2025/11/19 by Gupta, Purvi, Sahu, Rudranil
#32E20 #53D12 #Complex Variables (math.CV) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2511.15306
Given a compact smooth totally real immersed n-submanifold M⊂\mathbb Cn with only finitely many transverse double points, it is known that if M is Lagrangian with respect to some Kähler form on \mathbb Cn, then it is rationally convex in \mathbb Cn (Gayet, 2000), but the converse is not true (Mitrea, 2020). We show that M is Lagrangian with respect to some Kähler form on \mathbb Cn 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 Cn that maps the pair to (\mathbb Rn,(D+i)\mathbb Rn) for some real diagonal n× n matrix D.