Thom conjecture
From Wikipedia, the free encyclopedia
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, 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
.
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 of 1994 using the then new Seiberg-Witten invariants.
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 Peter Ozsváth and Zoltán Szabó[1]), which states that a symplectic submanifold of a symplectic manifold is genus minimizing within its homology class.
[edit] See also
[edit] References
- ^ Ozsváth and Szabó's paper, arXiv:math.DG/9811087
