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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. Kollár) 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