The Envelope of Holomorphy of a Model Third-Degree Surface and the Rigidity Phenomenon

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.