Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 755differential form λ obeyingdλdu= ω + exact form in x, (4.247)√2 dx8π y .with ω =For N = 4 the structure is the same, except that 8π is replaced by 4π. Fshould be such that the residues of λ are linear in the quark bare masses.This is a severe restriction on F; we see that it determines F uniquely (upto the usual changes of variables) independently of most of the argumentsthat we have used up to this point.Let us write (4.247) in a more symmetrical form. If λ = dx a(x, u), then(4.247) means√2 dx ∂a= dx8π y ∂u + dx ∂ f(x, u); (4.248)∂xthe arbitrary total x−derivative dx ∂f/∂x is allowed here because it doesnot contribute to the periods. (4.248) can be understood much better ifwritten symmetrically in x and u. Henceforth, instead of using a 1–formω = ( √ 2/8π) · dx/y, we use a 2–formω =√28πdx du.ySimilarly, we combine the functions a, f appearing in (4.248) into a 1–formλ = −a(x, u)dx + f(x, u)du. The change in notation for ω and λ shouldcause no confusion. Then equation (4.248) can be more elegantly writtenasω = dλ. (4.249)The meaning of the problem of finding λ can now be stated. Let X bethe (noncompact) complex surface defined by the equation y 2 = F(x, u)(we suppress the parameters m i and τ). Being closed, ω defines an element[ω] ∈ H 2 (X, C). A smooth differential λ obeying (4.249) exists if and onlyif [ω] = 0. Moreover, by standard theorems, in the absence of restrictionson the growth of λ at infinity, if λ exists it can be chosen to be holomorphicand of type (1, 0).If on the other hand [ω] ≠ 0, then (4.249) has no smooth, much lessholomorphic, solution. However, X has the property that if one throwsaway a sufficient number of complex curves C a , then X ′ = X − ∪ a C a