Around the Mukai conjecture for Fano manifolds (Q253155)

Let \(X\) be a Fano manifold, i.e., a smooth complex projective variety whose anticanonical bundle \(-K_X\) is ample, and define its \textit{index} \(r_X\) and \textit{pseudoindex} \(i_X\) as \(r_X := \max \{m \in \mathbb N\;|-K_X = mL {\mathrm{\;for \;some \;line \;bundle \;}} L \},\) \(i_X := \min \{m \in \mathbb N\;|-K_X \cdot C = m {\mathrm{\;for \;some \;rational \;curve \;}} C \subset X \}.\) A conjecture of Mukai predicts that, denoted by \(\rho_X\) the Picard number of \(X\), and by \(n\) its dimension, then \[ \rho_X(r_X - 1) \leq n, \] with equality if and only if \(X = (\mathbb P^{r_X-1})^{\rho_X}\). A generalized version of this conjecture has been proposed by replacing the index with the pseudoindex. Both conjectures are known to hold if \(\rho_X \leq 3\) or \(n \leq 5\). In the paper under review the author conjectures that manifolds which are ``close to the bound'', namely those for which \( \rho_X(r_X - 1) \geq n -1\) (resp. \( \rho_X(i_X - 1) \geq n -1\)) are very special (explicit lists are given), and he proves the conjecture for \(\rho_X \leq 3\) or \(n \leq 5\). In the version with the pseudoindex, the proof relies on the numerical characterization of hyperquadrics as varieties of pseudoindex equal to the dimension.