Polynomially convex hulls of singular real manifolds (Q2787977)

The authors consider polynomial convexity of \(n\)-dimensional real manifolds in \(\mathbb C^n\) with only (transverse) self-intersections and open Whitney umbrellas as possible singularities, motivated by the work of \textit{A. B. Givental'} [Funct. Anal. Appl. 20, 197--203 (1986; Zbl 0621.58025); translation from Funkts. Anal. Prilozh. 20, No. 3, 35--41 (1986)] who proved that any compact real surface can be realized as a Lagrangian submanifold in \(\mathbb C^2\) with isolated singularities of the above type -- moreover, by Givental [loc. cit.] and \textit{G.-o Ishikawa} [Invent. Math. 126, No. 2, 215--234 (1996; Zbl 0869.58002)], a generic Lagrangian inclusion has those two types of singularities only -- and by the work of \textit{M. Gromov} [Invent. Math. 82, 307--347 (1985; Zbl 0592.53025)] on properties of Lagrangian inclusions into \(\mathbb C^n\), in particular on the existence of a nonconstant Riemann surface with boundary attached to a Lagrangian embedding. The latter -- considered from the complex analysis point of view -- is a problem on polynomial convexity.NEWLINENEWLINEDenote by \(\Sigma\) the standard open Whitney umbrella. The first result (Theorem 1.2) of the present paper states that the image \(\phi(\Sigma)\) under \(\phi: \mathbb C^2 \rightarrow \mathbb C^2,\) a smooth generic symplectomorphism near \(0,\) is locally polynomially convex near \(\phi(0)\). The next result (Theorem 1.3) states that if two smooth Lagrangian manifolds in \(\mathbb C^n\) intersect transversally at \(p\) then their union is locally polynomially convex near \(p\). Theorem 1.4 is a generalization of Gromov's above-mentioned result for Lagrangian embeddings and states that a smooth compact Lagrangian immersion \(L\) in \(\mathbb C^n\) with finitely many self-intersection points near which \(L\) is locally polynomially convex, admits a nonconstant analytic disc, continuous to the boundary, with boundary attached to \(L\). In particular, the conclusion holds for such immersions with finitely many transverse double self-intersection points and this generalizes a result by Ivashkovich and Shevchishin.