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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
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