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Correction to: On the Relationship Between D’Angelo q-Type and Catlin q-Type

  • Vasile Brinzanescu
  • Andreea C. NicoaraEmail author
Correction
  • 51 Downloads

Keywords

Orders of contact D’Angelo finite q-type Catlin finite q-type Finite type domains in \({\mathbb {C}}^n\) pseudoconvexity 

Mathematics Subject Classification

Primary 32F18 32T25 Secondary 32V35 13H15 

1 Correction to: J Geom Anal (2015) 25:1701–1719,  https://doi.org/10.1007/s12220-014-9490-5

John D’Angelo brought to the authors’ attention the following counterexample to the claim in Corollary 2.11 of [1] that for any ideal \(\mathcal {I}\) of holomorphic germs the infimum in the definition of the D’Angelo q-type is achieved and equal to the generic value: \(\mathcal {I}=(z_1^3-z_2 z_3, z_2^2) \subset \mathcal {O}_0\) [2]. This example is due to John D’Angelo and Martino Fassina; see also [3]. Here the D’Angelo 1-type \(\Delta _1(\mathcal {I},0)=+\infty ,\) the D’Angelo 2-type
$$\begin{aligned} \Delta _2(\mathcal {I},0)= \inf _{\{a_1,a_2,a_3 \}} \,\, \Delta _1 \left( (z_1^3-z_2 z_3, z_2^2,a_1 z_1+a_2 z_2 + a_3 z_3), 0\right) = 3, \end{aligned}$$
whereas
$$\begin{aligned} \hbox {g}en.val_{\{a_1,a_2,a_3 \}} \,\, \Delta _1 \left( (z_1^3-z_2 z_3, z_2^2,a_1 z_1+a_2 z_2 + a_3 z_3), 0\right) = 4. \end{aligned}$$
The corresponding claim for real hypersurfaces found in Proposition 2.10 of [1] is likewise invalidated by writing the sum of squares hypersurface corresponding to \(\mathcal {I}\) in \({\mathbb {C}}^4\) given by \(r=Re\{z_4\}+ ||z_1^3-z_2 z_3||^2 + ||z_2^2||^2.\)

As the results in [1] are proven with respect to the generic value, we introduce the following definition:

Definition 0.1

Let \(2 \le q \le n.\) If \(\mathcal {I}\) is an ideal in \(\mathcal {O}_{x_0},\)
$$\begin{aligned} \tilde{\Delta }_q(\mathcal {I}, x_0)=\hbox {g}en.val_{\{w_1, \dots , w_{q-1} \}} \,\,\ \Delta _1\Big ( (\mathcal {I}, w_1, \dots , w_{q-1}), x_0 \Big ), \end{aligned}$$
where the generic value is taken over all non-degenerate sets \(\{w_1, \dots , w_{q-1} \}\) of linear forms in \(\mathcal {O}_{x_0},\)\((\mathcal {I}, w_1, \dots , w_{q-1})\) is the ideal in \(\mathcal {O}_{x_0}\) generated by \(\mathcal {I}, w_1, \dots , w_{q-1},\) and \(\Delta _1\) is the D’Angelo 1-type. Likewise, if M is a real hypersurface in \({\mathbb {C}}^n\) and \(x_0 \in M,\)
$$\begin{aligned} \tilde{\Delta }_q (M, x_0) =\hbox {g}en.val_{\{w_1, \dots , w_{q-1} \}} \,\,\ \Delta _1\Big ( (\mathcal {I}(M), w_1, \dots , w_{q-1}), x_0 \Big ), \end{aligned}$$
where \((\mathcal {I}(M), w_1, \dots , w_{q-1})\) is the ideal in \(C^\infty _{x_0}\) generated by all smooth functions \(\mathcal {I}(M)\) vanishing on M along with \(w_1, \dots , w_{q-1}.\)

By substituting the D’Angelo q-type \(\Delta _q\) by \(\tilde{\Delta }_q,\) all the results and proofs [1] remain true.

Notes

References

  1. 1.
    Brinzanescu, V., Nicoara, A.C.: On the relationship between D’Angelo \(q\)-type and Catlin \(q\)-type. J. Geom. Anal. 25(3), 1701–1719 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    D’Angelo, J.P.: Private communicationGoogle Scholar
  3. 3.
    Fassina, M.: A remark on two notions of order of contact. J. Geom. Anal. 29(1), 707–716 (2019)MathSciNetCrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit 3BucharestRomania
  2. 2.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA

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