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On the rigidity of differential systems modelled on Hermitian symmetric spaces and disproofs of a conjecture concerning modular interpretations of configuration spaces. (English) Zbl 0908.17013

Akahori, Takao (ed.) et al., CR-geometry and overdetermined systems. Based on the lectures delivered at the conference, Osaka, Japan, December 19–21, 1994. On the occasion of Kuranishi’s 70th birthday. Tokyo: Kinokuniya Company. Adv. Stud. Pure Math. 25, 318-354 (1997).
Let \(E(k,n;\alpha)\) be the hypergeometric system of differential equations of type \((k,n)\) defined on the configuration space \(X(k,n)\) on \(n\) hyperplanes in general position of the projective space \({\mathbb{P}}^{k-1}\), where \(\alpha \) is a system of parameters: \(\alpha=(\alpha_1,\ldots,\alpha_n)\), \(\alpha_1+\ldots+\alpha_n=n-k\). The rank of the system \(E(k,n;\alpha)\) is \(r=\binom{n-2}{k-1}\). Let \(\phi: X(k,n)\rightarrow \mathbb{P}^{r-1}\), \(x\mapsto (u_1(x):\ldots:u_r(x))\), where the \(u_j\)’s are linearly independent solutions of the system. The authors prove that if \(k>2\), \(n-k>2\) and \((k,n)\neq (3,6)\), then the image of \(\phi\) does not lie in \(\text{Gr}_{k-1,n-2}\subset \mathbb{P}^{r-1}\) for any \(\alpha_i\).
For the entire collection see [Zbl 0869.00033].
Reviewer: V.L.Popov (Moskva)

MSC:

17B56 Cohomology of Lie (super)algebras
32M99 Complex spaces with a group of automorphisms
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