Thom conjecture

In mathematics, the Thom conjecture is a proven conjecture that goes back to René Thom about surfaces in the complex-projective plane. The conjecture and its generalization to symplectic manifolds were an important motivation in the development of analytical-topological methods such as the Seiberg-Witten invariants .

background

Smooth algebraic curves in the complex-projective plane are given by homogeneous polynomials. They are complex 1-dimensional manifolds, i.e. topological surfaces . The gender of an algebraic curve given by a polynomial of degree is calculated according to the formula ${\displaystyle C}$ ${\displaystyle d}$

${\displaystyle g=(d-1)(d-2)/2}$.

guess

The Thom conjecture, named after René Thom, says: If there is a differentiable surface embedded in the complex-projective plane that represents the same homology class as a smooth algebraic curve given by a homogeneous polynomial of degree , then the gender of the surface satisfies the inequality ${\displaystyle \Sigma }$ ${\displaystyle d}$ ${\displaystyle g}$ ${\displaystyle \Sigma }$

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

In particular, every algebraic curve is an area of ​​minimal gender ( Thurston norm- minimizing area) in its homology class. ${\displaystyle C}$

It is easy to see that the 2nd homology of the complex-projective plane is isomorphic to the integers ; smooth algebraic curves of gender correspond to the number under this isomorphism . The Thom conjecture thus calculates the Thurston norm (the minimum gender) for all homology classes in . ${\displaystyle \mathbb {Z} }$ ${\displaystyle d}$ ${\displaystyle d\in \mathbb {Z} }$ ${\displaystyle H_{2}(\mathbb {C} P^{2})}$

proof

A few weeks after Edward Witten had introduced the Seiberg-Witten invariants to mathematics, Kronheimer - Mrowka proved the Thom conjecture in October 1994 with the help of these new invariants.

generalization

The symplectic Thom conjecture says that symplectic surfaces in symplectic 4-manifolds are surfaces of minimal gender in their homology class. The Thom conjecture is a special case because the smooth algebraic curves are symplectic submanifolds with respect to the canonical symplectic structure on the complex-projective level.

The symplectic Thom conjecture was proved with the help of Seiberg-Witten invariants by Morgan - Szabó - Taubes for symplectic surfaces with non-negative self-intersection numbers. Ozsváth and Szabó finally gave the general proof for the symplectic Thom conjecture with the help of Seiberg-Witten invariants.

However, it is generally a difficult question which homology classes of a symplectic manifold can be represented by symplectic submanifolds.

Individual evidence

1. ↑ Kronheimer, PB; Mrowka, TS: The genus of embedded surfaces in the projective plane. Math. Res. Lett. 1 (1994) no. 6, 797-808
2. ↑ Morgan, JW; Szabó, Z .; Taubes, CH: A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture. J. Differential Geom. 44 (1996) no. 4, 706-788
3. ↑ Ozsváth, P .; Szabó, Z .: The symplectic Thom conjecture. Ann. of Math. (2) 151 (2000), no. 1, 93-124