Dual elliptic structures on CP2

TL;DR

This paper explores elliptic structures on CP2, demonstrating that E-lines form a CP2 with a dual structure, and establishing Plücker formulas that constrain the singularities of E-curves, generalizing holomorphic curve theory.

Contribution

It introduces the concept of elliptic structures on CP2, shows the space of E-lines forms a CP2 with a dual structure, and derives Plücker formulas for E-curves, extending classical results.

Findings

E-lines form a CP2 with a tame elliptic structure E^*

Each E-curve has an associated dual E^*-curve

E-curves satisfy Plücker formulas restricting singularities

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.