Thom conjecture explained

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

The Thom conjecture, named after French mathematician René Thom, states that if

\Sigma is any smoothly embedded connected curve representing the same class in homology as C , then the genus g of \Sigma satisfies the inequalityg \geq (d -1 )(d -2 )/ 2 .

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,^[1] using the then-new Seiberg–Witten invariants.

Assuming that

\Sigma has nonnegative self intersection number this was generalized to Kähler manifolds (an example being the complex projective plane) by John Morgan, Zoltán Szabó, and Clifford Taubes,^[2] also using the 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 Szabó in 2000^[3]). 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.

See also

• Adjunction formula
• Milnor conjecture (topology)

References

1. The Genus of Embedded Surfaces in the Projective Plane. Mathematical Research Letters. 1. 6. 797–808. 10.4310/mrl.1994.v1.n6.a14. Peter B.. Kronheimer. Peter B. Kronheimer. Tomasz S.. Mrowka. Tomasz Mrowka. 1994. free.
2. John . Morgan. John Morgan (mathematician). Szabó . Zoltán . Zoltán Szabó (mathematician) . Clifford. Taubes. Clifford Taubes. A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture. Journal of Differential Geometry. 44 . 1996. 4. 706–788. 1438191. 10.4310/jdg/1214459408. free.
3. Ozsváth . Peter. Peter Ozsváth . Szabó . Zoltán . Zoltán Szabó (mathematician) . 2000 . The symplectic Thom conjecture . . 151 . 1 . 93–124 . math.DG/9811087 . 10.2307/121113. 121113. 5283657 .