Rank of ordinary webs in codimension one. An effective method

Original · EN

We are interested by holomorphic d-webs W of codimension one in a complex n-dimensional manifold M. If they are ordinary, i.e. if they satisfy to some condition of genericity (whose precise definition is recalled), we proved in [CL] that their rank ρ(W) is upper-bounded by a certain number π'(n,d)(which, for n≥ 3, is stictly smaller than the Castelnuovo-Chern's bound π(n,d)). In fact, denoting by c(n,h) the dimension of the space of homogeneous polynomials of degree h with n unknowns, and by h₀ the integer such that c(n,h₀-1)<d≤ c(n,h₀), π'(n,d) is just the first number of a decreasing sequence of positive integers π'(n,d)=ρₕ₀₋₂≥ ρₕ₀₋₁≥ ≥ ρₕ≥ ρₕ₊₁≥≥ ρ∞=ρ(W)≥ 0 becoming stationary equal to ρ(W) after a finite number of steps. This sequence is an interesting invariant of the web, refining the data of the only rank. The method is effective: theoretically, we can compute ρₕ for any given h; and, as soon as two consecutive such numbers are equal (ρₕ=ρₕ₊₁, h≥ h₀-2), we can construct a holomorphic vector bundle Rₕ→ M of rank ρₕ, equipped with a tautological holomorphic connection ∇ʰ whose curvature Kʰ vanishes iff the above sequence is stationary from there. Thus, we may stop the process at the first step where the curvature vanishes. Examples will be given.