Polynomially convex hulls of singular real manifolds

Abstract

We obtain local and global results on polynomially convex hulls of Lagrangian and totally real submanifolds of C n with self-intersections and open Whitney umbrella points.

On regular Stein neighborhoods of a union of two maximally totally real subspaces in $$\mathbb {C}^n$$ C n

“…At the end we also give the application of this result to totally real immersions of real n-manifolds in C n with only finitely many double points, and such that the union of the tangent spaces at each intersection in some local coordinates coincides with M (A) ∪ N , described above. In connection to this we also note that Weinstock's result has been recently generalised by Gorai [6] and Shafikov and Sukhov [12,Theorems 1.3 and 4.2], to the effect that a union of two maximally totally real submanifolds in C n , intersecting transversally at the origin, is polynomially convex near the origin, if the union of their tangent spaces at the origin is polinomially convex near the origin.…”

Open Whitney umbrellas are locally polynomially convex

“…This result was proved for a generic real-analytic φ in [12] and for a generic smooth φ in [13]. Our theorem establishes polynomial convexity in full generality in this context.…”

Discs in hulls of real immersions into Stein manifolds

“…For this reason his approach considerably relies on the affine structure of C n . In [19] we extended his result to the case of certain totally real immersions to C n . The goal of the present paper is to extend Alexander's result and the results of [19] to the case of totally real immersions to some Stein manifolds (the integrability of complex structure in fact, is not needed for some of our results).…”

“…It would be interesting to extend part (ii) of Theorem 2.4 to the almost complex case. (2) Corollary 2.5 is well-known in the case when M = C n and E is a smooth (or even topological) submanifold, see [4,21]; for totally real immersions in C n it is obtained in [19]. (3) It is well-known that there exist compact totally real manifolds (for example, some ntori in C n ) which do not contain the whole boundary of a nonconstant complex disc, see Alexander [2] and Duval-Gayet [7].…”

“…Our main observation is that the removal of boundary singularity is connected with "complex" convexity properties of the singular set of the immersed manifold. In [19] we used the polynomial convexity working in C n ; the notion of p.s.h. convexity is suitable in the Stein case.…”