The Genus of Embedded Surfaces in the Projective Plane

Abstract

A b s t r a c t . We show how the new invariants of 4-manifolds resulting from the Seiberg-Witten monopole equation lead quickly to a proof of the 'Thom conjecture'. Statement of the resultThe genus of a smooth algebraic curve of degree d in CP 2 is given by the formula g = (d − 1)(d − 2)/2. A conjecture sometimes attributed to Thom states that the genus of the algebraic curve is a lower bound for the genus of any smooth 2-manifold representing the same homology class. The conjecture has previously been proved for d ≤ 4 and for d = 6, and less sharp lower bounds for the genus are known for all degrees [5,6,7,8,10]. In this note we confirm the conjecture. Very recently, Seiberg and Witten [11,12,13] introduced new invariants of 4-manifolds, closely related to Donaldson's polynomial invariants [2], but in many respects much simpler to work with. The new techniques have led to more elementary proofs of many theorems in the area. Given the monopole equation and the vanishing theorem which holds when the scalar curvature is positive (something which was pointed out by Witten), the rest of the argument presented here is not hard to come by. A slightly different proof of the Theorem, based on the same techniques, has been found by Morgan, Szabo and Taubes.It is also possible to prove a version of Theorem 1 for other complex surfaces, without much additional work. This and various other applications will be treated in a later paper, with joint authors. The monopole equation and the Seiberg-Witten invariantsLet X be an oriented, closed Riemannian 4-manifold. Let a spin c structure on X be given. We write c for the spin c structure and write W + = W

PU (2) monopoles and links of top-level Seiberg-Witten moduli spaces

“…In §2.1, we review the construction of the moduli spaces of PU(2) monopoles M t from [24], but now phrased in the convenient and more compact framework of "spin u structures" t. In §2.2 we recall our Uhlenbeck compactness and transversality results for the moduli spaces of PU(2) monopoles from [24] and [15]. In §2.3, we recall the construction of the moduli space of Seiberg-Witten monopoles M s as in [43], [45], [56], but with non-standard perturbations so we can directly identify these moduli spaces with strata of reducible PU(2) monopoles. Our transversality result [15] ensures that the natural stratification of the Uhlenbeck-compactified moduli space of PU (2) monopoles is smooth.…”

“…We note that in the usual presentation of the Seiberg-Witten equations [43], [60], one takes τ = id Λ + and ϑ = 0, while η is a generic perturbation. However, we shall see in Lemma 3.12 that in order to identify solutions to the Seiberg-Witten equations (2.55) with reducible solutions to the PU(2) monopole equations (2.32), we need to employ the perturbations given in equation (2.55) and choose…”

Finite group actions on 4-manifolds

“…We sum up the basic properties of the Seiberg-Witten invariants, which have been developed by many people: see for example [KM,Tl,T2] and [W]. (K x ) = ±1.…”

Liouville vortex and φ4 kink solutions of the Seiberg–Witten equations

“…The Seiberg-Witten equations [1] do not admit nonsingular solutions unless the curvature of the four dimensional base manifold M happens to be negative over some regions of M. In particular,if M = R 4 , the Weitzenbock formula implies that the modulus squared of the spinor field ψ must either vanish everywhere, or exhibit singularities instead of local maxima [2]. There is also a global restriction on flat-space Seiberg-Witten solutions: Integrating the Weitzenbock formula, Witten [1] showed that all nontrivial flat solutions, including dimensionally reduced ones based on R 3 , R 2 or R 1 , are all necessarily non-L 2 .…”