# zbMATH — the first resource for mathematics

A finiteness theorem for elliptic Calabi-Yau threefolds. (English) Zbl 0838.14033
For the purposes of this paper:
We define a Calabi-Yau threefold to be an algebraic threefold $$X$$ over the field of complex numbers which is birationally equivalent to a threefold $$Y$$ with $$\mathbb{Q}$$-factorial terminal singularities, $$K_Y = 0$$, and $$\chi ({\mathcal O}_Y) = h^1 (Y, {\mathcal O}_Y) = h^2 (Y, {\mathcal O}_Y) = 0$$. In this case, we call $$Y$$ a minimal Calabi-Yau threefold. — Calabi-Yau threefolds can be thought of as a generalization of K3 surfaces. If one considers possibly nonalgebraic K3 surfaces, they are all Kähler, and one obtains an irreducible 20-dimensional moduli space. Thus in particular, all K3 surfaces are homeomorphic. If one restricts one’s attention to algebraic K3 surfaces, the situation becomes much more complicated: one obtains a countable number of 19-dimensional components in the 20-dimensional space of Kähler K3s. In the case of Calabi-Yau threefolds, however, any deformation of an algebraic Calabi-Yau is algebraic, so it makes sense to restrict one’s attention to algebraic Calabi-Yaus. — Given this, one could ask whether there are a finite number of topological types of algebraic Calabi-Yau threefolds. (This is known not to be true if one allows non-Kähler Calabi-Yau threefolds.) A stronger question to ask would be whether there are a finite number of families of algebraic minimal Calabi-Yau threefolds. Our main theorem is the following.
Theorem 0.1. There exists a finite number of triples $$({\mathcal X}_i, {\mathcal S}_i, {\mathcal T}_i)$$ of quasi-projective varieties with maps $\begin{matrix} {\mathcal X}_i \\ \downarrow f_i & & \overset {\pi_i} {\searrow} \\ {\mathcal S}_i & @>g_i>> & & {\mathcal T}_i \end{matrix}$ where $$\pi_i$$ is smooth and proper with each fibre a Calabi-Yau threefold, $$f_i$$ proper with generic fibre an elliptic curve, and $$g_i$$ smooth and proper with each fibre a rational surface, such that for any elliptic fibration $$X \to S$$ wtih $$X$$ Calabi-Yau and $$S$$ rational there exists a $$t \in {\mathcal T}_i$$ for some $$i$$ such that there are birational maps $$X \to ({\mathcal X}_i)_t$$, $$S\to ({\mathcal S}_i)_t$$ with the following diagram commutative: $\begin{matrix} X & \to & ({\mathcal X}_i)_t \\ \downarrow & & \downarrow \\ S & \to & ({\mathcal S}_i)_t. \end{matrix} .$

##### MSC:
 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations 14J30 $$3$$-folds
Full Text:
##### References:
 [1] P. Deligne, Courbes elliptiques: formulaire d’après J. Tate , Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1975, 53-73. Lecture Notes in Math., Vol. 476. · Zbl 1214.11075 · doi:10.1007/BFb0097583 [2] P. Deligne, J.-F. Boutot, L. Illusie, and J.-L. Verdier, Cohomologie Étale , Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin, 1977. · Zbl 0345.00010 [3] I. Dolgachev and M. Gross, Elliptic threefolds. I. Ogg-Shafarevich theory , J. Algebraic Geom. 3 (1994), no. 1, 39-80. · Zbl 0803.14021 [4] A. Grassi, On minimal models of elliptic threefolds , Math. Ann. 290 (1991), no. 2, 287-301. · Zbl 0719.14006 · doi:10.1007/BF01459246 · eudml:164819 [5] A. Grassi, The singularities of the parameter surface of a minimal elliptic threefold , Internat. J. Math. 4 (1993), no. 2, 203-230. · Zbl 0802.14020 · doi:10.1142/S0129167X93000121 [6] A. Grassi, Log contractions and equidimensional models of elliptic threefolds , MSRI preprint #055-93, 1993. · Zbl 0840.14026 [7] M. Gross, Elliptic three-folds II: Multiple fibres , MSRI preprint #018-93, 1992. JSTOR: · Zbl 0884.14017 · doi:10.1090/S0002-9947-97-01845-X · links.jstor.org [8] 1 A. Grothendieck, Le groupe de Brauer. I. Algèbres d’Azumaya et interprétations diverses , Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 46-66. · Zbl 0193.21503 [9] 2 A. Grothendieck, Le groupe de Brauer. II. Théorie cohomologique , Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 67-87. · Zbl 0198.25803 [10] 3 A. Grothendieck, Le groupe de Brauer. III. Exemples et compléments , Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 88-188. · Zbl 0198.25901 [11] A. Grothendieck and J. Dieudonné, Éléments de Géométrie Algébrique, I , Gtundlehren Math. Wiss., vol. 166, Springer-Verlag, Berlin, 1971. · Zbl 0203.23301 [12] B. Hunt, A bound on the Euler number for certain Calabi-Yau $$3$$-folds , J. Reine Angew. Math. 411 (1990), 137-170. · Zbl 0745.14016 · doi:10.1515/crll.1990.411.137 · crelle:GDZPPN002207990 · eudml:153271 [13] Y. Kawamata, Kodaira dimension of certain algebraic fiber spaces , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), no. 1, 1-24. · Zbl 0516.14026 [14] J. Kollár and S. Mori, Classification of three-dimensional flips , J. Amer. Math. Soc. 5 (1992), no. 3, 533-703. JSTOR: · Zbl 0773.14004 · doi:10.2307/2152704 · links.jstor.org [15] J. Kollàr, et al., Flips and abundance for algebraic threefolds , · Zbl 1246.01050 [16] J. S. Milne, Étale cohomology , Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. · Zbl 0433.14012 [17] R. Miranda, Smooth models for elliptic threefolds , The birational geometry of degenerations (Cambridge, Mass., 1981), Progr. Math., vol. 29, Birkhäuser Boston, Mass., 1983, pp. 85-133. · Zbl 0583.14014 [18] R. Miranda and U. Persson, On extremal rational elliptic surfaces , Math. Z. 193 (1986), no. 4, 537-558. · Zbl 0652.14003 · doi:10.1007/BF01160474 · eudml:173804 [19] M. Miyanishi and D.-Q. Zhang, Gorenstein log del Pezzo surfaces of rank one , J. Algebra 118 (1988), no. 1, 63-84. · Zbl 0664.14019 · doi:10.1016/0021-8693(88)90048-8 [20] D. Mumford and K. Suominen, Introduction to the theory of muldi , Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer-School in Math.), Wolters-Noordhoff, Groningen, 1972, pp. 171-222. · Zbl 0242.14004 [21] N. Nakayama, On Weierstrass models , Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo, 1988, pp. 405-431. · Zbl 0699.14049 [22] N. Nakayama, Local structure of an elliptic fibration , preprint, Univ. of Tokyo, 1991. · Zbl 1059.14015 [23] K. Oguiso, On algebraic fiber space structures on a Calabi-Yau threefold , preprint, 1992. · Zbl 1107.14013 [24] F. Sakai, The structure of normal surfaces , Duke Math. J. 52 (1985), no. 3, 627-648. · Zbl 0699.14049 · doi:10.1215/S0012-7094-85-05233-0 [25] C. Schoen, On fiber products of rational elliptic surfaces with section , Math. Z. 197 (1988), no. 2, 177-199. · Zbl 0631.14032 · doi:10.1007/BF01215188 · eudml:183725
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.