The projective plane, \(J\)-holomorphic curves and Desargues' theorem (Q392620)

Let \((\mathbb C P^2,\omega)\) be the complex projective plane with the natural symplectic structure, and let \(J\) be an almost complex structure on \(\mathbb C P^2\) tamed by \(\omega\). By a result of Gromov, the points of \(\mathbb C P^2\) and the \(J\)-holomorphic curves such that the corresponding homology class in \(H_2(CP^2,Z)\) is the positive generator define a projective plane. Ghys posed the question if the Theorem of Desargues for this projective plane implies that \(J\) is integrable; the main result of this paper gives a positive answer to this question. The proof is an extension of classical results of Geometric Algebra.