On some special Fano manifolds (Q1922682)

In this paper we deal with Fano manifolds of special kind, such as products and \(\mathbb{P}\)-bundles with some extra conditions. For a Fano manifold of product type all factors are Fano. So we simply call such a manifold a Fano product. Let \(X=Y_1\times \dots\times Y_s\) be a Fano product. By investigating some inequalities involving the coindex \(c\) of \(X\), the coindexes of the factors and their number \(s\), we prove that \(s\leq c+1\) and characterize the \(X\) for which \(c-2\leq s\leq c+1\). In some sense our results are best possible. The case \(s=c-2\) involves at worst Mukai manifolds as factors; but in some cases with \(s=c-3\) one of the \(Y_i\) belongs to the still unexplored class of Fano manifolds of coindex 4. Besides products, some \(\mathbb{P}\)-bundles are Fano manifolds. Recently, several papers appeared on the classification of Fano \(\mathbb{P}\)-bundles of low dimensions, while other recent papers related to a conjecture of Mukai drew attention to ample vector bundles \({\mathcal E}\) on a projective manifold \(B\) such that \(K_B+\text{det }{\mathcal E}\) is not nef (or not ample, or trivial). The obvious link between these subjects is that if \(-(K_B+\text{det} {\mathcal E})\) is nef, then \(P=\mathbb{P}_B({\mathcal E})\) is a Fano manifold. In this paper we also investigate some properties of Fano \(\mathbb{P}\)-bundles of the above type \(\mathbb{P}_B({\mathcal E})\) satisfying the condition \(\text{det }{\mathcal E}=-K_B\). We define these to be special. Up to replacing \({\mathcal E}\) with \({\mathcal E}\otimes L\), for a suitable line bundle \(L\) on \(B\), they simply coincide with Fano \(\mathbb{P}\)-bundles of maximal index. If \(P\) is a special Fano \(\mathbb{P}\)-bundle as above, letting \(r=\text{rk }{\mathcal E}\) and \(b=\dim B\), it follows from a paper by \textit{T. Fujita} [in: Complex algebraic varieties, Proc. Conf., Bayreuth 1990, Lect. Notes Math. 1507, 105-112 (1992; Zbl 0782.14018)] that \(\text{cork }P\geq 0\), where, by definition, \(\text{cork }P=b+1-r\), and several results scattered in the literature provide the classification of special Fano \(\mathbb{P}\)-bundles with \(\text{cork }P\leq 2\), up to a still undecided case when \(\text{cork }P=2\). -- In this paper, after providing several examples in the range \(\text{cork }P\geq 3\) we classify a special class of Fano \(\mathbb{P}\)-bundles of corank 3 on product manifolds, which we call standard. Roughly speaking, this means that \({\mathcal E}\) comes from vector bundles on the factors. There are several such \(\mathbb{P}\)-bundles of corank 3, while there are very few for \(\text{cork }P=2\) and do not appear at all when \(\text{cork }P\leq 1\). Although this classification relies on the known results for Fano \(\mathbb{P}\)-bundles with \(\text{cork }P\leq 2\), we would like to stress that our result is effective: actually the undecided case mentioned above in this context is irrelevant.