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Abelian differentials, Castelnuovo bounds and geometry of webs. (Formes différentielles abéliennes, bornes de Castelnuovo et géométrie des tissus.) (French) Zbl 1064.53011
Let $$d\geq k\geq 2$$ and $$n\geq 1$$ be integers. A $$d$$-web $$W(d,k,n)$$ is a collection of $$d$$ complex analytic foliations of codimension $$n$$ of $$(\mathbb{C}^{kn},0)$$. The leaves of each foliation are the fibres of a germ of a complex analytic submersion $$F_i: (\mathbb{C}^{kn},0)\to (\mathbb{C}^n,0)$$, $$i= 1,\dots, d$$. The author assumes that the leaves of $$W(d,k,n)$$ are in general position, which means that, for every $$1\leq i_1<\cdots< i_k\leq d$$, the tangent map at $$0$$ of $$(F_{i_1},\dots, F_{i_k}): (\mathbb{C}^{kn},0)\to (\mathbb{C}^n, 0)^k$$ is bijective. For $$0\leq p\leq n$$ let $$A^p$$ denote the $$\mathbb{C}$$-vector space of $$d$$-tuples $$(\omega_1,\dots, \omega_d)$$ of germs of complex analytic differential $$p$$-forms on $$(\mathbb{C}^n,0)$$ such that $$F^*_1\omega_1+\cdots+ F^*_d\omega_d= 0$$ for $$p> 0$$ and = const for $$p= 0$$. The exterior derivative transforms $$(A^p)_{0\leq p\leq n}$$ into a complex $$A^0$$ of $$\mathbb{C}$$-vector spaces. The author shows that $$A^p$$ is of finite dimension over $$\mathbb{C}$$, $$0\leq p\leq n$$, and gives optimal bounds $$\dim_{\mathbb{C}} A^p\leq \pi_p(d,k,n)$$, generalizing a result of S.-S. Chern and P. A. Griffiths [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 5, 539–557 (1978; Zbl 0402.57001)] who treated the case $$p= n$$.
$$d$$-webs occur in projective geometry as follows: let $$X$$ be an $$n$$-dimensional subvariety of $$\mathbb{P}^{n+k-1}$$ of degree $$d$$, contained in no hyperplane. Assume $$X$$ smooth, for simplicity. Let $$G$$ denote the Grassmannian of $$(k-1)$$-dimensional linear subspaces of $$\mathbb{P}^{n+k-1}$$, let $$Y\subset X\times G$$ consist of the pairs $$(x,[L])$$ with $$x\in L$$, and let $$\pi$$ and $$\rho$$ denote the projections of $$Y$$ on $$X$$ and $$G$$, respectively. Let $$[L]$$ denote a general point of $$G$$ such that $$L\cap X$$ consists of $$d$$ points and any $$k$$ of these points generate $$L$$. For a small open neighborhood $$W$$ of $$[L]$$, $$\rho^{-1}(W)= V_1\cup\cdots\cup V_d$$ with $$\rho| V_i\overset\sim\rightarrow W$$. Let $$U_i= \pi(V_i)$$ and $$F_i: W\to U_i$$ be the composition of $$\pi$$ with the inverse of $$\rho| V_i\overset\sim\rightarrow W$$. If $$\omega\in H^0(X,\Omega^p_X)$$ then $$\text{Tr}_\rho(\pi^*\omega)\in H^0(G,\Omega^p_G)$$ which is $$0$$ for $$p> 0$$ and equals $$\mathbb{C}$$ for $$p= 0$$. This implies that $$F^*_1(\omega| U_1)+\cdots+ F^*_d(\omega| U_d)= 0$$ for $$p> 0$$ and is constant for $$p= 0$$. In this way one gets a $$d$$-web $$W(d,k,n)$$ and injections $$H^0(X,\Omega^p_X)\subseteq A^p$$. It follows that $$\dim_{\mathbb{C}} H^0(X,\Omega^p_X)\leq \pi_p(d,k,n)$$. For $$p= n= 1$$ this is the well-known Castelnuovo bound for the genus of curves in $$\mathbb{P}^k$$.
In fact, the author assumes only that $$X$$ is reduced of pure dimension $$n$$, replacing $$\Omega^p_X$$ by the sheaves $$\omega^p_X$$ of D. Barlet [Lect. Notes Math. 679, 187–204 (1978; Zbl 0398.32009)]. He also constructs certain arrangements of $$n$$-planes in $$\mathbb{P}^{n+k-1}$$ such that the associated webs satisfy $$\dim_{\mathbb{C}} A^p= \pi_p(d,k,n)$$, $$0\leq p\leq n$$.

##### MSC:
 53A60 Differential geometry of webs 14C21 Pencils, nets, webs in algebraic geometry
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