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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

We show how to lift Lagrangian immersions in Pn1 to produce Lagrangian cones in 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 P2 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.

lagrangian, cone, SYZ, Calabi–Yau, knot, Harvey–Lawson, hypercube diagram, grid diagram, contact homology
Mathematical Subject Classification
Primary: 53D17, 53D35, 57R17
Received: 31 January 2021
Revised: 11 November 2021
Accepted: 26 November 2021
Published: 27 October 2022
Scott Baldridge
Department of Mathematics
Louisiana State University
Baton Rouge, LA
United States
Ben McCarty
Department of Mathematical Sciences
University of Memphis
Memphis, TN
United States
David Vela-Vick
Department of Mathematics
Louisiana State University
Baton Rouge, LA
United States