Abstract
This article is concerned with the geometry of germs of real analytic surfaces in \(({\mathbb {C}}^2,0)\) having an isolated Cauchy–Riemann (CR) singularity at the origin. These are perturbations of Bishop quadrics. There are two kinds of CR singularities stable under perturbation: elliptic and hyperbolic. Elliptic case was studied by Moser–Webster (Acta Math 150(3–4), 255–296, 1983) who showed that such a surface is locally, near the CR singularity, holomorphically equivalent to normal form from which lots of geometric features can be read off. In this article we focus on perturbations of hyperbolic quadrics. As was shown by Moser and Webster (1983), such a surface can be transformed to a formal normal form by a formal change of coordinates that may not be holomorphic in any neighborhood of the origin. Given a non-degenerate real analytic surface M in \(({\mathbb {C}}^2,0)\) having a hyperbolic CR singularity at the origin, we prove the existence of a non-constant Whitney smooth family of connected holomorphic curves intersecting M along holomorphic hyperbolas. This is the very first result concerning hyperbolic CR singularity not equivalent to quadrics. This is a consequence of a non-standard KAM-like theorem for pair of germs of holomorphic involutions \(\{\tau _1,\tau _2\}\) at the origin, a common fixed point. We show that such a pair has large amount of invariant analytic sets biholomorphic to \(\{z_1z_2=const\}\) (which is not a torus) in a neighborhood of the origin, and that they are conjugate to restrictions of linear maps on such invariant sets.
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Notes
Through the paper, for any \({{\mathcal {S}}}\subset {{\mathbb {R}}}\), \(|{{\mathcal {S}}}|\) denotes its Lebesgue measure.
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Acknowledgements
The authors thank X. Gong for his interest, stimulating discussions that help them to substantially improve the exposition of the main results. The authors would also like to thank L. Lempert and M. Procesi for their interests and sharp comments which help them to improve the text. The authors thank also the anonymous referees for the careful review and helpful suggestions.
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Communicated by Ngaiming Mok.
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This work has been supported by the French government through the ANR grant ANR-15-CE40-0001-03 for the project Bekam and through the UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01.
Appendix A: Proof of Lemma 6.11
Appendix A: Proof of Lemma 6.11
Let \(\varsigma :=\max \{\Vert f_1-f_2\Vert _{{{\mathcal {O}}},\beta '',r''},\ \Vert g_1-g_2\Vert _{{{\mathcal {O}}},\beta '',r''}\}\), which is, by (98), smaller than \(\frac{\beta '^2}{16}\). Let \(\omega \in {{{\mathcal {O}}}}(r'', \beta '')\). In order to estimate its norm, let us first decompose the following expression:
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Terms in (204)
Expanding \(h_{l,0}\) around \(\xi \eta +e^{-{\mathrm{i}}b\alpha }\eta f_2+e^{{\mathrm{i}}b\alpha }\xi g_2+f_2 g_2\) in (204), we obtain:
Combining Corollary 6.5 together with Remark 6.6 and (98), we obtain, for \((\xi ,\eta )\in {{\mathcal {C}}}^{r''}_{\omega ,\beta ''}\) and \(m=1,2\):
By Cauchy’s inequality, we have, for all \( (\xi ,\eta )\in {{\mathcal {C}}}_{\omega ,\beta ''}\) and \({{\tilde{k}}}\ge 1\),
We recall that, for \(|z|<1\),
Hence, developing \(h^{(k)}_{l,0}\) around \(\xi \eta \), we have, for \(k\ge 1\),
Recalling that \(|\omega |<r''^2-\beta ''\), by Lemma 3.3, we have, for \(-1\le b\le 1\),
Since \(\frac{\varsigma }{\beta '}<\frac{\beta '}{8}\) and according to (85), we have \(1-\frac{\beta '}{8}>\frac{99}{100}\) and \(1-\frac{26\varsigma }{25\beta '}\left( 1-\frac{\beta '}{8}\right) ^{-1}>\frac{99}{100}\), so that \(\frac{26}{25}\left( 1-\frac{\beta '}{8}\right) ^{-2}\left( 1-\frac{26\varsigma }{25\beta '}\left( 1-\frac{\beta '}{8}\right) ^{-1}\right) ^{-1}<\frac{27}{25}.\) Combining (209) and (210), together with (19), we obtain
Hence, according to Lemma 3.3, for \(l\ge 0\),
On the other hand, (95) implies \(2r''{{\tilde{\beta }}} <\frac{r'-r''}{8}\). Indeed, since \(0<r''<r'<\frac{1}{4}\) and \(0<\beta <1\), then \(8\beta ^{\frac{1}{2}}<(r'-r'')r''\) implies
Therefore, according to (86), we have
As a consequence, we have
Thus, under (95), the \(\Vert \cdot \Vert _{{{\mathcal {O}}},\beta '',r''}\)-norm of (204) is bounded by
To show (204)\(\in {{{\mathcal {A}}}}_{\beta '', r''}\left( {{{\mathcal {O}}}}\right) \) provided that \(f_1,f_2,g_1,g_2\in {{{\mathcal {A}}}}_{\beta '', r''}\left( {{{\mathcal {O}}}}\right) \), it remains to verify the analyticity on \({{{\mathcal {O}}}}(r'',\beta '') \) for the coefficients (204)\(_{l,j}\), \(l,j\ge 0\), \(lj=0\). According to (207), (209) and (210), for \({{\tilde{l}}}\ge 0\),
and, by (23) and (211), for \(\tilde{l}\ge 0\), for \(l,j\ge 0\) with \(lj=0\),
Note that, for \(l\ge 0\),
and for \(j\ge 1\),
where, by Lemma 6.5 and by (23), for \(l,j\ge 0\) with \(lj=0\),
Then we see that, for \(\omega \in {{{\mathcal {O}}}}(r'',\beta '') \),
which implies the analyticity of (204)\(_{l,0}\) under (95), and it is similar for that of (204)\(_{0,j}\). Hence, (204)\(\in {{{\mathcal {A}}}}_{\beta '', r''}\left( {{{\mathcal {O}}}}\right) \).
-
Terms in (205)
Note that (19) and (208) imply that, for \(\omega \in {{{\mathcal {O}}}}(r'',\beta '')\), for \(l\ge 0\),
In (205), we have, for \(l\ge 1\),
Furthermore, by (95), we have \(\beta ^{\frac{5}{4}}<\beta ^{\frac{1}{2}}<\frac{r'-r''}{32}\). Recalling that \(\varsigma <\frac{\beta '^2}{8}\le \frac{(16\beta ^\frac{5}{4})^2}{8}\) and using (212), we have
Therefore, the \(\Vert \cdot \Vert _{{{\mathcal {O}}},\beta '', r''}\)-norm of (205) is bounded by
For \(l\ge 0\), we have that (205)\(_{l,0}\) equals to
and for \(j\ge 1\), (205)\(_{0,j}\) equals to
If \(f_1,f_2,g_1,g_2\in {{{\mathcal {A}}}}_{\beta '', r''}\left( {{{\mathcal {O}}}}\right) \), then by (214)–(216) and (19), we see the analyticity of (205)\(_{l,0}\), since for \(\omega \in {{{\mathcal {O}}}}(r'',\beta '') \),
The proof for (205)\(_{0,j}\) is similar, hence (205)\(\in {{{\mathcal {A}}}}_{\beta '',r''}\left( {{{\mathcal {O}}}}\right) \).
Combining (213), (217) and similar estimates obtained for expressions (206), we obtain
since (95) implies that \(\beta '\le {{\tilde{\beta }}}<\frac{r'-r''}{64}\). This finishes the proof of Lemma 6.11. \(\square \)
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Stolovitch, L., Zhao, Z. Geometry of hyperbolic Cauchy–Riemann singularities and KAM-like theory for holomorphic involutions. Math. Ann. 386, 587–672 (2023). https://doi.org/10.1007/s00208-022-02408-6
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DOI: https://doi.org/10.1007/s00208-022-02408-6