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

### Scott Baldridge, Ben McCarty and David Vela-Vick

Vol. 5 (2022), No. 1, 43–79
##### Abstract

We show how to lift Lagrangian immersions in $ℂ{P}^{n-1}$ to produce Lagrangian cones in ${ℂ}^{n}$, and use this process to produce several families of examples of Lagrangian cones and special Lagrangian cones. As an application of this theorem, for $n=3$ we show how to produce Lagrangian cones that are isotopic to the Harvey–Lawson special Lagrangian cone and the trivial cone. The projections of the Legendrian links of both of these cones to $ℂ{P}^{2}$ are immersions with four and seven transverse double points. We expect that these double points represent the chord generators of the $0$-filtration level of a suitably defined version of Legendrian contact homology of the links.

##### Keywords
lagrangian, cone, SYZ, Calabi–Yau, knot, Harvey–Lawson, hypercube diagram, grid diagram, contact homology
##### Mathematical Subject Classification
Primary: 53D17, 53D35, 57R17