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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 \(-6\).

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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[1] Aspinwall, P.S., Green, B.R., Kirklin, K.H., Miron, P.J. Searching for three-generation Calabi-Yau Manifolds. Harvard University Theoretical Physics (Preprint)
[2] Beauville, A.: Some remarks on K?hler manifolds withc 1=0. In: Ueno, K. (ed.) Classification of Algebraic and Analytic Manifolds. Progress in Math.39. Boston: Birkh?user 1983
[3] Beauville, A.: Variet?s k?hl?rienes dont la premi?re classe de Chern est nulle. J Differ. Geom.18, 755-782 (1983) · Zbl 0537.53056
[4] Beauville, A.: Les familles stables de courbes elliptiques surP 1 admettant quatre fibres singuli?res. C.R. Acad. Sci. Paris294, 657-660 (1982) · Zbl 0504.14016
[5] Bogomolov, F.: On the decomposition of K?hler manifolds with trivial canonical class. Math. USSR Sbornik22, 580-583 (1974) · Zbl 0304.32016 · doi:10.1070/SM1974v022n04ABEH001706
[6] Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 4, Berlin Heidelberg New York: Springer 1984 · Zbl 0718.14023
[7] Coombes, K., Harbarter, D.: Hurwitz Families and Arithmetic Galois Groups. Duke Math. J.52, 821-839 (1985) · Zbl 0601.14023 · doi:10.1215/S0012-7094-85-05243-3
[8] Fulton, W.: Hurwitz schemes and irreducibility of moduli of algebraic curves. Ann. Math.90, 542-575 (1969) · Zbl 0194.21901 · doi:10.2307/1970748
[9] Fulton, W.: Intersection Theory, Berlin Heidelberg New York: Springer 1984 · Zbl 0541.14005
[10] Friedman, R.: Simultaneous resolution of threefold double points. Math. Ann.275 (1986) · Zbl 0576.14013
[11] Hartshorne, R.: Algebraic Geometry, Graduate Texts in Math.52, Berlin Heidelberg New York: Springer 1977 · Zbl 0367.14001
[12] Hirzebruch, F.: Threefolds withc 1=0, in 26. mathematische Arbeitstagung, Preprint series, Max-Plank-Institut f?r Mathematik, Bonn (1986, no. 26)
[13] H?bsch, T.: Calabi-Yau Manifolds-Motivations and Constructions. Commun. Math. Phys.108, 291-318 (1987) · Zbl 0602.53061 · doi:10.1007/BF01210616
[14] Hirzebruch, F.: Some examples of threefolds with trivial canonical bundle. Notes by J. Werner, Preprint series, Max-Plank-Institut f?r Mathematik, Bonn (1985), no. 58)
[15] Lejarraga, P.: Thesis Brandeis (1985)
[16] Lang, W.: An analog of the logarithmic transform in characteristicp. In: Carell, J., Geramita, A.V., Russell, P. (eds.) Can. Math. Soc. Conf. Proc.6, 1986
[17] Milnor, J.: Morse Theory. Ann. Math. Stud.51, Princeton University Press, Princeton (1963) · Zbl 0108.10401
[18] Miranda, H.P., Persson, U.: On extremal rational elliptic surfaces. Math. Z.193, 537-558 (1986) · Zbl 0652.14003 · doi:10.1007/BF01160474
[19] Ogg, A.: Cohomology of abelian, varieties over function fields. Ann. Math.76, 185-212 (1962) · Zbl 0121.38002 · doi:10.2307/1970272
[20] Safarevic, I.R.: Principal homogeneous spaces over a function field. Am. Math. Soc. Translations, series 2,37, 85-114 (1964)
[21] Schoen, C.: On the geometry of a special determinantal hypersurface associated to the Mumford-Horrocks vector bundle. J. Reine Angew. Math.364, 85-111 (1986) · Zbl 0568.14022 · doi:10.1515/crll.1986.364.85
[22] Schoen, C.: Complex multiplication cycles on elliptic modular threefolds. Duke Math. J.53 771-794 (1986) · Zbl 0623.14018 · doi:10.1215/S0012-7094-86-05343-3
[23] Shioda, T.: On elliptic modular surfaces. J. Math. Soc. Jpn24, 20-59 (1972) · Zbl 0226.14013 · doi:10.2969/jmsj/02410020
[24] Strominger, A., Witten, E.: New manifolds for superstring compactification. Commun. Math. Phys.101, 341-361 (1986) · doi:10.1007/BF01216094
[25] Ueno, K.: Classification Theory of Algebraic Varieties and Compact Complex Spaces (Lect. Notes Math., vol. 439) Berlin Heidelberg New York: Springer 1975 · Zbl 0299.14007
[26] Werner, J.: Thesis, Universit?t Bonn (In preparation)
[27] Yau, Shing-Tung: On the Ricci curvature of a compact Kaehler manifold and the complex Monge-Amp?re equation, I. Comm. Pure Apl. Math.31, 339-411 (1978) · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
[28] Yau, Shing-Tung: Compact three dimensional K?hler manifolds with zero Ricci curvature. With an appendix by S.-T. Yau and G. Tian. (Preprint) · Zbl 0643.53050
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