On Fano 4-folds of index 1 and homogeneous vector bundles over Grassmannians
Besides complete intersections of hypersurfaces in projective space, Grassmannians belong to the classical examples of Fano manifolds, by which we mean smooth algebraic varieties \(X\) defined over the field \(\mathbb{C}\) of complex numbers with ample anticanonical divisor \(- K_X\). The coindex of \(X\) is defined as \(\dim (X) - r + 1\) where \(r\), the index of \(X\), is the greatest integer which divides \(- K_X\) in \(\text{Pic} (X)\). -- Contrarily to the case of coindex \(\leq 3\), where the classification, due to Fujita, Iskovskih, Mori and Mukai, is complete up to a minor assumption, very little is known for larger coindex. We consider homogeneous vector bundles in the study of Fano manifolds of coindex 4: the Fano manifolds arising as zero schemes of sections in homogeneous vector bundles form a large class of examples and should be typical in some sense. We restrict ourselves to the case of bundles over Grassmannians of \(k\)-dimensional quotient spaces of \(\mathbb{C}^n\) which are homogeneous with respect to \(SL (n, \mathbb{C})\). The main result is the classification of all Fano 4-folds \(X\) of index 1 that are zero schemes of general global sections in direct sums of globally generated irreducible \(SL (n, \mathbb{C})\)-homogeneous vector bundles over Grassmannians. Moreover we compute the Hodge numbers \(h^{p,q} (X)\) as well as the Hilbert polynomials \(h^0 (X, {\mathcal O}_X (- tK_X))\) of these manifolds.
scientific article; zbMATH DE number 784089
scientific article