In mathematics, a smooth algebraic curve
The Thom conjecture, named after French mathematician René Thom, states that if
In particular, C is known as a genus minimizing representative of its homology class. It was first proved by Kronheimer–Mrowka in October 1994, using the then-new Seiberg–Witten invariants.
Assuming that
There is at least one generalization of this conjecture, known as the symplectic Thom conjecture (which is now a theorem, as proved for example by Ozsváth and Szabó in 2000). 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 (complex dimension 1, real dimension 2) are symplectic surfaces within the complex projective plane, which is a symplectic 4-manifold.