Nondegenerate locally tame complete intersection varieties and geometry of nonisolated hypersurface singularities
Authors:
Christophe Eyral and Mutsuo Oka
Journal:
J. Algebraic Geom. 31 (2022), 561-591
DOI:
https://doi.org/10.1090/jag/784
Published electronically:
May 5, 2022
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Abstract |
References |
Additional Information
Abstract: We give a criterion to test geometric properties such as Whitney equisingularity and Thom’s $a_f$ condition for new families of (possibly nonisolated) hypersurface singularities that “behave well” with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.
References
- Joël Briançon, Le théorème de Kouchnirenko, unpublished lecture notes.
- Joël Briançon, Philippe Maisonobe, and Michel Merle, Localisation de systèmes différentiels, stratifications de Whitney et condition de Thom, Invent. Math. 117 (1994), no. 3, 531–550 (French). MR 1283729, DOI 10.1007/BF01232255
- Christophe Eyral and Mutsuo Oka, Non-compact Newton boundary and Whitney equisingularity for non-isolated singularities, Adv. Math. 316 (2017), 94–113. MR 3672903, DOI 10.1016/j.aim.2017.06.003
- Christophe Eyral and Mutsuo Oka, Whitney regularity and Thom condition for families of non-isolated mixed singularities, J. Math. Soc. Japan 70 (2018), no. 4, 1305–1336. MR 3868208, DOI 10.2969/jmsj/77437743
- Christopher G. Gibson, Klaus Wirthmüller, Andrew A. du Plessis, and Eduard J. N. Looijenga, Topological stability of smooth mappings, Lecture Notes in Mathematics, Vol. 552, Springer-Verlag, Berlin-New York, 1976. MR 0436203, DOI 10.1007/BFb0095244
- Helmut Hamm, Lokale topologische Eigenschaften komplexer Räume, Math. Ann. 191 (1971), 235–252 (German). MR 286143, DOI 10.1007/BF01578709
- Helmut A. Hamm and Lê Dũng Tráng, Un théorème de Zariski du type de Lefschetz, Ann. Sci. École Norm. Sup. (4) 6 (1973), 317–355. MR 401755, DOI 10.24033/asens.1250
- A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), no. 1, 1–31 (French). MR 419433, DOI 10.1007/BF01389769
- John Mather, Notes on topological stability, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 4, 475–506. MR 2958928, DOI 10.1090/S0273-0979-2012-01383-6
- John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
- Mutsuo Oka, On the topology of the Newton boundary. III, J. Math. Soc. Japan 34 (1982), no. 3, 541–549. MR 659622, DOI 10.2969/jmsj/03430541
- Mutsuo Oka, Canonical stratification of nondegenerate complete intersection varieties, J. Math. Soc. Japan 42 (1990), no. 3, 397–422. MR 1056828, DOI 10.2969/jmsj/04230397
- Mutsuo Oka, Non-degenerate complete intersection singularity, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1997. MR 1483897
- Mutsuo Oka, On Milnor fibrations of mixed functions, $a_f$-condition and boundary stability, Kodai Math. J. 38 (2015), no. 3, 581–603. MR 3417523, DOI 10.2996/kmj/1446210596
- Mutsuo Oka, On the Milnor fibration for $f(z)\overline g(z)$, Eur. J. Math. 6 (2020), no. 3, 998–1019. MR 4151727, DOI 10.1007/s40879-019-00380-1
- Mutsuo Oka, On the Milnor fibration for $f(z)\overline {g}(z)$ II, J. Math. Soc. Japan 73 (2021), no. 2, 649–669. MR 4255080, DOI 10.2969/jmsj/83328332
- Adam Parusiński, Limits of tangent spaces to fibres and the $w_f$ condition, Duke Math. J. 72 (1993), no. 1, 99–108. MR 1242881, DOI 10.1215/S0012-7094-93-07205-5
- R. Thom, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969), 240–284 (French). MR 239613, DOI 10.1090/S0002-9904-1969-12138-5
- Hassler Whitney, Tangents to an analytic variety, Ann. of Math. (2) 81 (1965), 496–549. MR 192520, DOI 10.2307/1970400
- Oscar Zariski, Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481–491. MR 277533, DOI 10.1090/S0002-9904-1971-12729-5
References
- Joël Briançon, Le théorème de Kouchnirenko, unpublished lecture notes.
- Joël Briançon, Philippe Maisonobe, and Michel Merle, Localisation de systèmes différentiels, stratifications de Whitney et condition de Thom, Invent. Math. 117 (1994), no. 3, 531–550 (French). MR 1283729, DOI 10.1007/BF01232255
- Christophe Eyral and Mutsuo Oka, Noncompact Newton boundary and Whitney equisingularity for nonisolated singularities, Adv. Math. 316 (2017), 94–113. MR 3672903, DOI 10.1016/j.aim.2017.06.003
- Christophe Eyral and Mutsuo Oka, Whitney regularity and Thom condition for families of nonisolated mixed singularities, J. Math. Soc. Japan 70 (2018), no. 4, 1305–1336. MR 3868208, DOI 10.2969/jmsj/77437743
- Christopher G. Gibson, Klaus Wirthmüller, Andrew A. du Plessis, and Eduard J. N. Looijenga, Topological stability of smooth mappings, Lecture Notes in Mathematics, Vol. 552, Springer-Verlag, Berlin-New York, 1976. MR 0436203
- Helmut A. Hamm, Lokale topologische Eigenschaften komplexer Räume, Math. Ann. 191 (1971), 235–252 (German). MR 286143, DOI 10.1007/BF01578709
- Helmut A. Hamm and Lê Dũng Tráng, Un théorème de Zariski du type de Lefschetz, Ann. Sci. École Norm. Sup. (4) 6 (1973), 317–355. MR 401755
- A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), no. 1, 1–31 (French). MR 419433, DOI 10.1007/BF01389769
- John Mather, Notes on topological stability, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 4, 475–506. MR 2958928, DOI 10.1090/S0273-0979-2012-01383-6
- John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
- Mutsuo Oka, On the topology of the Newton boundary, III, J. Math. Soc. Japan 34 (1982), no. 3, 541–549. MR 659622, DOI 10.2969/jmsj/03430541
- Mutsuo Oka, Canonical stratification of nondegenerate complete intersection varieties, J. Math. Soc. Japan 42 (1990), no. 3, 397–422. MR 1056828, DOI 10.2969/jmsj/04230397
- Mutsuo Oka, Nondegenerate complete intersection singularity, Actualités Mathématiques [Current Mathematical Topics], Hermann, Paris, 1997. MR 1483897
- Mutsuo Oka, On Milnor fibrations of mixed functions, $a_f$-condition and boundary stability, Kodai Math. J. 38 (2015), no. 3, 581–603. MR 3417523, DOI 10.2996/kmj/1446210596
- Mutsuo Oka, On the Milnor fibration for $f(\mathbf {z})\overline g(\mathbf {z})$, Eur. J. Math. 6 (2020), no. 3, 998–1019. MR 4151727, DOI 10.1007/s40879-019-00380-1
- Mutsuo Oka, On the Milnor fibration for $f(z)\bar g (z)$ II, J. Math. Soc. Japan 73 (2021), no. 2, 649–669. MR 4255080, DOI 10.2969/jmsj/83328332
- Adam Parusiński, Limits of tangent spaces to fibres and the $w_f$ condition, Duke Math. J. 72 (1993), no. 1, 99–108. MR 1242881, DOI 10.1215/S0012-7094-93-07205-5
- René Thom, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969), 240–284 (French). MR 239613, DOI 10.1090/S0002-9904-1969-12138-5
- Hassler Whitney, Tangents to an analytic variety, Ann. of Math. (2) 81 (1965), 496–549. MR 192520, DOI 10.2307/1970400
- Oscar Zariski, Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481–491. MR 277533, DOI 10.1090/S0002-9904-1971-12729-5
Additional Information
Christophe Eyral
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland
MR Author ID:
621468
ORCID:
0000-0003-4729-3843
Email:
cheyral@impan.pl
Mutsuo Oka
Affiliation:
3-19-8 Nakaochiai, Shinjuku-ku, Tokyo 161-0032, Japan
MR Author ID:
193915
Email:
okamutsuo@gmail.com
Received by editor(s):
August 25, 2020
Received by editor(s) in revised form:
February 3, 2021
Published electronically:
May 5, 2022
Article copyright:
© Copyright 2022
University Press, Inc.
