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 below), we proved in [CL] that their rank $\rho (W)$ is upper-bounded by a certain number $\pi^\prime (n, d)$ (which, for $n \geq 3$, is strictly smaller than the Castelnuovo–Chern’s bound $\pi (n, d)$). In fact, denoting by $c(n, h)$ the dimension of the space of homogeneous polynomials of degree $h$ with $n$ unknowns, and by $h_0$ the integer such that\[c(n, h_0 - 1) \lt d \leq c(n, h_0),\]$\pi^\prime (n, d)$ is just the first number of a decreasing sequence of positive integers\[\pi^\prime (n, d) = \rho_{h_0 - 2} \geq \rho_{h_0 - 1} \geq \dotsc \geq \rho_h \geq \rho_{h+1} \geq \dotsc \geq \rho_\infty = \rho (W) \geq 0\]becoming stationary equal to $\rho (W)$ 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 $\rho_h$ for any given $h$; and, as soon as two consecutive such numbers are equal ($\rho_h = \rho_{h+1} , h \geq h_0 - 2$), we can construct a holomorphic vector bundle $R_h \to M$ of rank $\rho_h$, equipped with a tautological holomorphic connection $\nabla^h$ whose curvature $K^h$ vanishes iff the above sequence is stationary from there. Thus, we may stop the process at the first step where the curvature vanishes, and compute the rank without to have to exhibit explicitly independant abelian relations. Examples will be given.

中文翻译：

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普通卷筒纸的维数排序是一种有效的方法

我们对复维$ n $维流形$ M $中全维一的全维$ d $ -webs $ W $感兴趣。如果它们是普通的，即，如果它们满足某种通用性条件（下面将回顾其精确定义），则我们在[CL]中证明了它们的等级$ \ rho（W）$上限为某个数字$ \ pi ^ \ prime（n，d）$（对于$ n \ geq 3 $，它严格小于Castelnuovo–Chern约束的$ \ pi（n，d）$）。实际上，用$ c（n，h）$表示度为$ h $且具有$ n $未知数的齐次多项式空间的维数，而用$ h_0 $表示整数，使得\ [c（n，h_0-1 ）\ lt d \ leq c（n，h_0），\] $ \ pi ^ \ prime（n，d）$只是正整数递减序列的第一个数字\ [\ pi ^ \ prime（n，d）= \ rho_ {h_0-2} \ geq \ rho_ {h_0-1} \ geq \ dotsc \ geq \ rho_h \ geq \ rho_ {h + 1} \ geq \ dotsc \ geq \ rho_ \ infty = \ rho（ W）\ geq 0 \]在经过有限步数后变为等于$ \ rho（W）$的平稳状态。该序列是Web的一个有趣的不变式，它完善了唯一等级的数据。该方法是有效的：理论上，我们可以为任何给定的$ h $计算$ \ rho_h $；并且，一旦两个连续的此类数相等（$ \ rho_h = \ rho_ {h + 1}，h \ geq h_0-2 $），我们就可以构造全纯矢量束$ R_h \到等级$ \的M $ rho_h $，配备了重言同音同形连接$ \ nabla ^ h $，只要上述序列从此处开始是静止的，其曲率$ K ^ h $就会消失。因此，我们可以在曲率消失的第一步停止该过程，并计算等级，而不必表现出明显独立的阿贝尔关系。