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Calabi-Yau hypersurfaces in products of semi-ample surfaces. (English) Zbl 0658.14020
This paper continues the quest of string theorists for a Calabi-Yau 3- manifold with small Euler characteristic. One of the features of these manifolds is the fact that the Euler characteristic determines the difference $$b_{1,1}-b_{1,2}$$ of Hodge numbers but the individual numbers themselves are not so easy to compute and yet have great geometrical significance. In this case, where $$c_ 1=0$$, $$b_{1,2}$$ is isomorphic to the infinitesimal deformation number $$\dim (H^ 1(X,\theta))$$ and physicists have already encountered the fact that K. Kodaira, too, noticed thirty years ago [see “Complex manifolds and deformation of complex structures” (1986; Zbl 0581.32012)] namely that the number of moduli is usually easier to compute than the number $$b_{1,2}.$$
Here, the authors aim to produce Calabi-Yau manifolds as anti-canonical divisors in the product of two surfaces with ample (or semi-ample) anti- canonical bundle. Representing such surfaces as blow-ups of $${\mathbb{C}}P^ 2$$, they obtain information about moduli and finally calculate Euler characteristics.
In fact, there are certain features which escape their attention. One (which may be of physical interest) is the relationship between the intersection form on the orthogonal complement of the anti-canonical divisor of a surface with ample anti-canonical bundle and the Dynkin diagrams of type $$E_ k$$. - The second one is the basic geometric fact that all cubics which pass through eight points in the plane pass through a ninth and consequently an anti-canonical divisor for the product of two planes blown up eight times is singular. - Some small Euler characteristics which seem to be attainable from their formula (3.1) do not therefore exist.
Reviewer: N.J.Hitchin

##### MSC:
 14J30 $$3$$-folds 81T99 Quantum field theory; related classical field theories 32G13 Complex-analytic moduli problems 14J15 Moduli, classification: analytic theory; relations with modular forms 32J15 Compact complex surfaces
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