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Corrigendum to “Thom irregularity and Milnor tube fibrations” [Bull. Sci. Math. 143 (2018) 58–72]
Bull. Sci. math. 153 (2019) 120–123
Contents lists available at ScienceDirect
Bulletin des Sciences Mathématiques www.elsevier.com/locate/bulsci
Corrigendum
Corrigendum to “Thom irregularity and Milnor tube fibrations” [Bull. Sci. Math. 143 (2018) 58–72] A.J. Parameswaran a,∗ , Mihai Tibăr b,∗ a
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India b Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, F-59000 Lille, France
a r t i c l e
i n f o
Article history: Received 4 April 2019 Available online 30 April 2019
a b s t r a c t This note corrects a computation in [3, Lemma 2.5] reformulating the criterion for “Disc f g¯ ⊂ {0}” in terms of Disc(f, g). © 2019 Elsevier Masson SAS. All rights reserved.
MSC: 14D06 14P15 32S20 58K05 57R45 58K15 Keywords: Singularities of real polynomial maps Fibrations Bifurcation locus
Let f, g : (C n , 0) → (C, 0) be holomorphic function germs. We have studied in [3] the Thom regularity of the real map germ f g¯ : (C n , 0) → (C, 0) in relation to the Thom
DOI of original article: https://doi.org/10.1016/j.bulsci.2017.12.001.
* Corresponding authors. E-mail addresses: [email protected] (A.J. Parameswaran), [email protected] (M. Tibăr). https://doi.org/10.1016/j.bulsci.2019.04.002 0007-4497/© 2019 Elsevier Masson SAS. All rights reserved.
A.J. Parameswaran, M. Tibăr / Bull. Sci. math. 153 (2019) 120–123
121
regularity of the holomorphic map germ (f, g) : (C n , 0) → (C 2 , 0). These maps have singular loci Sing f g¯ and Sing (f, g), and discriminants Disc f g¯ and Disc(f, g) defined as the images of the singular loci by the respective maps.1 The first result of [3] is a criterion for the condition “Disc f g¯ ⊂ {0}” in terms of Disc(f, g) produced by [3, Lemma 2.5], and it needs to be revised due to a computation failure which happened in the second part of the proof of loc.cit. We thank Mutsuo Oka for bringing this problem to our attention. This corrigendum rectifies the computation and provides the adjusted formulation of our criterion. Briefly, instead of “Disc f g¯ ⊂ {0} if and only if Disc(f, g) contains no lines different from the axes” one should read: “Disc f g¯ ⊂ {0} if and only if Disc(f, g) contains only curve components which are tangent to the coordinate axes.” According to this adjusted criterion, we need to reformulate three other statements: Theorems 2.3 and 3.1, and Corollary 4.1. Their proofs remain the same, as we detail in the following. Subsection 4.1 of [3], which contains a discussion about the genericity of the condition “Disc f g¯ ⊂ {0}”, needs to be removed completely since it has no more object after the modification of the above criterion; more precisely, the class of maps which satisfies the modified criterion is far from being generic. We recall our strategy based on the natural relation of f g¯ with the holomorphic map (f,g)
u¯ v
(f, g), as follows. The map f g¯ decomposes as C n → C 2 → C. We have proved in [3] that Disc f g¯ is the union over all irreducible complex curve components C ⊂ Disc(f, g) of the images by u¯ v of the singular loci of the restrictions (u¯ v )|C . Lemma 2.5 of [3] considers such a restriction, and here is its reformulation together with its corrected proof. Lemma 2.5. Let C be an irreducible complex curve component of the germ (Disc(f, g), 0) ⊂ v )|C\{0} is a submersion if and only if C is tangent to (C 2 , 0). Then the restriction (u¯ one of the coordinate axes without coinciding with it. Proof. We continue to work with set germs at 0. In case the irreducible curve germ C is one of the coordinate axes, say {u = 0} (and note that {u = 0} ⊂ Disc(f, g) if and only if Sing f \{g = 0} = ∅), the image u¯ v ({u = 0}) v )|C\{0} is totally singular. is the origin and the restriction (u¯ Let the irreducible curve C be different from the coordinate axes, and let u = tp , v = a1 tq + h.o.t., be a Puiseux expansion of C, with a1 = 0. We may assume without loss of generality that p ≤ q since the singular locus of u¯ v coincides to that of v¯ u, thus we may interchange the coordinates u and v. The equality which defines the singular locus of the function u(t)¯ v(t) is vdu = λudv, where |λ| = 1, and note that λ depends on t. Taking the modulus both sides gives 1
These discriminants are well-defined as set-germs, see [1,2].
122
A.J. Parameswaran, M. Tibăr / Bull. Sci. math. 153 (2019) 120–123
|vdu| = |udv|, and thus |pa1 tp+q−1 + h.o.t.| = |qa1 tp+q−1 + h.o.t.|. Dividing out by |a1 ||t|p+q−1 , and taking the limit for t → 0, one gets the equality p = q. This implies that if C is tangent to the coordinate axes, without coinciding with one of them, then the restriction (u¯ v )|C\{0} is a submersion. The change with respect to the proof in [3] starts from now: we prove that, reciprocally, if p = q then the equation vdu = λudv has solutions, as follows. If v = a1 tp is a one-term expansion, then all points t are solutions of this equation, and with λ ≡ 1; consequently the corresponding critical value set contributes with a real half-line in Disc f g¯, since C is a line different from the coordinate axes by our assumption a1 = 0. If the Puiseux expansion v = a1 tp + a2 tp+j + h.o.t. has at least 2 terms, where j ≥ 1, then the equivalent equation |vdu| = |udv| reduces, after dividing out by p|a1 ||t|2p−1 , to an equation of the form |1 + btj + h.o.t.| = |1 + ctj + h.o.t.|, with b, c = 0 and b = c. By inverting one of the holomorphic series, we obtain the equation |1 + αtj + h.o.t.| = 1 for some α = 0, which has solutions for all small enough t. More precisely, there is at least one2 real semi-analytic arc γ(s) parametrised by s ∈ [0, ε[, for some small enough ε > 0, which is solution of this equation. Then its image (u¯ v )(γ) is included in the discriminant Disc f g¯. Since by hypothesis neither u nor v are constant along this arc, it follows that this image is not reduced to the point 0, hence it must be a non-trivial continuous real arc. This ends the proof of our statement. 2 Theorem 2.3. Let f, g : (C n , 0) → (C, 0), n > 1, be some non-constant holomorphic function germs. Then the discriminant Disc f g¯ of the mixed function germ f g¯ : (C n , 0) → (C, 0) is either {0} or a union of finitely many real semi-analytic arcs at the origin. Moreover, Disc f g¯ ⊂ {0} if and only if the discriminant Disc(f, g) contains only curve components which are tangent to the coordinate axes. Proof. The first assertion of Theorem 2.3 is a consequence of the proof of Lemma 2.5 together with Łojasiewicz’ result that the image by an analytic map of a real semi-analytic arc is a semi-analytic arc. The second assertion also follows from the same Lemma 2.5 since it gives the precise characterisation of the condition Disc f g¯ ⊂ {0} in terms of Disc(f, g). 2 The hypothesis of our main result [3, Theorem 3.1] is equivalent to “Disc f g¯ ⊂ {0}”, so its statement must be also reformulated now according to the rectified criterion stated in the above Theorem 2.3, as follows: Theorem 3.1. Let f, g : (C n , 0) → (C, 0) be any holomorphic germs such that Disc(f, g) contains only curves tangent to the coordinate axes. Then 2
Actually an even number of semi-analytic arcs.
A.J. Parameswaran, M. Tibăr / Bull. Sci. math. 153 (2019) 120–123
N Tf g¯ ⊂ N T(f,g) .
123
2
The proof of Theorem 3.1 in [3] remains unchanged since it already starts with the assumption Disc f g¯ ⊂ {0} which is equivalent to the hypothesis in the above statement. By the same reason, the proof of the following adjustment of the statement of Corollary 4.1 remains unchanged too: Corollary 4.1. Assume that Disc(f, g) contains only curves tangent to the coordinate axes. If the map (f, g) is Thom regular (in particular, if (f, g) is an ICIS), then f g¯ is Thom regular. 2 Conflict of interest statement No conflict of interest. References [1] R.N. Araújo dos Santos, M. Ribeiro, M. Tibăr, Fibrations of highly singular map germs, arXiv: 1711.07544. [2] C. Joiţa, M. Tibăr, Images of analytic map germs, arXiv:1810.05158. [3] A.J. Parameswaran, M. Tibăr, Thom irregularity and Milnor tube fibrations, Bull. Sci. Math. 143 (2018) 58–72.
Contents lists available at ScienceDirect
Bulletin des Sciences Mathématiques www.elsevier.com/locate/bulsci
Corrigendum
Corrigendum to “Thom irregularity and Milnor tube fibrations” [Bull. Sci. Math. 143 (2018) 58–72] A.J. Parameswaran a,∗ , Mihai Tibăr b,∗ a
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India b Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, F-59000 Lille, France
a r t i c l e
i n f o
Article history: Received 4 April 2019 Available online 30 April 2019
a b s t r a c t This note corrects a computation in [3, Lemma 2.5] reformulating the criterion for “Disc f g¯ ⊂ {0}” in terms of Disc(f, g). © 2019 Elsevier Masson SAS. All rights reserved.
MSC: 14D06 14P15 32S20 58K05 57R45 58K15 Keywords: Singularities of real polynomial maps Fibrations Bifurcation locus
Let f, g : (C n , 0) → (C, 0) be holomorphic function germs. We have studied in [3] the Thom regularity of the real map germ f g¯ : (C n , 0) → (C, 0) in relation to the Thom
DOI of original article: https://doi.org/10.1016/j.bulsci.2017.12.001.
* Corresponding authors. E-mail addresses: [email protected] (A.J. Parameswaran), [email protected] (M. Tibăr). https://doi.org/10.1016/j.bulsci.2019.04.002 0007-4497/© 2019 Elsevier Masson SAS. All rights reserved.
A.J. Parameswaran, M. Tibăr / Bull. Sci. math. 153 (2019) 120–123
121
regularity of the holomorphic map germ (f, g) : (C n , 0) → (C 2 , 0). These maps have singular loci Sing f g¯ and Sing (f, g), and discriminants Disc f g¯ and Disc(f, g) defined as the images of the singular loci by the respective maps.1 The first result of [3] is a criterion for the condition “Disc f g¯ ⊂ {0}” in terms of Disc(f, g) produced by [3, Lemma 2.5], and it needs to be revised due to a computation failure which happened in the second part of the proof of loc.cit. We thank Mutsuo Oka for bringing this problem to our attention. This corrigendum rectifies the computation and provides the adjusted formulation of our criterion. Briefly, instead of “Disc f g¯ ⊂ {0} if and only if Disc(f, g) contains no lines different from the axes” one should read: “Disc f g¯ ⊂ {0} if and only if Disc(f, g) contains only curve components which are tangent to the coordinate axes.” According to this adjusted criterion, we need to reformulate three other statements: Theorems 2.3 and 3.1, and Corollary 4.1. Their proofs remain the same, as we detail in the following. Subsection 4.1 of [3], which contains a discussion about the genericity of the condition “Disc f g¯ ⊂ {0}”, needs to be removed completely since it has no more object after the modification of the above criterion; more precisely, the class of maps which satisfies the modified criterion is far from being generic. We recall our strategy based on the natural relation of f g¯ with the holomorphic map (f,g)
u¯ v
(f, g), as follows. The map f g¯ decomposes as C n → C 2 → C. We have proved in [3] that Disc f g¯ is the union over all irreducible complex curve components C ⊂ Disc(f, g) of the images by u¯ v of the singular loci of the restrictions (u¯ v )|C . Lemma 2.5 of [3] considers such a restriction, and here is its reformulation together with its corrected proof. Lemma 2.5. Let C be an irreducible complex curve component of the germ (Disc(f, g), 0) ⊂ v )|C\{0} is a submersion if and only if C is tangent to (C 2 , 0). Then the restriction (u¯ one of the coordinate axes without coinciding with it. Proof. We continue to work with set germs at 0. In case the irreducible curve germ C is one of the coordinate axes, say {u = 0} (and note that {u = 0} ⊂ Disc(f, g) if and only if Sing f \{g = 0} = ∅), the image u¯ v ({u = 0}) v )|C\{0} is totally singular. is the origin and the restriction (u¯ Let the irreducible curve C be different from the coordinate axes, and let u = tp , v = a1 tq + h.o.t., be a Puiseux expansion of C, with a1 = 0. We may assume without loss of generality that p ≤ q since the singular locus of u¯ v coincides to that of v¯ u, thus we may interchange the coordinates u and v. The equality which defines the singular locus of the function u(t)¯ v(t) is vdu = λudv, where |λ| = 1, and note that λ depends on t. Taking the modulus both sides gives 1
These discriminants are well-defined as set-germs, see [1,2].
122
A.J. Parameswaran, M. Tibăr / Bull. Sci. math. 153 (2019) 120–123
|vdu| = |udv|, and thus |pa1 tp+q−1 + h.o.t.| = |qa1 tp+q−1 + h.o.t.|. Dividing out by |a1 ||t|p+q−1 , and taking the limit for t → 0, one gets the equality p = q. This implies that if C is tangent to the coordinate axes, without coinciding with one of them, then the restriction (u¯ v )|C\{0} is a submersion. The change with respect to the proof in [3] starts from now: we prove that, reciprocally, if p = q then the equation vdu = λudv has solutions, as follows. If v = a1 tp is a one-term expansion, then all points t are solutions of this equation, and with λ ≡ 1; consequently the corresponding critical value set contributes with a real half-line in Disc f g¯, since C is a line different from the coordinate axes by our assumption a1 = 0. If the Puiseux expansion v = a1 tp + a2 tp+j + h.o.t. has at least 2 terms, where j ≥ 1, then the equivalent equation |vdu| = |udv| reduces, after dividing out by p|a1 ||t|2p−1 , to an equation of the form |1 + btj + h.o.t.| = |1 + ctj + h.o.t.|, with b, c = 0 and b = c. By inverting one of the holomorphic series, we obtain the equation |1 + αtj + h.o.t.| = 1 for some α = 0, which has solutions for all small enough t. More precisely, there is at least one2 real semi-analytic arc γ(s) parametrised by s ∈ [0, ε[, for some small enough ε > 0, which is solution of this equation. Then its image (u¯ v )(γ) is included in the discriminant Disc f g¯. Since by hypothesis neither u nor v are constant along this arc, it follows that this image is not reduced to the point 0, hence it must be a non-trivial continuous real arc. This ends the proof of our statement. 2 Theorem 2.3. Let f, g : (C n , 0) → (C, 0), n > 1, be some non-constant holomorphic function germs. Then the discriminant Disc f g¯ of the mixed function germ f g¯ : (C n , 0) → (C, 0) is either {0} or a union of finitely many real semi-analytic arcs at the origin. Moreover, Disc f g¯ ⊂ {0} if and only if the discriminant Disc(f, g) contains only curve components which are tangent to the coordinate axes. Proof. The first assertion of Theorem 2.3 is a consequence of the proof of Lemma 2.5 together with Łojasiewicz’ result that the image by an analytic map of a real semi-analytic arc is a semi-analytic arc. The second assertion also follows from the same Lemma 2.5 since it gives the precise characterisation of the condition Disc f g¯ ⊂ {0} in terms of Disc(f, g). 2 The hypothesis of our main result [3, Theorem 3.1] is equivalent to “Disc f g¯ ⊂ {0}”, so its statement must be also reformulated now according to the rectified criterion stated in the above Theorem 2.3, as follows: Theorem 3.1. Let f, g : (C n , 0) → (C, 0) be any holomorphic germs such that Disc(f, g) contains only curves tangent to the coordinate axes. Then 2
Actually an even number of semi-analytic arcs.
A.J. Parameswaran, M. Tibăr / Bull. Sci. math. 153 (2019) 120–123
N Tf g¯ ⊂ N T(f,g) .
123
2
The proof of Theorem 3.1 in [3] remains unchanged since it already starts with the assumption Disc f g¯ ⊂ {0} which is equivalent to the hypothesis in the above statement. By the same reason, the proof of the following adjustment of the statement of Corollary 4.1 remains unchanged too: Corollary 4.1. Assume that Disc(f, g) contains only curves tangent to the coordinate axes. If the map (f, g) is Thom regular (in particular, if (f, g) is an ICIS), then f g¯ is Thom regular. 2 Conflict of interest statement No conflict of interest. References [1] R.N. Araújo dos Santos, M. Ribeiro, M. Tibăr, Fibrations of highly singular map germs, arXiv: 1711.07544. [2] C. Joiţa, M. Tibăr, Images of analytic map germs, arXiv:1810.05158. [3] A.J. Parameswaran, M. Tibăr, Thom irregularity and Milnor tube fibrations, Bull. Sci. Math. 143 (2018) 58–72.