Thom conjecture

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In mathematics, a smooth algebraic curve $C$ in the complex projective plane, of degree $d$ , has genus given by the formula

g = (d - 1)(d - 2) / 2

The Thom conjecture, named after the 20th century mathematician René Thom, states that if $Σ$ is any smoothly embedded connected curve representing the same class in homology as $C$ , then the genus $g$ of $Σ$ satisfies

$g \geq (d-1)(d-2)/2$ .

In particular, C is known as a genus minimizing representative of its homology class. There are proofs for this conjecture in certain cases such as when $Σ$ has nonnegative self intersection number, and assuming this number is nonnegative, this generalizes to Kähler manifolds (an example being the complex projective plane). It was first proved by Kronheimer-Mrowka and Morgan-Szabó-Taubes in October 1994, using the then-new Seiberg-Witten invariants.

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 Peter Ozsváth and Zoltán Szabó^[1]). 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.