Lifting Lagrangian immersions in $\mathbb{C} P^{n-1}$ to Lagrangian cones in $\mathbb{C}^n$

TL;DR

This paper presents a method to lift Lagrangian immersions in complex projective space to construct Lagrangian cones in complex Euclidean space, leading to new examples including special Lagrangian cones with controlled singularities.

Contribution

It introduces a novel lifting technique for Lagrangian immersions and constructs new families of Lagrangian and special Lagrangian cones with specific geometric properties.

Findings

Constructed new Lagrangian cones from immersions in projective space.

Produced examples of special Lagrangian cones with few double points.

Demonstrated isotopies to Harvey-Lawson and trivial cones.

Abstract

In this paper we show how to lift Lagrangian immersions in $\mathbb{C} P^{n-1}$ to produce Lagrangian cones in $\mathbb{C} ^n$, and use this process to produce several families of examples of Lagrangian cones and special Lagrangian cones. Moreover we show how to produce Lagrangian cones, isotopic to the Harvey-Lawson and trivial cones, whose projections to $\mathbb{C} P^{n-1}$ are immersions with few transverse double points.