is any smoothly embedded connected curve representing the same class in homology as
satisfies the inequality In particular, C is known as a genus minimizing representative of its homology class.
It was first proved by Peter Kronheimer and Tomasz Mrowka in October 1994,[1] using the then-new Seiberg–Witten invariants.
has nonnegative self intersection number this was generalized to Kähler manifolds (an example being the complex projective plane) by John Morgan, Zoltán Szabó, and Clifford Taubes,[2] also using the Seiberg–Witten invariants.
This would imply the previous result because algebraic curves (complex dimension 1, real dimension 2) are symplectic surfaces within the complex projective plane, which is a symplectic 4-manifold.