Thom conjecture
In mathematics, a smooth algebraic curve in the complex projective plane, of degree, has genus given by the genus–degree formula
The Thom conjecture, named after French mathematician René Thom, states that if is any smoothly embedded connected curve representing the same class in homology as, then the genus of 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, using the then-new Seiberg–Witten invariants.
Assuming that has nonnegative self intersection number this was generalized to Kähler manifolds by John Morgan, Zoltán Szabó, and Clifford Taubes, also using the Seiberg–Witten invariants.
There is at least one generalization of this conjecture, known as the symplectic Thom conjecture. It states that a symplectic surface of a symplectic 4-manifold is genus minimizing within its homology class. This would imply the previous result because algebraic curves are symplectic surfaces within the complex projective plane, which is a symplectic 4-manifold.
