Dual elliptic structures on CP2

Abstract

We consider an almost complex structure J on CP2, or more generally an elliptic structure E which is tamed by the standard symplectic structure. An E-curve is a surface tangent to E (this generalizes the notion of J(holomorphic)-curve), and an E-line is an E-curve of degree 1. We prove that the space of E-lines is again a CP2 with a tame elliptic structure E^*, and that each E-curve has an associated dual E^*-curve. This implies that the E-curves, and in particular the J-curves, satisfy the Pl\"ucker formulas, which restricts their possible sets of singularities.Comment: 18 pages The only difference with the first version is the mention of the thesis of Benjamin MacKay ("Duality and integrable systems of pseudoholomorphic curves", Duke University, 1999), which I did not know at the time, and which contains a large part of the results of my pape