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

(d - 1)(d - 2) / 2

The Thom conjecture, of René Thom, states that if $Σ$ is any smoothly embedded 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).

There is at least one other version of this conjecture known as the symplectic Thom conjecture (which is now a theorem, as proved for example by P. Ozsváth and Z. Szabó), which states that a symplectic submanifold of a symplectic manifold is genus minimizing within its homology class.