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

TL;DR

The paper characterizes when a totally real immersion $\iota:M\to\mathbb{C}^n$ with isolated transverse double points is Lagrangian for some Kähler form. The core idea is to reduce each double point to the linear model $\mathbb{R}^n\cup S(A)$ with $S(A)=(A+i)\mathbb{R}^n$ and require $A$ to be diagonalizable over $\mathbb{R}$; this diagonalizability, together with rational convexity, yields a global Kähler form making $\iota(M)$ Lagrangian. The construction combines a local potential near diagonalizable points, solved via a tailored PDE for a correction term, with Mitrea’s global degenerate form to patch into a global positive $(1,1)$-form. The result provides a precise analytic-geometry criterion linking rational convexity and a linear-algebraic condition at self-intersections, delivering a complete characterization of Lagrangian immersions with transverse double points in $\mathbb{C}^n$.

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{ä}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{ä}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 $D$.

Theorems & Definitions (7)

• Theorem 1.1
• Lemma 3.1
• Proposition 3.2
• Lemma 3.3: We88
• proof
• Lemma 4.1
• proof