. 61 (1988), 1163

    1988

  47. [55]

    Greene and M.R

    B.R. Greene and M.R. Plesser, Duality in Calabi-Yau moduli space , Nuclear Physics B 338 (1990), no. 1, 15-37

    1990

  48. [56]

    Greene, D.R

    B.R. Greene, D.R. Morrison, and A. Strominger, Black hole condensa- tion and the unification of string vacua , Nuclear Physics B 451, no. 1-2 (1995), 109-120. [ arXiv:hep-th/9504145]

    1995 arXiv

  49. [57]

    Gross, Primitive Calabi–Yau threefolds , J

    M. Gross, Primitive Calabi–Yau threefolds , J. Differential Geom. 45 (1997), 288–318. [ arXiv:alg-geom/9512002]

    1997 arXiv

  50. [58]

    Harvey and H.B

    R. Harvey and H.B. Lawson, Calibrated geometries, Acta Mathematica 148 (1982), 47-157

    1982

  51. [59]

    Hein and S

    H.J. Hein and S. Sun, Calabi-Yau manifolds with isolated coni- cal singularities , Publ. Math. IHES 126 (2017), no. 1, 73–130. 31 [arXiv:1607.02940]

    2017 arXiv

  52. [60]

    H¨ ubsch,Calabi-Yau Manifolds: A Bestiary for Physicists , World Scientific (1992)

    T. H¨ ubsch,Calabi-Yau Manifolds: A Bestiary for Physicists , World Scientific (1992)

    1992

  53. [61]

    Hull, Compactifications of the Heterotic Superstring, Phys.Lett

    C. Hull, Compactifications of the Heterotic Superstring, Phys.Lett. B178 (1986) 357

    1986

  54. [62]

    C. M. Hull and P. K. Townsend, The two loop beta function for sigma models with torsion , Phys. Lett. B 191, 115 (1987)

    1987

  55. [63]

    Jacob, T

    A. Jacob, T. Walpuski, Hermitian-Yang-Mills metrics on re- flexive sheaves over asymptotically cylindrical Kahler manifolds , Comm. Partial Differential Equations 43 (2018), no. 11, 1566–1598. [arXiv:1603.07702]

    2018 arXiv

  56. [64]

    Kahler, Uber eine bemerkenswerte Hermitesche Metrik , Abh

    E. Kahler, Uber eine bemerkenswerte Hermitesche Metrik , Abh. Math. Semin. Univ. Hambg. 9 (1933), 173–186

    1933

  57. [65]

    Kas and M

    A. Kas and M. Schlessinger, On the versal deformation of a complex space with an isolated singularity , Math. Ann. 196 (1972), 23–29

    1972

  58. [66]

    Katz, On the finiteness of rational curves on quintic threefolds , Com- positio Mathematica 60 (1986), no

    S. Katz, On the finiteness of rational curves on quintic threefolds , Com- positio Mathematica 60 (1986), no. 2, 151-162

    1986

  59. [67]

    Deformations of manifolds with torsion of negative canonical bundle

    Y. Kawamata, Unobstructed deformations. A remark on a paper of Z. Ran: “Deformations of manifolds with torsion of negative canonical bundle”, J. Algebraic Geom. 1 (1992), no. 2, 183–190

    1992

  60. [68]

    Kodaira, Complex Manifolds and Deformation of Complex Struc- tures, Grundlehren der Math

    K. Kodaira, Complex Manifolds and Deformation of Complex Struc- tures, Grundlehren der Math. Wiss. 283, Springer (1986)

    1986

  61. [69]

    Kontsevich, Homological Algebra of Mirror Symmetry , Proceedings of the International Congress of Mathematicians (1995), 120-139

    M. Kontsevich, Homological Algebra of Mirror Symmetry , Proceedings of the International Congress of Mathematicians (1995), 120-139

    1995

  62. [70]

    Kontsevich, Mirror symmetry in dimension 3 , Asterique 237 (1996), Seminaire Bourbaki no

    M. Kontsevich, Mirror symmetry in dimension 3 , Asterique 237 (1996), Seminaire Bourbaki no. 801, 275-293

    1996

  63. [71]

    Lee, Finite distance problem on the moduli of non-Kahler Calabi- Yau ∂ ¯∂-threefolds, preprint

    T.-J. Lee, Finite distance problem on the moduli of non-Kahler Calabi- Yau ∂ ¯∂-threefolds, preprint. [arXiv:2404.19125]

    arXiv

  64. [72]

    Lee, H.-W

    Y.-P. Lee, H.-W. Lin, and C.-L. Wang, Towards A+B theory in conifold transitions for Calabi–Yau threefolds, J. Differential Geom. 110 (2018), no. 3, 495-541. [ arXiv:1502.03277]

    2018 arXiv

  65. [73]

    Lerche, C

    W. Lerche, C. Vafa and N.P. Warner, Chiral rings in N = 2 supercon- formal theories, Nucl. Phys. B 324 (1989), no. 2, 427–474

    1989

  66. [74]

    Li, Polarized Hodge structures for Clemens manifolds , Mathematis- che Annalen 389 (2024), 525-541

    C. Li, Polarized Hodge structures for Clemens manifolds , Mathematis- che Annalen 389 (2024), 525-541. [ arXiv:2202.10353]

    2024 arXiv

  67. [75]

    Li and Y

    A. Li and Y. Ruan, Symplectic surgery and Gromov–Witten invari- ants of Calabi–Yau 3-folds , Invent. Math. 145 (2001), no. 1, 151–218. [arXiv:math/9803036] 32 S. PICARD

    2001 arXiv

  68. [76]

    Li and S.-S

    Y. Li and S.-S. Wang, Gromov–Witten/Pandharipande–Thomas corre- spondence via conifold transitions , preprint. [arXiv:2310.18170]

    arXiv

  69. [77]

    Lu and G

    P. Lu and G. Tian, The complex structure on a connected sum of S3 × S3 with trivial canonical bundle , Mathematische Annalen 298 (1994), 761–764

    1994

  70. [78]

    Lu and G

    P. Lu and G. Tian, Complex structures on connected sums of S3 × S3, Manifolds and geometry, 284-293, Sympos. Math., XXXVI, Cambridge Univ. Press, Cambridge, 1996

    1996

  71. [79]

    Martelli and J

    D. Martelli and J. Sparks, Non-Kahler heterotic rotations , Advances in Theoretical and Mathematical Physics 15, no. 1 (2011), 131-174. [arXiv:1010.4031]

    2011 arXiv

  72. [80]

    McOrist and E.E

    J. McOrist and E.E. Svanes, Heterotic quantum cohomology , J. High Energ. Phys. 11 (2022). [ arXiv:2110.06549]

    2022 arXiv

  73. [81]

    Melnikov, R

    I. Melnikov, R. Minasian, and S. Sethi, Heterotic fluxes and supersym- metry, J. High Energ. Phys. 6 (2014). [ arXiv:1403.4298]

    2014 arXiv

  74. [82]

    Michelsohn, On the existence of special metrics in complex geom- etry, Acta Math

    M.L. Michelsohn, On the existence of special metrics in complex geom- etry, Acta Math. 149 (1982), 261-295

    1982

  75. [83]

    B. G. Moishezon, On n-dimensional compact complex manifolds having n algebraically independent meromorphic functions. I , Izv. Math., 30:1 (1966)

    1966

  76. [84]

    Pardon, Universally counting curves in Calabi–Yau threefolds , preprint

    J. Pardon, Universally counting curves in Calabi–Yau threefolds , preprint. [arXiv:2308.02948]

    arXiv

  77. [85]

    Picard, The Strominger system and flows by the Ricci tensor, Surveys in Differential Geometry 27, no

    S. Picard, The Strominger system and flows by the Ricci tensor, Surveys in Differential Geometry 27, no. 1 (2022), 103-145. [arXiv:2402.17770]

    2022 arXiv

  78. [86]

    Picard and P.-L

    S. Picard and P.-L. Wu, Balanced and Aeppli Parameters for the Het- erotic Moduli, preprint. [arXiv:2401.05331]

    arXiv

  79. [87]

    Popovici, Non-K¨ ahler Hodge Theory and Deformations of Complex Structures, Book available on the author’s website

    D. Popovici, Non-K¨ ahler Hodge Theory and Deformations of Complex Structures, Book available on the author’s website. [ hodge-def.pdf]

  80. [88]

    Popovici, Holomorphic Deformations of Balanced Calabi-Yau ∂ ¯∂- Manifolds, Annales de l’Institut Fourier 69 (2019), no

    D. Popovici, Holomorphic Deformations of Balanced Calabi-Yau ∂ ¯∂- Manifolds, Annales de l’Institut Fourier 69 (2019), no. 2, 673-728. [arXiv:1304.0331]

    2019 arXiv

  81. [89]

    Ran, Deformations of manifolds with torsion or negative canonical bundle, J

    Z. Ran, Deformations of manifolds with torsion or negative canonical bundle, J. Algebraic Geom. 1 (1992), no. 2, 279–291

    1992

  82. [90]

    Reid, The moduli space of 3-folds with K = 0 may nevertheless be irreducible, Math

    M. Reid, The moduli space of 3-folds with K = 0 may nevertheless be irreducible, Math. Ann. 278 (1987), no. 1-4, 329–334

    1987

  83. [91]

    Rollenske and R

    S. Rollenske and R. Thomas, Smoothing nodal Calabi-Yau n-folds, Jour- nal of Topology 2 (2009), no. 2, 405-421. [ arXiv:0810.0141] 33

    2009 arXiv

  84. [92]

    Rong and Y

    X. Rong and Y. Zhang, Continuity of extremal transitions and flops for Calabi-Yau manifolds, J. Differential Geom. 89 (2011), no. 2, 233–269. [arXiv:1012.2940]

    2011 arXiv

  85. [93]

    Rossi, Geometric transitions, J

    M. Rossi, Geometric transitions, J. Geom. Phys. 56 (2006), 1940-1983. [arXiv:0412514]

    2006

  86. [94]

    Ruan and Y

    W.D. Ruan and Y. Zhang, Convergence of Calabi–Yau mani- folds, Advances in Mathematics 228 (2011), no. 3, 1543–1589. [arXiv:0905.3424]

    2011 arXiv

  87. [95]

    Siu, Lectures on Hermitian-Einstein metrics for stable bundles and Kahler-Einstein metrics , DMV Seminar, 8

    Y.-T. Siu, Lectures on Hermitian-Einstein metrics for stable bundles and Kahler-Einstein metrics , DMV Seminar, 8. Birkhauser Verlag, Basel, 1987

    1987

  88. [96]

    Smith, R.P

    I. Smith, R.P. Thomas, S.-T. Yau, Symplectic Conifold Transitions, J. Differential Geom. 62 (2002), no. 2, 209-242. [ arXiv:0209319]

    2002

  89. [97]

    Song, On a Conjecture of Candelas and de la Ossa , Commun

    J. Song, On a Conjecture of Candelas and de la Ossa , Commun. Math. Phys. 334 (2015), no. 2, 697–717. [ arXiv:1201.4358]

    2015 arXiv

  90. [98]

    Sparks, Sasaki-Einstein manifolds , Surveys in Differential Geometry 16 (2011), 265-324

    J. Sparks, Sasaki-Einstein manifolds , Surveys in Differential Geometry 16 (2011), 265-324. [ arXiv:1004.2461]

    2011 arXiv

  91. [99]

    Strominger, Superstrings with torsion, Nuclear Physics B 274 (1986), no

    A. Strominger, Superstrings with torsion, Nuclear Physics B 274 (1986), no. 2, 253–284

    1986

  92. [100]

    Strominger, Massless black holes and conifolds in string theory , Nuclear Physics B 451 (1995), no

    A. Strominger, Massless black holes and conifolds in string theory , Nuclear Physics B 451 (1995), no. 1-2, 96–108. [ hep-th/9504090]

    1995 arXiv

  93. [101]

    Strominger, S.-T

    A. Strominger, S.-T. Yau, E. Zaslow, Mirror symmetry is T-duality , Nuclear Physics B 479 (1996), 243–259

    1996

  94. [102]

    G. Tian, Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric , Mathematical as- pects of string theory (San Diego, Calif., 1986), World Sci. Publishing, Singapore, 1987, Adv. Ser. Math. Phys., vol. 1, 629–646

    1986

  95. [103]

    Tian, Smooth 3-folds with trivial canonical bundle and ordinary double points , Essays on mirror manifolds, 458–479, Int

    G. Tian, Smooth 3-folds with trivial canonical bundle and ordinary double points , Essays on mirror manifolds, 458–479, Int. Press, Hong Kong, 1992

    1992

  96. [104]

    I , Comm

    A Todorov, The Weil-Petersson geometry of the moduli space of SU(n) (Calabi-Yau) manifolds. I , Comm. Math. Phys. 126 (1989), no. 2, 325–346

    1989

  97. [105]

    Tomasiello, Reformulating supersymmetry with a generalized Dol- beault operator, JHEP02 (2008) 010

    A. Tomasiello, Reformulating supersymmetry with a generalized Dol- beault operator, JHEP02 (2008) 010. [ arXiv:0704.2613]

    2008 arXiv

  98. [106]

    Tosatti, Non-K¨ ahler Calabi-Yau manifolds, Contemp

    V. Tosatti, Non-K¨ ahler Calabi-Yau manifolds, Contemp. Math. 644 (2015), 261-277. [ arXiv:1401.4797] 34 S. PICARD

    2015 arXiv

  99. [107]

    Tosatti and B

    V. Tosatti and B. Weinkove, The Monge-Amp` ere equation for (n-1)- plurisubharmonic functions on a compact K¨ ahler manifold , J. Amer. Math. Soc. 30 (2017), no. 2, 311-346. [ arXiv:1305.7511]

    2017 arXiv

  100. [108]

    Tseng and S.-T

    L.-S. Tseng and S.-T. Yau, Non-Kaehler Calabi-Yau manifolds , Proc. Sympos. Pure Math. 85 (2012), 241-254

    2012

  101. [109]

    Uhlenbeck and S.T

    K. Uhlenbeck and S.T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles , Comm. Pure Appl. Math. 39-S (1986), 257–293; 42 (1989), 703–707

    1986

  102. [110]

    C. T. C. Wall, Classification problems in topology V : On certain 6- manifolds, Invent. Math. 1 (1966), 355-374

    1966

  103. [111]

    Wang, On the connectedness of the standard web of Calabi-Yau 3-folds and small transitions , Asian J

    S.-S. Wang, On the connectedness of the standard web of Calabi-Yau 3-folds and small transitions , Asian J. Math. 22 (2018), no. 6, 981-1004. [arXiv:1603.03929]

    2018 arXiv

  104. [112]

    Wu, On the Geometry of Superstrings with Torsion , Thesis in the Department of Mathematics, Harvard University (2006)

    C.-C. Wu, On the Geometry of Superstrings with Torsion , Thesis in the Department of Mathematics, Harvard University (2006)

    2006

  105. [113]

    Yau, On the Ricci curvature of a compact K¨ ahler manifold and the complex Monge-Amp` ere equation

    S.-T. Yau, On the Ricci curvature of a compact K¨ ahler manifold and the complex Monge-Amp` ere equation. I, Comm. Pure Appl. Math. 31 (1978), 339-411

    1978

  106. [114]

    Yau and S

    S.T. Yau and S. Nadis, The shape of inner space: String theory and the geometry of the universe’s hidden dimensions , Basic Books, 2010

    2010

  107. [115]

    Zhang, On polynomial convergence to tangent cones for singular Kahler-Einstein metrics , preprint

    J. Zhang, On polynomial convergence to tangent cones for singular Kahler-Einstein metrics , preprint. [arXiv:2407.07382] Email address: spicard@math.ubc.ca Department of Mathematics, The University of British Columbia, 1984 Math- ematics Road, V ancouver BC Canada V6T 1Z2

    1984 arXiv

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