Pith. sign in

REVIEW 3 major objections 4 minor 46 references

Invisible singularities in complex algebraic geometry

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Singular fibers of a morphism from a smooth projective variety need not be detectable by the topology of the total space.

desk verdict A high-quality counterexample paper that settles four open questions in equisingularity theory; the main proof is solid, with one terse spot in the PL-sphere claim that is fixable. read the letter →

arxiv 2608.10973 v1 pith:LNX2YX64 submitted 2026-08-11 math.AG math.CV

Maurício Corrêa , János Kollár , Stefan Schreieder , Botong Wang This is my paper · ORCID
classification math.AGmath.CV MSC 14D0614F4532Q5532S1555R10
keywords topologyofvarietiesvanishingcycleequisingularityfiberbundlesmoothnessconjectureuniversalcoverholomorphicone-formAomotocomplex
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

This paper constructs explicit morphisms $g: Y \to \mathbb{P}^1$ from smooth complex projective varieties in which some fibers are singular yet the singularity is invisible to the topology of $Y$. The maps satisfy the strongest regularity conditions short of smoothness: the direct images $R^i g_* \mathbb{Z}$ are constant sheaves, $g$ is a homotopy fiber bundle with simply connected fibers, every fiber is a piecewise-linear manifold, and $g$ is homotopic to a $C^\infty$ fiber bundle. Despite this, $g$ is not a $C^0$ fiber bundle, so the singular fibers are topologically indistinguishable from smooth ones without being homeomorphic to them. The same construction gives negative answers to four open problems about equisingularity, universal covers, real versus holomorphic one-forms, and the exactness of Aomoto complexes.

What carries the argument

The load-bearing mechanism is a quotient-and-blow-up recipe. One starts with a family $Z \to \Delta$ whose Milnor monodromy $T$ satisfies $\det(T-I) = \pm 1$, quotients $Z \times A$ by a diagonal finite group action where $A$ is an abelian variety, and then blows up the resulting family inside a smooth projective fibration with simply connected fibers. The two identities that make this work are the blow-up decomposition $R\pi_* \mathbb{Z}_M \simeq \mathbb{Z}_M \oplus \bigoplus_{r=1}^{c-1} \iota_* \mathbb{Z}_Z[-2r]$, which preserves the homology-fiber-bundle property in both directions, and the eigenvalue computation $\chi_{\vec c}(1) = \pm 1$ for the monodromy of a sum of powers, which guarantees that the quotient by the abelian variety does not change the homology of the fibers. The final smoothing step uses the h-cobordism theorem to trivialize the cobordism over a square and a pseudo-isotopy theorem to make the trivialization compatible with the projection, producing the $C^\infty$ fiber bundle homotopic to $g$.

What would settle it

Compute the link of the singular locus of $\{x^2+y^3-u'w=0\}$ in $\mathbb{A}^5$ directly and check whether every such link is PL-homeomorphic to a standard sphere; a single non-spherical link would invalidate the claim that all fibers are piecewise-linear manifolds.

Watch

Extended reading notes

Core claim

The central discovery is a five-dimensional morphism $g: Y \to \mathbb{P}^1$, with $Y$ smooth and projective, that has singular fibers but is a topological fiber bundle in almost every sense: the local systems $R^i g_* \mathbb{Z}$ are constant, all fibers are simply connected and are piecewise-linear manifolds, and $g$ is homotopic to a locally trivial $C^\infty$ fiber bundle that agrees with $g$ outside a small neighborhood of the singular fibers. Yet $g$ is not a $C^0$ fiber bundle, because the Milnor fiber over a critical value has $H^3 \cong \mathbb{Z}^2$, so vanishing cycles appear even though they do not change the homology of the total space. The construction begins with a rational elliptic surface with six type-II singular fibers, forms a diagonal quotient with an elliptic curve under a sixth-order automorphism, and blows up the resulting threefold inside a rationally connected fivefold fibration over $\mathbb{P}^1$; the blow-up makes the fibers simply connected, upgrading the homology fibration to a homotopy fibration, while the quotient makes the monodromy unimodular so the homology does not jump.

Load-bearing premise

The proof that the bad fibers are still nice manifolds relies on the assumption that a certain non-isolated singular point in the local model has a link that is a standard sphere, even though the cited sphere theorem was proved only for isolated singularities.

Editorial extensions

If this is right

  • There are morphisms from smooth projective fivefolds to $\mathbb{P}^1$ that are homotopy fiber bundles with simply connected fibers but are not $C^0$ fiber bundles, so the singular locus can be invisible to every cohomological or homotopical invariant of the total space.
  • Taking the fiber product of such a map with an elliptic curve $E \to \mathbb{P}^1$ yields a smooth projective fivefold $X$ whose Albanese map is a homotopy fiber bundle and is homotopic to a $C^\infty$ fiber bundle but is not a submersion; consequently the universal cover of $X$ has the homotopy type of a finite CW complex even though $X$ is not smoothly fibered over $E$.
  • On that fivefold every real cohomology class in $H^1(X,\mathbb{R})$ is represented by a closed one-form without zeros, yet the harmonic representative of every such class has a zero and no holomorphic one-form is nowhere zero, giving a negative answer to the 1-form conjecture.
  • There is a smooth projective fivefold for which every Aomoto complex $(H^*(X',\mathbb{C}), \wedge \omega)$ is exact for every finite étale cover $X' \to X$ and every nonzero holomorphic one-form $\omega$, yet every real closed one-form on $X$ has a zero, so exactness of all Aomoto complexes does not force a nowhere-zero holomorphic one-form.
  • In dimension three, the paper proves the opposite rigidity: a morphism with simply connected fibers from a smooth projective threefold to $\mathbb{P}^1$ that is a $\mathbb{Q}$-homology fiber bundle must be smooth, leaving dimension four as the only open case.

Reading between the lines

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

  • The quotient-and-blow-up template may generalize to other starting families and abelian varieties, potentially producing invisible singularities in every dimension at least five, or with singular loci of higher dimension.
  • Since the counterexample to the 1-form conjecture is rationally connected, the paper leaves open whether the conjecture holds for varieties of general type or with nef canonical class; the same two questions are posed in the paper, and testing them would require new constructions.
  • The exactness of the Aomoto complex for all finite étale covers makes the constructed variety a natural test space for other proposed topological detectors of holomorphic one-forms without zeros, such as restrictions on the Albanese or on the cohomology ring.
  • The unproved step in the PL-manifold argument could be checked numerically or conceptually by computing the link of the non-isolated singular locus of the local model $\{x^2+y^3-u'w=0\}$; if a non-spherical link exists, the main theorem's PL conclusion would fail even if the other conclusions survive.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper constructs explicit morphisms from smooth complex projective varieties to the projective line that have singular fibers while being topologically indistinguishable from smooth fibrations: the direct images of the constant integral sheaf are constant, the map is a homotopy fiber bundle with simply connected fibers, all fibers are PL-manifolds, and the map is homotopic to a C-infinity fiber bundle, yet it is not a C^0 fiber bundle. The same construction is used to give counterexamples to four conjectures: the smoothness conjecture of Fernández de Bobadilla–Kollár, a question of Kollár–Pardon on universal covers, Kotschick's conjecture on one-forms, and Schreieder's conjecture on Aomoto complexes. The paper also proves a positive equisingularity theorem for fibrations over Riemann surfaces and a rigidity result in dimension three.

Significance. If the results are correct, they settle several open problems and show that singular fibers of morphisms from smooth projective varieties can be invisible to the topology of the total space. The main construction is explicit and relatively low-dimensional (dimension five), and the paper connects it to substantial existing machinery: Milnor and Brieskorn theory, Smale and Cerf smoothing theory, blow-up arguments, and the decomposition theorem. The paper is well organized and the main lines of the proofs are credible. However, a few load-bearing justifications are asserted rather than demonstrated; they appear repairable, but they must be addressed before the main claims are fully established.

major comments (3)
  1. [Proof of Theorem 1, near Eq. (13)] The conclusion that the singular fibers are PL-manifolds depends on the assertion that the link of every point of the singular locus of the local model {x^2+y^3-u'w=0} in A^5 is a standard PL sphere, justified by [Bri66, Satz 1]. This is not a direct application of that theorem as written: (13) has a one-dimensional singular locus (the v-axis), and the equation is not in Pham-Brieskorn form. The missing steps are (a) reducing the transverse isolated singularity x^2+y^3-u'w=0 in C^4, after the linear change u'w = u^2+v^2 (or u^2-v^2), to the Pham-Brieskorn singularity x^2+y^3+u^2+v^2 with exponents (2,3,2,2), and (b) verifying the Brieskorn sphere criterion for this exponent vector, for instance by computing the characteristic polynomial of the Milnor monodromy and checking chi(1)=±1. Without this verification, item (iii) of Theorem 1 is not established. Please supply the reduction and the check.
  2. [Proposition 20, Steps 3 and 4] In the proof of Proposition 20, the transition from Step 3 to Step 4 asserts that 'every Phi_t preserves F times {1}', and this is used to conclude that Psi' is fiber-preserving on the right vertical face, i.e., g(Psi'(z,1,y)) = (1,y). The pseudo-isotopy group P(F) defined at the beginning of Step 3 consists of diffeomorphisms of F times I fixing F times {0}; Cerf's theorem does not imply that the connecting path can be chosen to preserve the opposite face F times {1}. This matters because the right-face condition is needed to glue ~g_Q to g on the boundary and to obtain a globally defined submersion. Please either prove that the path can be chosen in the subgroup preserving F times {1} (for example by a relative version of Cerf's theorem), or modify the argument by prescribing the top-face trivialization directly in Theorem 21.
  3. [Section 8.4, proof of Theorem 4] The proof of Theorem 4 uses the claim that R^4 tilde h_* Z fails to be locally constant at every point of tilde E lying over the branch locus of p_C: C -> E, and this is asserted without proof. This does not follow from Corollary 37, which only shows that the rational direct image is a trivial local system; the failure is an integral-coefficient phenomenon. Since Proposition 32 is then applied to these integral sheaves to conclude that H_i(tilde X, Z) is not finitely generated, the integral-coefficient statement needs a direct justification, for instance by computing the specialization map F -> F/<sigma_1> on integral cohomology and showing that it is not an isomorphism. Please add this argument.
minor comments (4)
  1. [Corollary 16] The cases n=4,5 are deferred to Example 23, but the embedding construction is not spelled out there for those cases; please make the reduction explicit or state precisely how Example 23 supplies the required embedding into a rationally connected fourfold bundle.
  2. [Example 23] The claim that P' is smooth along the image of V is used to lift the embedding V -> P' to V -> P; a short local-coordinate verification of this smoothness would make the argument easier to check.
  3. [Remark 25] Remark 25 states without proof that the singular fibers admit C-infinity manifold structures agreeing with the algebraic structure outside small neighborhoods; since this remark is not needed for the main theorems, it would be helpful either to indicate the argument or to mark it as a separate open point.
  4. [Theorem 26 and Corollary 27] The statement of Theorem 26 says b_1(X)=2, but this is justified only through the assertion that f induces an isomorphism on fundamental groups; please state that deduction explicitly at the statement or immediately after the construction of X.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the counterexamples are assembled from explicit computations and independent external theorems, not from their own conclusions.

full rationale

The derivation chain is self-contained against external, independently established ingredients. The main construction (Section 5) starts from the explicit rational elliptic surface S in Example 23, forms V=(S×A)/⟨(ξ6,τ6)⟩, and uses Theorem 7 to prove V→P1 is a Z-homology fiber bundle. Theorem 7's hypothesis χ(1)=±1 is verified by an explicit Milnor-fiber computation (χ(x)=x^2−x+1, from Pham/Brieskorn), not by assuming the desired conclusion. The blow-up Y=Bl_V P then satisfies R^i g_*Z constant and simply connected fibers by Proposition 22, which is derived from the blow-up formula (6) and Whitehead's theorem. Proposition 20's smoothing up to homotopy uses Smale's h-cobordism theorem and Cerf's pseudo-isotopy theorem, both external. The PL-manifold claim for fibers is the only flagged point: the link of the local model (13) is asserted to be a standard PL sphere via [Bri66, Satz 1] plus a join with S^1. This is a terse application of an external singularity theorem to a non-isolated singularity, and the paper does not spell out the transverse reduction to the isolated x^2+y^3−u'w=0 singularity; that is a proof-completeness/correctness risk, not a circularity, because the cited theorem is not derived from the paper's conclusions. Self-citations ([Sch21], [HS21], [LMW21], [SY25]) are background, conjectures being disproved, or independently published positive results (e.g., Theorem 40 relies on [HS21, Thm 1.4] without assuming the target statement). No fitted parameter is renamed as a prediction, and no equation is equivalent to its input by construction.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

No numbers are fitted to data and no parameters are hand-tuned. The construction rests on standard published theorems (Brieskorn, Milnor, Smale, Cerf, Whitehead, Tischler, decomposition theorem, Thom-Sebastiani, Artin vanishing) and on two published threefold results involving an author of this paper. No new physical or geometric entities are postulated; the varieties built are the results, not inputs.

assumptions (9)
  • standard math Brieskorn's theorem: the links of the relevant Brieskorn-type hypersurface singularities are standard PL spheres, [Bri66, Satz 1].
    Invoked in Example 14 and in the proof of Theorem 1 to conclude that the local models of the singular fibers are PL-manifolds.
  • standard math Smale's h-cobordism theorem and Cerf's pseudo-isotopy theorem in real dimension at least 8.
    Used in Proposition 20, Steps 1-4, to produce the C-infinity submersion g-tilde homotopic to g and agreeing with it off a neighborhood of the critical values.
  • standard math Whitehead's theorem: a homology isomorphism between simply connected spaces is a homotopy equivalence.
    Used in Corollary 6 and in Step 1 of Proposition 20 to upgrade Z-homology fiber bundles to homotopy fiber bundles.
  • standard math Decomposition theorem of Beilinson, Bernstein and Deligne [BBD82].
    Used in Lemma 36 and Theorem 40 to obtain the direct-sum decompositions of Rp_*Q and Rf_*Q.
  • standard math Thom-Sebastiani computation of Milnor fiber monodromy for sums of powers, after Pham [Pha65].
    Used in Lemma 12 to determine the characteristic polynomial of the monodromy and its value at 1.
  • standard math Milnor's fibration theorem and the existence of vanishing cycles at isolated critical points [Mil68].
    Used in Theorem 7 and Section 5 to compute the local behavior of the quotient family and to detect vanishing cycles.
  • standard math Tischler's theorem: existence of a closed real one-form without zeros implies a smooth S^1-fibration.
    Used in Corollary 27, Theorem 4 and Remark 39 to convert one-form information to bundle information.
  • standard math Artin vanishing for constructible sheaves on Stein manifolds, [KS90, Theorem 10.3.8].
    Used in Proposition 32 to control the Leray spectral sequence and reduce finite generation to local-system data.
  • domain assumption The dimension-three results of Hao-Schreieder [HS21, Theorem 1.4] and Pietig [Pie25a] on holomorphic one-forms without zeros.
    Used in Theorem 40 to conclude that exactness of Aomoto complexes forces f^*omega to be nowhere vanishing in dimension three. These are published theorems by authors partially overlapping with the present paper, and they are not used for the main negative results.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Invisible singularities in complex algebraic geometry." pith.science (2026). https://pith.science/paper/LNX2YX64

@misc{pith2026260810973,
  author       = {Pith},
  title        = {Pith review of: Invisible singularities in complex algebraic geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LNX2YX64}},
  note         = {Machine review of arXiv:2608.10973}
}
read the original abstract

We construct morphisms between smooth complex projective varieties that have singular fibers, but look topologically smooth. We use this to give negative answers to the following four conjectures and questions: the smoothness conjecture of Fern\'andez~de~Bobadilla and Koll\'ar, a question of Koll\'ar and Pardon on universal covers, Kotschick's conjecture on 1-forms, and a conjecture of Schreieder on Aomoto complexes.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

46 extracted references · 44 canonical work pages

  1. [1]

    Kazuhiko Aomoto and Michitake Kita, Theory of hypergeometric functions , Springer Monographs in Mathematics, Springer-Verlag, Tokyo (2011), xvi+317

    work page 2011

  2. [2]

    Alexander Beĭlinson, Joseph Bernstein and Pierre Deligne, Faisceaux pervers , Astérisque 100 (1982), 5--171

    work page 1982

  3. [3]

    Egbert Brieskorn, Beispiele zur D ifferentialtopologie von S ingularit\"aten , Invent. Math. 2 (1966), 1--14. 206972

    work page 1966

  4. [4]

    Nathan Chen, Benjamin Church, and Feng Hao, Nowhere vanishing holomorphic one-forms and fibrations over abelian varieties , Adv. Math. 480 (2025), Paper No. 110463, 16 pp

    work page 2025

  5. [5]

    Jean Cerf, La stratification naturelle des espaces de fonctions diff\'erentiables r\'eelles et le th\'eor\`eme de la pseudo-isotopie, Publ. Math. Inst. Hautes \'Etudes Sci. 39 (1970), 5--173

    work page 1970

  6. [6]

    Ciro Ciliberto, Antonella Grassi, Rick Miranda, Alessandro Verra, and Aline Zanardini, Rational elliptic surfaces with six singular double fibres, 2026, arXiv:2602.11725

    work page arXiv 2026

  7. [7]

    Benjamin Church, Nowhere vanishing 1 -forms on varieties admitting a good minimal model , arXiv:2410.22753, 2024

    work page Pith review arXiv 2024

  8. [8]

    Benjamin Church, Nowhere vanishing 1 -forms on four-folds , Int. Math. Res. Not. IMRN (2026), no. 11, Paper No. rnag102

    work page 2026

Show all 46 references
  1. [9]

    Yajnaseni Dutta, Feng Hao, and Yongqiang Liu, Generic vanishing, 1-forms, and topology of Albanese maps , Math. Z. 306 (2024), article no. 56

    2024

  2. [10]

    Jiabin Du, Feng Hao, Haoyuan Li, and Zichang Wang, Geometry of holomorphic one-forms on smooth projective varieties , Preprint 2026, arXiv:2606.08185

    2026 arXiv

  3. [11]

    Olivier Debarre, Zhi Jiang, and Martí Lahoz, Rational cohomology tori , Geom. Topol. 21 (2017), 1095--1130

    2017

  4. [12]

    Javier Fern\'andez de Bobadilla, Answers to some equisingularity questions, Invent. Math. 161 (2005), no. 3, 657--675. 2181471

    2005

  5. [13]

    Greuel and G

    Javier Fern\'andez de Bobadilla, Topological equisingularity: old problems from a new perspective (with an appendix by G.-M. Greuel and G. Pfister on Singular), in: Handbook of geometry and topology of singularities III, Springer, Cham, 2022, 145--202,

    2022

  6. [14]

    Javier Fern\'andez de Bobadilla and J\'anos Koll\'ar, Homotopically trivial deformations , J. Singul. 5 (2012), 85--93

    2012

  7. [15]

    Javier Fern\'andez de Bobadilla and Tomasz Pe ka, Symplectic monodromy at radius zero and equimultiplicity of -constant families, Ann. of Math. (2) 200 (2024), no. 1, 153--299

    2024

  8. [16]

    Feng Hao, Nowhere vanishing holomorphic one-forms on varieties of Kodaira codimension one , Int. Math. Res. Not. IMRN 2024, no. 6, 4501--4515

    2024

  9. [17]

    Allen Hatcher, Algebraic topology , Cambridge University Press, Cambridge, 2002

    2002

  10. [18]

    Feng Hao and Stefan Schreieder, Holomorphic one-forms without zeros on threefolds , Geom. Topol. 25 (2021), no. 1, 409--444

    2021

  11. [19]

    Feng Hao, Zichang Wang, and Lei Zhang, Good minimal models with nowhere vanishing holomorphic 1 -forms , Ann. Sc. Norm. Super. Pisa Cl. Sci. 20 (2026), 20 pp., https://doi.org/10.2422/2036-2145.202501_007 doi:10.2422/2036-2145.202501\_007

    2026

  12. [20]

    Kervaire and John W

    Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres. I , Ann. of Math. (2) 77 (1963), 504--537

    1963

  13. [21]

    134, Cambridge University Press, Cambridge, 1998

    J\'anos Koll\'ar and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998

    1998

  14. [22]

    Dieter Kotschick, Holomorphic one-forms, fibrations over the circle, and characteristic numbers of Kähler manifolds , Math. Proc. Cambridge Philos. Soc. 172 (2022), no. 1, 95--103

    2022

  15. [23]

    J\'anos Koll\'ar and John Pardon, Algebraic varieties with semialgebraic universal cover , J. Topol. 5 (2012), no. 1, 199--212

    2012

  16. [24]

    Kirby and Laurence C

    Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies, vol. 88, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1977

    1977

  17. [25]

    292, Springer-Verlag, Berlin, 1990

    Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der mathematischen Wissenschaften, vol. 292, Springer-Verlag, Berlin, 1990

    1990

  18. [26]

    Reine Angew

    Yongqiang Liu, Lauren t iu Maxim, and Botong Wang, Aspherical manifolds, Mellin transformation and a question of Bobadilla--Koll \'a r , J. Reine Angew. Math. 781 (2021), 1--18

    2021

  19. [27]

    L\^e D\ ung Tr\'ang and Chakravarthi Padmanabhan Ramanujam, The invariance of Milnor's number implies the invariance of the topological type, Amer. J. Math. 98 (1976), no. 1, 67--78

    1976

  20. [28]

    Massey, The L \^e varieties

    David B. Massey, The L \^e varieties. II , Invent. Math. 104 (1991), no. 1, 113--148. 1094048

    1991

  21. [29]

    61, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1968, iii+122 pp

    John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, vol. 61, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1968, iii+122 pp

    1968

  22. [30]

    Fr\'ed\'eric Pham, Formules de P icard- L efschetz g\'en\'eralis\'ees et ramification des int\'egrales , Bull. Soc. Math. France 93 (1965), 333--367. 195868

    1965

  23. [31]

    Simon Pietig, Holomorphic 1-forms without zeros on K\"ahler threefolds , Preprint 2025, arXiv:2506.22067

    2025 arXiv

  24. [32]

    Simon Pietig, Holomorphic one-forms without zeros on K\"ahler manifolds of Kodaira codimension one , Preprint 2025, arXiv:2512.08395

    2025

  25. [33]

    Bjorn Poonen, Rational Points on Varieties, Grad. Stud. Math. 186, Amer. Math. Soc., Providence, RI, 2017

    2017

  26. [34]

    Mihnea Popa and Christian Schnell, Kodaira dimension and zeros of holomorphic one-forms , Ann. of Math. 179 (2014), no. 3, 1109--1120

    2014

  27. [35]

    Stefan Schreieder, Zeros of holomorphic one-forms and topology of Kähler manifolds , Int. Math. Res. Not. IMRN 2021, no. 8, 6169--6183

    2021

  28. [36]

    Fernando Serrano, Divisors of bielliptic surfaces and embeddings in P^4 , Math. Z. 203 (1990), 527--533

    1990

  29. [37]

    589, Springer-Verlag, Berlin--New York, 1977

    Alexandre Grothendieck, Cohomologie -adique et fonctions L (SGA 5) , Séminaire de Géométrie Algébrique du Bois-Marie 1965--1966, édité par Luc Illusie, Lecture Notes in Mathematics, vol. 589, Springer-Verlag, Berlin--New York, 1977

    1965

  30. [38]

    Dirk Siersma, The vanishing topology of non isolated singularities, in: New developments in singularity theory (Cambridge, 2000), NATO Sci. Ser. II Math. Phys. Chem., vol. 21, Kluwer Acad. Publ., Dordrecht, 2001, 447--472

    2000

  31. [39]

    Stephen Smale, A Vietoris mapping theorem for homotopy, Proc. Amer. Math. Soc. 8 (1957), 604--610

    1957

  32. [40]

    Stephen Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387--399

    1962

  33. [41]

    Masahiro Shiota and Masataka Yokoi, Triangulations of subanalytic sets and locally subanalytic manifolds, Trans. Amer. Math. Soc. 286 (1984), no. 2, 727--750

    1984

  34. [42]

    Stefan Schreieder and Ruijie Yang, Zeros of one-forms and homologically trivial fibrations , Michigan Math. J. 75 (2025), no. 5, 917--926

    2025

  35. [43]

    777, Springer, Berlin, 1980, 71--146

    Bernard Teissier, R\'esolution simultan\'ee I--II, in: Michel Demazure, Henry Charles Pinkham, and Bernard Teissier (eds.), S\'eminaire sur les singularit\'es des surfaces, Lecture Notes in Math., vol. 777, Springer, Berlin, 1980, 71--146

    1980

  36. [44]

    David Tischler, On fibering certain foliated manifolds over S^1 , Topology 9 (1970), 153--154

    1970

  37. [45]

    76, Cambridge University Press, Cambridge, 2002

    Claire Voisin, Hodge theory and complex algebraic geometry I , Cambridge Studies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2002

    2002

  38. [46]

    Oscar Zariski, Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481--491. 277533

    1971

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.