Homogeneous vector bundles and families of Calabi-Yau threefolds. II.

*(English)*Zbl 0748.14015
Several complex variables and complex geometry, Proc. Summer Res. Inst., Santa Cruz/CA (USA) 1989, Proc. Symp. Pure Math. 52, Part 2, 83-91 (1991).

[For the entire collection see Zbl 0732.00008.]

Let \(X_ 4\) denote the variety of lines contained in a generic complete intersection of two quadrics in \(\mathbb{P}^ 6\). The author computes the cohomology of the Fano 4-fold \(X_ 4\) using the fact that it is the zero locus of a homogeneous vector bundle. It is shown that a generic hyperplane section \(X_ 3\) of \(X_ 4\) is a Calabi-Yau 3-fold with Betti numbers \(b_ 0=1\), \(b_ 1=0\), \(b_ 2=8\), \(b_ 3=50\) and Euler number \(e=-32\). The cohomology ring of \(X_ 3\) is also completely determined. It is proved that the cone of effective curves on \(X_ 3\) is the convex hull of finitely many extremal rays defined by lines and it maps onto the corresponding cone of \(X_ 4\).

The paper contains an appendix (due to J. KollĂˇr) with the following theorem. Let \(X\) be a smooth Fano variety of dimension \(\geq 4\). Let \(Y\subset X\) be a smooth divisor in \(| -K_ X|\) (in fact \(Y\) can have arbitrary singularities). Then the cones of effective 1-cycles of \(X\) and \(Y\) are naturally isomorphic. Another family of Calabi-Yau 3- folds (with Euler number \(-6\)) associated to homogeneous vector bundles related to a quadric in \(\mathbb{P}^ 9\) was studied by the author in part I of this paper [Duke Math. J. 61, 395-415 (1990: Zbl 0739.14026)].

Let \(X_ 4\) denote the variety of lines contained in a generic complete intersection of two quadrics in \(\mathbb{P}^ 6\). The author computes the cohomology of the Fano 4-fold \(X_ 4\) using the fact that it is the zero locus of a homogeneous vector bundle. It is shown that a generic hyperplane section \(X_ 3\) of \(X_ 4\) is a Calabi-Yau 3-fold with Betti numbers \(b_ 0=1\), \(b_ 1=0\), \(b_ 2=8\), \(b_ 3=50\) and Euler number \(e=-32\). The cohomology ring of \(X_ 3\) is also completely determined. It is proved that the cone of effective curves on \(X_ 3\) is the convex hull of finitely many extremal rays defined by lines and it maps onto the corresponding cone of \(X_ 4\).

The paper contains an appendix (due to J. KollĂˇr) with the following theorem. Let \(X\) be a smooth Fano variety of dimension \(\geq 4\). Let \(Y\subset X\) be a smooth divisor in \(| -K_ X|\) (in fact \(Y\) can have arbitrary singularities). Then the cones of effective 1-cycles of \(X\) and \(Y\) are naturally isomorphic. Another family of Calabi-Yau 3- folds (with Euler number \(-6\)) associated to homogeneous vector bundles related to a quadric in \(\mathbb{P}^ 9\) was studied by the author in part I of this paper [Duke Math. J. 61, 395-415 (1990: Zbl 0739.14026)].

Reviewer: U.N.Bhosle (Bombay

##### MSC:

14J30 | \(3\)-folds |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14J45 | Fano varieties |