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On fiber products of rational elliptic surfaces with section. (English) Zbl 0631.14032
A large number of topologically distinct families of smooth, simply connected, projective 3-folds with $$K=0$$ having any predetermined Euler characteristic $$2n$$, $$-5<n<47$$, are constructed. The starting point for the construction are fiber products of relatively minimal, semi-stable, rational elliptic surfaces with section. Unlike most known families of 3-folds with trivial dualizing sheaf and ordinary double point singularities, one has in this case good control over the placement of large numbers of singularities, over the cohomology of the resolution, and over the delicate question of the existence of a small projective resolution. The construction yields numerous rigid varieties (which are roughly classified) and raises interesting questions concerning two dimensional Galois representations and algebraic cycles. Also intriguing geometric variations of Hodge structure with remarkably low Hodge numbers are constructed.
The impetus for this work was the desire of superstring theorists to find examples of Ricci flat, compact Kähler manifolds with small Euler characteristic, preferably $$6$$ or $$-6$$.

##### MSC:
 14J30 $$3$$-folds 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14C99 Cycles and subschemes 14J10 Families, moduli, classification: algebraic theory
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