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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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