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A characterization of products of projective spaces. (English) Zbl 1115.14034
By a conjecture proposed by Mukai in 1988, for a Fano manifold \(X\) covered by rational curves always should take place the inequality \(\rho_X(r_X-1) \leq \dim X\), where \(\rho_X\) is the rank of \(\operatorname{Pic}X\) over \({\mathbb Z}\), and \(r_X\) is the index of \(X\), defined as the greatest integer \(m\) such that \(-K_X = mL\) for some \(L \in \operatorname{Pic}X\). A recent progress in this direction is marked in the paper [L. Bonavero, C. Casagrande, O. Debarre and S. Druel, Comment. Math. Helv. 78, No. 3, 601–626 (2003; Zbl 1044.14019)] where the conjecture of Mukai is verified for \(\dim X \leq 4\) and for certain special Fano manifolds of greater dimensions [see also J. Wisniewski, Manuscr. Math. 68, No. 2, 135–141 (1990; Zbl 0715.14033); M. Andreatta, E. Chierici and G. Occhetta, Cent. Eur. J. Math. 2, No. 2, 272–293 (2004; Zbl 1068.14049)]. The main result of the present paper is the following:
Theorem 1.1. A smooth complex projective variety \(X\) of dimension \(n\) is isomorphic to a product of projective spaces \({\mathbb P}^{n(1)} \times\dots \times {\mathbb P}^{n(k)}\) if and only if there exist \(k\) unsplit covering families of rational curves \(V^1,\dots,V^k\) of degrees \(n(1)+1,\dots,n(k)+1\) with \(\Sigma n(i) = n\) such that the numerical classes of \(V^1,\dots, V^k\) are linearly independent in \(N_1(X)\).

14J45 Fano varieties
14J40 \(n\)-folds (\(n>4\))
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