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An introduction to real oriented blowups in toric, toroidal and logarithmic geometries

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Pith's one-line read The rounding of complex log spaces is a canonical real oriented blowup: it cuts any complex toroidal variety along its toroidal boundary and yields canonical representatives of links and Milnor fibers in singularity theory.

desk verdict A careful, genuinely useful survey of real oriented blowups and roundings; its main soft spot is a hand-waved boundary argument in Corollary 4.53. read the letter →

arxiv 2507.11982 v1 pith:VPLAWXCR submitted 2025-07-16 math.AG

Patrick Popescu-Pampu This is my paper · ORCID
classification math.AG MSC 14A2114B0514M2532S0532S55
keywords roundingKato-NakayamaspacesrealorientedblowupslogstructurestoroidalvarietiesMilnorfiberslinksofsingularitiespolarcoordinates
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 survey argues that Kato and Nakayama's rounding operation, also called Betti realization, is the right canonical form of A'Campo's real oriented blowup. The paper builds toric, toroidal, and logarithmic geometries from the single example of the passage to polar coordinates, and then presents rounding as a functor that replaces the algebro-geometric boundary of a complex log space by a topological boundary. If the claims are correct, every complex toroidal variety has a canonical manifold-with-boundary model, and every isolated complex analytic singularity and every smoothing has canonical representatives for its link and Milnor fiber once a toroidal resolution is chosen. The practical payoff is that these representatives come with canonical projections and are canonically decomposed into pieces indexed by the components of the exceptional divisor or central fiber, so no choices of tubular neighborhoods are involved.

What carries the argument

The central object is the rounding functor $W\mapsto W^\odot$, defined for a complex log space $W$ by $W^\odot := \{(x,u) : x\in |W|,\ u\in\mathrm{Hom}(M_{W,x},S^1),\ u(f)=\mathrm{sign}(f(x))\text{ for all units } f\}$, where $(M_{W,x},\alpha_{W,x})$ is the stalk of the log structure. Log structures—sheaves of monoids with an evaluation map to the structure sheaf—package the boundary divisor without choosing a tubular neighborhood. For the complex plane with the log structure of the origin, $W^\odot = \mathbb{R}_{\geq 0}\times S^1$ and $\tau_W$ is $\tau_{\mathbb{C}}$; for a toroidal variety, the same construction gives the real oriented blowup. Applied to a log morphism satisfying properness, separatedness, exactness, and relative log smoothness, rounding becomes a locally trivial fibration by the Nakayama–Ogus theorem. The polar log point $\Pi_{\mathbb{C}}^{\mathrm{pol}}$, i.e. the monoid $\mathbb{R}_{\geq 0}\times S^1$ acting on $\mathbb{C}$, serves as the universal 'rounding point': $W^\odot$ is exactly the set of log-space morphisms from $\Pi_{\mathbb{C}}^{\mathrm{pol}}$ to $W$.

What would settle it

Take a Brieskorn hypersurface such as $x^2 + y^3 + z^5 = 0$ with its standard toroidal resolution: compute the rounding of the restriction to the exceptional divisor $E$ of the divisorial log structure, and compare the resulting space, with its projection to $E$, to the classical link (a Brieskorn sphere). If the two are not homeomorphic as spaces over $E$, the claim that rounding gives a canonical representative of the link is false.

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Extended reading notes

Core claim

The central discovery presented is that rounding is a functorial generalization of the real oriented blowup and is already forced by the ordinary polar-coordinates map $\tau_{\mathbb{C}}: \mathbb{R}_{\geq 0}\times S^1 \to \mathbb{C}$. For any complex log space $W$, its rounding $W^\odot$ is the set of pairs $(x,u)$ with $x\in |W|$ and $u$ a monoid homomorphism from the stalk of the log structure at $x$ to $S^1$ extending the sign map; the rounding map $\tau_W: W^\odot \to |W|$ sends $(x,u)$ to $x$. On a toroidal variety $X$ with its canonical divisorial log structure, this $W^\odot$ is the real oriented blowup along the toroidal boundary, a topological manifold-with-boundary whose boundary is the preimage of $\partial X$ and is a canonical representative of the boundary of any tubular neighborhood of $\partial X$. In singularity theory, restricting the divisorial log structure to an exceptional divisor $E$ with simple normal crossings and rounding gives a representative of the link of an isolated singularity, while rounding the log enhancement of a resolution of a smoothing yields a locally trivial fibration isomorphic to the Milnor fibration. The engine for the fibration statement is the Nakayama–Ogus local triviality theorem.

Load-bearing premise

The load-bearing premise is that the quoted, unproved results are correct—Nakayama–Ogus's local triviality theorem for roundings, the identification of rounding with real oriented blowup, and Hironaka's resolution theorem—since the canonical links and Milnor fibers are built directly on them.

Editorial extensions

If this is right

  • Any complex toroidal variety $X$ comes with a canonical topological manifold-with-boundary $X^\odot$ whose boundary is homeomorphic to the boundary of every tubular neighborhood of $\partial X$.
  • Links of isolated complex analytic singularities have canonical representatives built from roundings of divisorial log structures; each representative carries a canonical projection to the exceptional divisor.
  • Milnor fibrations of smoothings of complex singularities are represented, up to isomorphism, by roundings of the log enhancement of $f\circ\pi$; the roundings are locally trivial fibrations and are canonically decomposed into pieces above the components of the central fiber.
  • Because rounding is functorial, any factorization of resolutions produces compatible maps between link representatives, yielding an inverse system of canonical representatives for the link.
  • The same machinery works when the resolution is only toroidal and not necessarily smooth, opening singularities to the use of tropical and toroidal modifications.

Reading between the lines

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

  • Beyond the paper: the functoriality of rounding suggests that one could define link invariants directly from log structures without fixing a resolution, provided a resolution-independent or derived version of the boundary is used.
  • Beyond the paper: the reinterpretation $W^\odot = \mathrm{Hom}(\Pi_{\mathbb{C}}^{\mathrm{pol}}, W)$ is a template: replacing the polar log point by other log points yields variants such as extended roundings with canonical geometric monodromy, which could be developed into a general theory of canonical boundary replacements for degenerations.
  • Beyond the paper: the canonical representatives may allow comparison of Milnor fibrations across different resolutions purely at the level of log structures, potentially simplifying proofs of resolution-independence conjectures for wider singularity classes.
  • Beyond the paper: the toroidal version of Corollary 4.53 can be tested explicitly on Newton non-degenerate singularities by computing roundings from local tropicalizations and checking that the resulting fibration is independent of the chosen tropical subdivision.
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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 / 6 minor

Summary. This expository paper develops the circle of ideas around real oriented blowups and Kato–Nakayama roundings, starting from polar coordinates and proceeding through affine and general toric varieties, toroidal varieties, A'Campo real oriented blowups, log spaces, prelog structures, charts, and the rounding functor. The final sections state Nakayama–Ogus local triviality and draw consequences for singularity theory: canonical representatives of links of isolated singularities and of Milnor fibrations of smoothings, obtained by rounding log enhancements of resolutions. The paper is explicitly a survey: all main theorems are attributed to the literature, with many proofs omitted or left as exercises.

Significance. If the external results it relies on are correct, the paper provides a useful and unusually pedagogical bridge from elementary polar coordinates to current log-geometric techniques in singularity theory. Its main conceptual claim — that rounding canonically replaces the algebraic boundary by a topological boundary, and that this gives canonical Milnor-fibration representatives — is significant for the intended audience. The manuscript is transparent about its sources, carefully attributes results to Kato, Kato–Nakayama, Nakayama–Ogus, Kajiwara–Nakayama, Argüz, and the author's joint work with Cueto and Stepanov, and it contains no circular derivations or fitted parameters. As a survey it does not prove the major external theorems, but that is appropriate. The main weakness is that one of its advertised applications, Corollary 4.53, contains a genuinely nontrivial boundary-with-corners step that is asserted rather than proved.

major comments (2)
  1. [§4.8, Corollary 4.53] The passage from Theorem 4.49/Corollary 4.51 to the Milnor-tube setting is load-bearing and is not actually supplied. Theorem 4.49 is stated for a proper morphism of log complex analytic spaces, while the Milnor tube representative has a topological boundary and f∘π is not proper on the closed tube; the text acknowledges this difference but only says that 'since f∘π is locally trivial near that boundary, it is straightforward to show that a version of Corollary 4.51 applies.' That is exactly the step that needs a collar or relative Ehresmann argument, and it is not given. Moreover, the final assertion that the restriction to the boundary circle is 'isomorphic to the Milnor fibration' is stronger than mere local triviality: it also requires a comparison of the rounding fibers over the circle with the classical Milnor fibers over the corresponding angles, including the monodromy identification. Because Corollary 4.53 is the stated mechanism for obtaining canonical representatives of Milnor fibrations, this gap should be closed by a proof, a precise reference, or a clearly marked reduction to an existing result in [19] or [57].
  2. [§1.4 and §4.9, applications to links] The link representative (E†)^⊙ is asserted to be a canonical representative of ∂(X,x) via Proposition 4.44, but the argument depends on the identification of rounding with the real oriented blowup for toroidal varieties, which is quoted from [42, Proposition A.1] and [6, Proposition 2.4]. This is acceptable for a survey, but the text should state more explicitly that the canonicity of the link representative is only relative to a chosen snc resolution and that the invariance under different resolutions is not established here; the current wording in §1.4 and §4.9 may leave a reader with the impression that the representative is independent of all choices.
minor comments (6)
  1. [§4.8, Theorem 4.49] In the statement of Theorem 4.49, the symbol f is used for the morphism, while the surrounding text and Corollary 4.51 use ϕ or φ; this should be unified to avoid confusion.
  2. [§4.4, Definition 4.25] The word 'pullbak' should be 'pullback'.
  3. [§3.2, Proposition 3.9] The proof of Proposition 3.9 invokes Proposition 4.43, which is stated much later and itself relies on log-geometric notions. A direct toric-chart argument, or an explicit statement that this is a forward reference in an expository ordering, would improve readability.
  4. [§2.7] The authors of the classical reference [48] are spelled 'Kempf, Knudsen, Mumford and Saint-Donat', but the text twice writes 'Kempf, Knudson, Mumford and Saint-Donat'; the spelling should be corrected.
  5. [§3.1, Definition 3.8] In the paragraph after Definition 3.8, 'real oriented real blowup' is a typo for 'real oriented blowup'.
  6. [§4.8, after Corollary 4.51] The sentence 'We will not give the definitions of the terms involved in the statement which we have not discussed until now' is helpful, but it would be even more useful to point the reader to the precise locations in [57] where relative coherence, exactness, relative log smoothness, and verticality are defined.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the paper is an expository survey whose concrete consequences are imported from external theorems; the only flagged weakness is an unproved boundary-collar step, which is a presentation gap, not circularity.

full rationale

This is a survey/introduction, not a paper that fits parameters to data or derives new predictions from its own assumptions. Its central claims about rounding, real oriented blowups, and canonical representatives of links and Milnor fibers are presented as consequences of prior results, chiefly Kato-Nakayama's rounding construction, Nakayama-Ogus's local triviality theorem (Theorem 4.49), and the identification of rounding with real oriented blowup (Proposition 4.43, cited from Kajiwara-Nakayama and Argüz). These are external, machine-checkable-style mathematical results with stated hypotheses that do not include the conclusions being drawn here. Self-citations appear (e.g., [19], [65]) but are used for terminology, context, and applications, not as the load-bearing justification for the expository claims; the uniqueness/canonicity statements are not forced by an author-imported uniqueness theorem. The one genuinely weak step is Corollary 4.53, where the paper acknowledges that the Milnor tube representative has a topological boundary and says 'it is straightforward to show that a version of Corollary 4.51 applies' without supplying the collar argument. That is an omitted proof or a gap in exposition, not a circularity: the assertion is not equivalent to an input by construction, nor does it rename a fitted quantity as a prediction. The survey's reliance on Hironaka resolution and on Nakayama-Ogus is legitimate external dependence. No circular step can be exhibited, so the circularity score is low.

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

This is a survey, so the axioms are the external theorems and background theories it relies on. The paper introduces no free parameters and no new entities.

assumptions (5)
  • domain assumption Hironaka's resolution of singularities: every complex singularity admits a resolution whose exceptional divisor has simple normal crossings.
    Invoked in Sections 1.4 and 4.8 to choose (X̃,E) with Z(f∘π) a simple normal crossings divisor.
  • domain assumption Nakayama-Ogus local triviality theorem: the rounding of a proper, separated, exact, relatively log smooth morphism of complex log spaces is a locally trivial fibration.
    Quoted as Theorem 4.49 from [57]; not proved in this survey, but the singularity-theory consequences in Corollaries 4.51 and 4.53 rest on it.
  • domain assumption Identification of rounding with real oriented blowup for toroidal varieties.
    Proposition 4.43 and Proposition 4.44, cited from [42] and [6]; used to connect log geometry to the classical real oriented blowup constructions.
  • standard math Kempf-Knudsen-Mumford-Saint-Donat theory of toroidal embeddings.
    Foundational combinatorial facts about fans of monoids in Section 2 are taken from [48] and [34].
  • standard math Kato's theory of log structures, following Fontaine and Illusie.
    Definitions in Section 4, including log spaces, charts, and fine log spaces, follow [43] and [59].

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Pith. "Pith review of An introduction to real oriented blowups in toric, toroidal and logarithmic geometries." pith.science (2026). https://pith.science/paper/VPLAWXCR

@misc{pith2026250711982,
  author       = {Pith},
  title        = {Pith review of: An introduction to real oriented blowups in toric, toroidal and logarithmic geometries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VPLAWXCR}},
  note         = {Machine review of arXiv:2507.11982}
}
abstract

This text is an introduction to the applications of rounding of complex log spaces (also known as Kato-Nakayama or Betti realization) to singularity theory. Log spaces in the sense of Fontaine and Illusie were first described in print by Kato, in a 1988 paper. Rounding of complex log spaces was introduced in a 1999 paper by Kato and Nakayama and is a functorial generalization of A'Campo's 1975 notion of a real oriented blowup. It allows to cut canonically any complex toroidal variety $X$ along its toroidal boundary $\partial X$, producing a topological manifold-with-boundary, whose boundary is a canonical representative of the boundary of any tubular neighborhood of $\partial X$ in $X$. In singularity theory, roundings may be used to get canonical representatives of links of isolated complex analytic singularities and of Milnor fibers of smoothings of complex singularities, once toroidal resolutions of the singularity or of the smoothing are chosen. The text starts with introductions to not necessarily normal toric varieties, it passes then to toroidal varieties and to their real oriented blowups. It continues with introductions to log spaces and to rounding of complex log spaces. It concludes with an important theorem of Nakayama and Ogus about the local triviality of the rounding of special types of log morphisms. The notions of affine toric variety, real oriented blowup, log structure and rounding are introduced by means of the classical passage to polar coordinates.

Figures

Figures reproduced from arXiv: 2507.11982 by the authors.

Figure 1
Figure 1. The real oriented blowup τC of C at the origin factors through its real blowup πC As in this simple case, A’Campo’s real oriented blowup replaces canonically the divisor D, which may be seen as an algebro-geometric boundary of M, by a topological boundary of the complement M \ D of the divisor, and the resulting manifold-with-boundary (and corners when the divisor has singular points) is homeomorphic to the compleme… view at source ↗
Figure 2
Figure 2. The real oriented blowup of a smooth surface along a divisor with simple normal crossings [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

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Works this paper leans on

81 extracted references · 79 canonical work pages

  1. [19]

    M. A. Cueto, P. Popescu-Pampu, D. Stepanov, The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof. In Essays in geometry–dedicated to Norbert A’Campo , 629–709, IRMA Lect. Math. Theor. Phys. 34, EMS Press, 2023. 6, 24, 27, 32, 38, 39

    work page 2023

  2. [57]

    Nakayama, A

    C. Nakayama, A. Ogus, Relative rounding in toric and logarithmic geometry . Geometry & Topology 14 (2010), 2189–2241. 5, 24, 34, 37

    work page 2010

  3. [1]

    A’Campo, La fonction zˆ eta d’une monodromie

    N. A’Campo, La fonction zˆ eta d’une monodromie. Comment. Math. Helv. 50 (1975), 233–248. 3, 6, 23, 37

    1975

  4. [2]

    Abramovich, Q

    D. Abramovich, Q. Chen, D. Gillam, Y. Huang, M. Olsson, M. Satriano, S. Sun, Logarithmic geometry and moduli . In Handbook of moduli. Vol. I, 1–61, Adv. Lect. Math. (ALM), 24, Int. Press, Somerville, MA, 2013. 24, 34

    work page 2013

  5. [3]

    Achinger, A

    P. Achinger, A. Ogus, Monodromy and log geometry . Tunis. J. Math. 2 (2020), no. 3, 455–534. 24, 34, 37

    work page 2020

  6. [4]

    Ambrosi, M

    E. Ambrosi, M. Manzaroli, Betti numbers of real semistable degenerations via real logarithmic geometry. ArXiv:2211.12134v1. 37

    arXiv

  7. [5]

    Arg¨ uz,Topological torus fibrations on Calabi-Yau manifolds via Kato-Nakayama spaces

    H. Arg¨ uz,Topological torus fibrations on Calabi-Yau manifolds via Kato-Nakayama spaces. Proceedings of the G¨ okova Geometry- Topology Conferences 2018/2019, 176–200, Int. Press, 2020. 24, 34

    work page 2018

  8. [6]

    Arg¨ uz,Real loci in (log) Calabi-Yau manifolds via Kato-Nakayama spaces of toric degenerations

    H. Arg¨ uz,Real loci in (log) Calabi-Yau manifolds via Kato-Nakayama spaces of toric degenerations . Eur. Journ. Maths. 7, 869–930 (2021). 27, 31, 34, 35, 37

    work page 2021

Show all 81 references
  1. [7]

    Barthel, L

    G. Barthel, L. Kaup, K.-H. Fieseler, Introduction to basic toric geometry. In Singularities in geometry and topology , 3–56, World Sci. Publ., 2007. 7

    2007

  2. [8]

    Brasselet, Introduction to toric varieties

    J.-P. Brasselet, Introduction to toric varieties. Publica¸ c˜ oes Matem´ aticas do IMPA, 2008. 7

    2008

  3. [9]

    Bultot, J

    E. Bultot, J. Nicaise, Computing motivic zeta functions on log smooth models . Math. Zeitschrift 295 (2020), 427–462. 6, 34

    2020

  4. [10]

    Cailotto, A note on Kato-Nakayama spaces

    M. Cailotto, A note on Kato-Nakayama spaces. C. R. Math. Acad. Sci. Paris 339 (2004), no. 4, 261–264. 34

    2004

  5. [11]

    Campesato, G

    J.-B. Campesato, G. Fichou, A. Parusi´ nski, Motivic, logarithmic, and topological Milnor fibrations . Adv. Math. 461 (2025), art. no. 110075. 6, 37, 38

    2025

  6. [12]

    Cannas da Silva, Lectures on symplectic geometry

    A. Cannas da Silva, Lectures on symplectic geometry. Lecture Notes in Maths 1764. Springer, 2001. 11

    2001

  7. [13]

    Cauwbergs, Logarithmic geometry and the Milnor fibration

    T. Cauwbergs, Logarithmic geometry and the Milnor fibration . C. R. Acad. Sci. Paris, Ser. I 354 (2016), 701–706. 37 40 P. POPESCU-PAMPU

    2016

  8. [14]

    Cerf, Topologie de certains espaces de plongements

    J. Cerf, Topologie de certains espaces de plongements. Bull. Soc. Math. France 89 (1961), 227–380. 36

    1961

  9. [15]

    J. L. Cisneros-Molina, J. Seade, Milnor’s fibration theorem for real and complex singularities. In Handbook of geometry and topology of singularities II , 309–359, Springer, 2021. 5, 38

    2021

  10. [16]

    Commelin, P

    J. Commelin, P. Habegger, A. Huber, Exponential periods and o-minimality. To appear in Annales de l’Institut Fourier. 36

  11. [17]

    Cox, What is a toric variety? In Topics in algebraic geometry and geometric modeling , 203–223, Contemp

    D. Cox, What is a toric variety? In Topics in algebraic geometry and geometric modeling , 203–223, Contemp. Math. 334, Amer. Math. Soc., Providence, RI, 2003. 7, 18

    2003

  12. [18]

    D. Cox, J. Little, H. Schenck, Toric varieties. Amer. Math. Soc., 2011. 7, 10, 18, 19

    2011

  13. [20]

    V. I. Danilov, Polyhedra of schemes and algebraic varieties. Math. USSR-Sb. 26 (1975), no. 1, 137–149 (1976); translated from Mat. Sb. (N.S.) 139 (1975), no. 1, 146–158, 160. 4

    1975

  14. [21]

    V. I. Danilov, The geometry of toric varieties. Russian Math. Surveys 33 (1978), no. 2, 97–154. 7

    1978

  15. [22]

    Delzant, Hamiltoniens p´ eriodiques et images convexes de l’application moment

    T. Delzant, Hamiltoniens p´ eriodiques et images convexes de l’application moment. Bulletin de la Soc. Math. France 116, no 3 (1988), 315–339. 11

    1988

  16. [23]

    Eisenbud, J

    D. Eisenbud, J. Harris, The geometry of schemes. Springer, 2000. 24, 26, 33, 36

    2000

  17. [24]

    Ewald, Combinatorial convexity and algebraic geometry

    G. Ewald, Combinatorial convexity and algebraic geometry. Springer, 1996. 7

    1996

  18. [25]

    Fern´ andez de Bobadilla, T

    J. Fern´ andez de Bobadilla, T. Pe lka,Symplectic monodromy at radius zero and equimultiplicity of µ-constant families. Ann. of Math. (2) 200 (1) (2024), 153–299. 6, 38

    2024

  19. [26]

    de Fernex, J

    T. de Fernex, J. Koll´ ar, C. Xu, The dual complex of singularities. In Higher dimensional algebraic geometry – in honour of Professor Yujiro Kawamata’s sixtieth birthday , 103–129, Adv. Stud. Pure Math. 74, Math. Soc. Japan, 2017. 4

    2017

  20. [27]

    Francis-Staite, D

    K. Francis-Staite, D. Joyce, C∞-Algebraic Geometry with Corners. Cambridge Univ. Press, 2024. 36

    2024

  21. [28]

    Fulton, Introduction to toric varieties

    W. Fulton, Introduction to toric varieties. Princeton Univ. Press, 1993. 7

    1993

  22. [29]

    E. R. Garc ´ ıa Barroso, P. D. Gonz´ alez P´ erez, P. Popescu-Pampu,The combinatorics of plane curve singularities: how Newton polygons blossom into lotuses. In Handbook of geometry and topology of singularities. I , 1–150, Springer, 2020. 7

    2020

  23. [30]

    I. M. Gelfand, M. Kapranov, A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants. Birkh¨ auser, 1994. 7, 18

    1994

  24. [31]

    Gillam, Oriented real blowup

    W. Gillam, Oriented real blowup. Manuscript of 2011. 24

    2011

  25. [32]

    Gillam, S

    W. Gillam, S. Molcho, Log differentiable spaces and manifolds with corners. arXiv:1507.06752. 36

    arXiv

  26. [33]

    R. E. Gompf, A. I. Stipsicz, 4 -Manifolds and Kirby Calculus. Amer. Math. Soc., 1999. 2

    1999

  27. [34]

    P. D. Gonz´ alez P´ erez, B. Teissier,Toric geometry and the Semple-Nash modification. Revista de la Real Academia de Ciencias Exactas, F ´ ısicas y Naturales, Serie A Matem´ aticas,108 (2014), 1–48. 15, 18

    2014

  28. [35]

    Gross, Tropical geometry and mirror symmetry

    M. Gross, Tropical geometry and mirror symmetry. Amer. Math. Soc., 2011. 24, 26, 27, 31, 33, 35

    2011

  29. [36]

    Grothendieck, Esquisse d’un programme

    A. Grothendieck, Esquisse d’un programme. In Geometric Galois actions 1, 5–48, Cambridge Univ. Press, 1997. English translation by L. and P. Lochak on pages 243–283 of the same volume. 2

    1997

  30. [37]

    Hubbard, P

    J. Hubbard, P. Papadopol, V. Veselov, A compactification of H´ enon mappings inC2 as dynamical systems . Acta Math. 184, No. 2 (2000), 203–270. 20

    2000

  31. [38]

    Illusie, Logarithmic spaces (according to K

    L. Illusie, Logarithmic spaces (according to K. Kato) . In Barsotti Symposium in Algebraic Geometry (Ed. V. Cristante, W. Messing), Perspectives in Math. 15, Academic Press, 1994, 183–203. 24, 26

    1994

  32. [39]

    Illusie, An overview of the work of K

    L. Illusie, An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic ´ etale cohomology. In Cohomologies p-adiques et applications arithm´ etiques, II.Ast´ erisque279 (2002), 271–322. 24

    2002

  33. [40]

    Illusie, K

    L. Illusie, K. Kato, C. Nakayama, Quasi-unipotent Logarithmic Riemann-Hilbert Correspondences. J. Math. Sci. Univ. Tokyo 12 (2005), 1–66. 34

    2005

  34. [41]

    Joyce, A generalization of manifolds with corners

    D. Joyce, A generalization of manifolds with corners . Adv. in Maths. 299 (2016), 760–862. 36

    2016

  35. [42]

    Kajiwara, C

    T. Kajiwara, C. Nakayama, Higher direct images of local systems in log Betti cohomology . J. Math. Sci. Univ. Tokyo 15 (2008) 291–323. 34, 35, 36

    2008

  36. [43]

    Kato, Logarithmic structures of Fontaine-Illusie

    K. Kato, Logarithmic structures of Fontaine-Illusie . In Algebraic Analysis, Geometry and Number Theory , the Johns Hopkins Univ. Press, Baltimore, 1988, 191–224. 4, 24, 26, 28, 29, 31, 33, 34

    1988

  37. [44]

    Kato, Toric singularities

    K. Kato, Toric singularities. Amer. Journ. of Maths. 116 (1994), 1073–1099. 16

    1994

  38. [45]

    Kato, Log smooth deformation theory

    F. Kato, Log smooth deformation theory. Tˆ ohoku Math. J.48 (1996), 317–354. 34

    1996

  39. [46]

    K. Kato, C. Nakayama, Log Betti cohomology, log ´ etale cohomology, and log de Rham cohomology of log schemes over C. Kodai Math. J. 22 (1999), 161–186. 5, 23, 24, 34, 37

    1999

  40. [47]

    Kawamata, On algebraic fiber spaces

    Y. Kawamata, On algebraic fiber spaces . In Contemporary trends in algebraic geometry and algebraic topology (Tianjin, 2000), vol. 5 of Nankai Tracts Math., World Sci. Publ., 2002, 135–154. 23, 37

    2000

  41. [48]

    Kempf, F

    G. Kempf, F. F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal embeddings. I. Lecture Notes in Maths. 339. Springer-Verlag,

  42. [49]

    A. G. Khovanskii, Newton polyhedra and toroidal varieties . Funkt. An. i Prilozhen. 11 (1977), 56–64; English translation, Funct. An. Appl. 11: 4 (1977), 289–296. 7

    1977

  43. [50]

    A. A. Kosinski, Differential manifolds. Pure and Applied Mathematics 138. Academic Press, 1993. 2

    1993

  44. [51]

    D. T. Lˆ e,Some Remarks on Relative Monodromy. Proc. of Nordic Summer School on Real and Complex Singularities, Oslo (1976), 397–403, 1977. 5, 38

    1976

  45. [52]

    Kottke, R

    C. Kottke, R. B. Melrose, Generalized blow-up of corners and fibre products . Trans. Amer. Math. Soc. 367 (2015), 651–705. 36

    2015

  46. [53]

    Majima, Asymptotic analysis for integrable connections with irregular singular points

    H. Majima, Asymptotic analysis for integrable connections with irregular singular points. Lecture Notes in Maths. 1075, Springer,

  47. [54]

    Milnor, A procedure for killing homotopy groups of differentiable manifolds

    J. Milnor, A procedure for killing homotopy groups of differentiable manifolds. Proc. Sympos. Pure Math., Vol. III, Amer. Math. Soc., 1961, 39–55. 2

    1961

  48. [55]

    Milnor, Singular points of complex hypersurfaces

    J. Milnor, Singular points of complex hypersurfaces . Princeton Univ. Press, 1968. 5, 38 AN INTRODUCTION TO REAL ORIENTED BLOWUPS 41

    1968

  49. [56]

    Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity

    D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity . Inst. Hautes ´Etudes Sci. Publ. Math. no. 9 (1961), 5–22. 4

    1961

  50. [58]

    Oda, Convex bodies and algebraic geometry

    T. Oda, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties. Springer-Verlag, 1988. 7

    1988

  51. [59]

    Ogus, Lectures on logarithmic algebraic geometry

    A. Ogus, Lectures on logarithmic algebraic geometry. Cambridge Univ. Press, 2018. 8, 13, 15, 16, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35

    2018

  52. [60]

    Parusi´ nski,Blow-Analytic Retraction onto the Central Fibre

    A. Parusi´ nski,Blow-Analytic Retraction onto the Central Fibre . In Real analytic and algebraic singularities , Proceedings of the Kuo Symposium, eds. Fukuda et al. Pitman Research Notes in Mathematics 381, Longman 1998, 43–61. 6

    1998

  53. [61]

    Persson, On degenerations of algebraic surfaces

    U. Persson, On degenerations of algebraic surfaces. Memoirs of the Amer. Math. Soc. 189, 1977. 23

    1977

  54. [62]

    Popescu-Pampu, On the analytical invariance of the semigroups of a quasi-ordinary hypersurface singularity

    P. Popescu-Pampu, On the analytical invariance of the semigroups of a quasi-ordinary hypersurface singularity. Duke Math. J. 124 (2004), no. 1, 67–104. 12

    2004

  55. [63]

    Popescu-Pampu, From singularities to graphs

    P. Popescu-Pampu, From singularities to graphs. In When form becomes substance. Power of Gestures, Diagrammatical Intuition and Phenomenology of Space . L. Boi, C. Lobo eds., Birkh¨ auser, 2022, 231–263. 4

    2022

  56. [64]

    Popescu-Pampu, Tropical and logarithmic techniques for the study of Milnor fibers

    P. Popescu-Pampu, Tropical and logarithmic techniques for the study of Milnor fibers. Slides of the mini-course with the same title given at the research school Logarithmic and non-archimedean methods in singularity theory , CIRM, Marseille, 27–31 January

  57. [65]

    Popescu-Pampu, D

    P. Popescu-Pampu, D. Stepanov, Local tropicalization. In Algebraic and combinatorial aspects of tropical geometry , 253–316, Contemp. Math. 589, Amer. Math. Soc., 2013. 6, 18

    2013

  58. [66]

    Popescu-Pampu, D

    P. Popescu-Pampu, D. Stepanov, An introduction to local tropicalization . To appear in Volume 8 of the Handbook of Geometry and Topology of Singularities , Springer, 2025. 6

    2025

  59. [67]

    Portilla Cuadrado, B

    P. Portilla Cuadrado, B. Sigurdsson, The total spine of the Milnor fibration of a plane curve singularity. ArXiv:2305.12555v2. 6, 38

  60. [68]

    Rau, Real Semi-Stable Degenerations, Real-Oriented Blow-Ups, and Straightening Corners

    J. Rau, Real Semi-Stable Degenerations, Real-Oriented Blow-Ups, and Straightening Corners. Int. Math. Res. Notices, Volume 2023, Issue 18, 15896–15927. 37

    2023

  61. [69]

    Riemann, Collected papers

    B. Riemann, Collected papers. Translated by R. Baker, C. Christenson, H. Orde. Kendrick Press, 2004. 41

    2004

  62. [70]

    Riemann, Grundlagen f¨ ur eine allgemeine Theorie der Functionen einer ver¨ anderlichen complexen Gr¨ osse

    B. Riemann, Grundlagen f¨ ur eine allgemeine Theorie der Functionen einer ver¨ anderlichen complexen Gr¨ osse. Inaugural dis- sertation, G¨ ottingen, 1851. English transl.: Foundations for a general theory of functions of a complex variable . In [69], 1–41. 2

  63. [71]

    Riemann, Theorie der Abel’schen Functionen

    B. Riemann, Theorie der Abel’schen Functionen. J. reine angew. Math. 54 (1857). English transl.: The theory of abelian functions. In [69], 79–134. 2

  64. [72]

    Seade, On Milnor’s fibration theorem and its offspring after 50 years

    J. Seade, On Milnor’s fibration theorem and its offspring after 50 years. Bull. Amer. Math. Soc. 56 No. 2 (2019), 281–348. 5, 38

    2019

  65. [73]

    L. C. Siebenmann, Deformation of homeomorphisms on stratified sets. I, II . Comment. Math. Helv. 47 (1972), 123–136; ibid. 137–163. 37

    1972

  66. [74]

    D. A. Stepanov, A remark on the dual complex of a resolution of singularities. Uspekhi Mat. Nauk 61 (2006), no. 1(367), 185–186. English transl., Russian Math. Surveys 61 (2006), no. 1, 181–183. 4

    2006

  67. [75]

    Sumihiro, Equivariant completion

    H. Sumihiro, Equivariant completion. J. Math. Kyoto Univ. 14-1 (1974), 1–28. 10

    1974

  68. [76]

    Talpo, A

    M. Talpo, A. Vistoli, The Kato-Nakayama space as a transcendental root stack . Int. Math. Res. Not. IMRN 2018, no. 19, 6145–

    2018

  69. [77]

    Teissier, Vari´ et´ es toriques et polytopes.Bourbaki Seminar, Vol

    B. Teissier, Vari´ et´ es toriques et polytopes.Bourbaki Seminar, Vol. 1980/81, 71–84, Lecture Notes in Math. 901, Springer, 1981. 7

    1980

  70. [78]

    Tu, Introduction to sheaf cohomology

    L. Tu, Introduction to sheaf cohomology. arXiv:2206.07512v1. 24, 33

    arXiv

  71. [79]

    Usui, Recovery of vanishing cycles by log geometry

    S. Usui, Recovery of vanishing cycles by log geometry. Tohoku Math. J. (2) 53 (1) (2001), 1–36. 37

    2001

  72. [80]

    A. H. Wallace, Modifications and cobounding manifolds. Canad. J. Math. 12 (1960), 503–528. 2 Author’s address: P. Popescu-Pampu, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlev´ e, F-59000 Lille, France. Email address: patrick.popescu-pampu@univ-lille.fr

    1960

  73. [1973]

    7, 10, 12, 13, 14, 19, 20

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