Special Lagrangian manifolds from G_p,p+qℂ

Abstract

This article provides a complete, detailed generalization of the construction of special Lagrangian submanifolds obtained from complex Grassmannians G_p,p+qℂ via real structures. We present explicit computations of the first Chern class using the adjunction formula and Poincaré residue. For each case G_2,4ℂ, G_2,5ℂ, G_3,6ℂ, and the general case, we determine the full topological invariants of the real locus: fundamental group, homology groups, cohomology ring, and the nature of the special Lagrangian fibration (circle bundle, torus bundle, or higher-dimensional torus bundle over real Grassmannians). All calculations are made explicit, and numerous illustrative tables and diagrams are provided.

1. Introduction

Let G = G_p,p+qℂ be the Grassmannian of complex p-planes in ℂ^p+q. Its Plücker embedding into ℂℙ^{C(p+q, p)−1} is given by the p × p minors of a p × (p+q) matrix. The Kähler form ω and the holomorphic volume form Ω on G satisfy c₁(G) = (p+q)σ₁, where σ₁ is the positive generator of H²(G, ℤ).

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Jelloul, R. (2026). Special Lagrangian manifolds from G_p,p+qℂ. Asia Mathematika, 10(2), 42–48. https://doi.org/10.5281/zenodo.23086664

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