Certain real surfaces in $\mathbb{C}^2$ with isolated singularities
[Sur certaines surfaces réelles dans $\mathbb{C}^2$ avec des singularités isolées]
Gorai, Sushil1
1Department of Mathematics and Statistics, Indian Institute of Science Education and Research Kolkata, Mohanpur, Nadia, West Bengal 741246 (India)
Annales de l'Institut Fourier, Online first, 45 p.

The surfaces in $\mathbb{C}^2$ with an isolated CR singularity at the origin and with cubic lowest degree homogeneous term in its graph near the origin, under certain geometric condition, can be reduced — up to biholomorphism of $\mathbb{C}^2$ — to a one-parameter family of the form

\[ M_t := \biggl \lbrace (z,w)\in \mathbb{C}^2: w=z^2\overline{z}+tz\overline{z}^2+\frac{t^2}{3} \overline{z}^3+o\bigl ({\vert z \vert ^3}\bigr )\biggr \rbrace , \quad t\in (0,\infty ), \]

near the origin. We prove that $M_t$ is not locally polynomially convex if $t<1$. The local hull contains a ball centred at the origin if $t<\sqrt{3}/2$. We also prove that $M_t$ is locally polynomially convex for $t\ge \sqrt{3/2}$. We show that, for $\sqrt{3}/2\le t<1$, the local hull of $M_t$ contains a one-parameter family of analytic discs passing through the origin. We also prove that, if we remove the higher order terms from the graphing function of $M_t$, it is locally polynomially convex for $t\ge \frac{\sqrt{15-\sqrt{33}}}{2\sqrt{2}}$. Some new results about the local polynomial convexity of the union of three totally-real planes are also reported.

Les surfaces dans $\mathbb{C}^2$ avec une singularité CR isolée à l’origine et avec un terme homogène de plus bas degré cubique dans son graphe près de l’origine, sous certaines conditions géométriques, peuvent être réduites — à un biholomorphisme près de $\mathbb{C}^2$ — à une famille à un paramètre de la forme

\[ M_t := \biggl \lbrace (z,w)\in \mathbb{C}^2: w=z^2\overline{z}+tz\overline{z}^2+\frac{t^2}{3} \overline{z}^3+o\bigl ({\vert z \vert ^3}\bigr )\biggr \rbrace , \quad t\in (0,\infty ), \]

près de l’origine. Nous prouvons que $M_t$ n’est pas localement polynomialement convexe si $t<1$. L’enveloppe locale contient une boule centrée à l’origine si $t<\sqrt{3}/2$. Nous prouvons également que $M_t$ est localement polynomialement convexe pour $t\ge \sqrt{3/2}$. Nous montrons que, pour $\sqrt{3}/2\le t<1$, l’enveloppe locale de $M_t$ contient une famille à un paramètre de disques analytiques passant par l’origine. Nous prouvons également que, si nous supprimons les termes d’ordre supérieur de la fonction graphique de $M_t$, elle est localement polynomialement convexe pour $t\ge \frac{\sqrt{15-\sqrt{33}}}{2\sqrt{2}}$. De nouveaux résultats sur la convexité polynomiale locale de l’union de trois plans totalement réels sont également rapportés.

Reçu le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3796
Classification : 32E20, 32D10, 30E10
Keywords: polynomially convex, totally real, CR singularity, polynomial hull, analytic disc
Mots-clés : polynomialement convexe, totalement réel, singularité CR, enveloppe polynomiale, disque analytique

Gorai, Sushil  1

1 Department of Mathematics and Statistics, Indian Institute of Science Education and Research Kolkata, Mohanpur, Nadia, West Bengal 741246 (India) 
Gorai, Sushil. Certain real surfaces in $\mathbb{C}^2$ with isolated singularities. Annales de l'Institut Fourier, Online first, 45 p.
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