A complex-analytic characterization of Lagrangian immersions in $\mathbb C^n$ with transverse double points

TL;DR

This paper characterizes when a compact totally real immersed submanifold with transverse double points in complex Euclidean space is Lagrangian, linking it to rational convexity and a specific diagonalizability condition at double points.

Contribution

It provides a complex-analytic criterion involving rational convexity and a diagonalizability condition that characterizes Lagrangian immersions with double points.

Findings

Lagrangian condition equivalent to rational convexity plus diagonalizability at double points

Introduces a complex linear transformation criterion for Lagrangian immersions

Connects geometric Lagrangian properties with algebraic diagonalizability condition

Abstract

Given a compact smooth totally real immersed $n$-submanifold $M\subset\mathbb C^n$ with only finitely many transverse double points, it is known that if $M$ is Lagrangian with respect to some K{\"a}hler form on $\mathbb C^n$, then it is rationally convex in $\mathbb C^n$ (Gayet, 2000), but the converse is not true (Mitrea, 2020). We show that $M$ is Lagrangian with respect to some K{\"a}hler form on $\mathbb C^n$ if and only if $M$ is rationally convex {\em and} at each double point, the pair of transverse tangent planes to $M$ satisfies the following diagonalizability condition: there is a complex linear transformation on $\mathbb C^n$ that maps the pair to $\left(\mathbb R^n,(D+i)\mathbb R^n\right)$ for some real diagonal $n\times n$ matrix $D$.