In mathematics, the Gromov invariant of Clifford Taubes counts embedded (possibly disconnected) pseudoholomorphic curves in a symplectic 4-manifold, where the curves are holomorphic with respect to an auxiliary compatible almost complex structure.
Much of the analytical complexity connected to this invariant comes from properly counting multiply covered pseudoholomorphic curves so that the result is invariant of the choice of almost complex structure.
Embedded contact homology is an extension due to Michael Hutchings of this work to noncompact four-manifolds of the form
ECH is a symplectic field theory-like invariant; namely, it is the homology of a chain complex generated by certain combinations of Reeb orbits of a contact form on Y, and whose differential counts certain embedded pseudoholomorphic curves and multiply covered pseudoholomorphic cylinders with "ECH index" 1 in
The ECH index is a version of Taubes's index for the cylindrical case, and again, the curves are pseudoholomorphic with respect to a suitable almost complex structure.