Geodesic
The Envelope of Holomorphy of a~Model Third-Degree Surface and the Rigidity Phenomenon
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis and applications, Tome 253 (2006), pp. 30-45

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The structures of the graded Lie algebra $\mathop{\mathrm{aut}}Q$ of infinitesimal automorphisms of a cubic (a model surface in $\mathbb C^N$) and the corresponding group $\mathop{\mathrm{Aut}}Q$ of its holomorphic automorphisms are studied. It is proved that for any nondegenerate cubic, the positively graded components of the algebra $\mathop{\mathrm{aut}}Q$ are trivial and, as a consequence, $\mathop{\mathrm{Aut}}Q$ has no subgroups consisting of nonlinear automorphisms of the cubic that preserve the origin (the so-called rigidity phenomenon). In the course of the proof, the envelope of holomorphy for a nondegenerate cubic is constructed and shown to be a cylinder with respect to the cubic variable whose base is a Siegel domain of the second kind. 
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     author = {R. V. Gammel' and I. G. Kossovskii},
     title = {The {Envelope} of {Holomorphy} of {a~Model} {Third-Degree} {Surface} and the {Rigidity} {Phenomenon}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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R. V. Gammel'; I. G. Kossovskii. The Envelope of Holomorphy of a~Model Third-Degree Surface and the Rigidity Phenomenon. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis and applications, Tome 253 (2006), pp. 30-45. http://geodesic.mathdoc.fr/item/TM_2006_253_a2/