The existence of elliptic fibre space structures on Calabi-Yau threefolds. II.

*(English)*Zbl 0923.14027[For part I of this paper see P. M. H. Wilson, Math. Ann. 300, No. 4, 693-703 (1994; Zbl 0819.14015).]

The existence of an elliptic fibre space (e.f.s.) structure on the smooth Calabi-Yau threefold \(X\) implies:

\((*)\) There exists a nef divisor \(D\) on \(X\) such that \(D^3 = 0\), \(D^2 \not\equiv 0\) and \(D.c_2 \geq 0\).

In Invent. Math. 107, No. 3, 561-583 (1992; Zbl 0766.14035); correction ibid. 114, No. 1, 231-233 (1993; Zbl 0794.14016) and in part I of this paper (loc. cit.), P. M. H. Wilson has shown that if \(D.c_2 > 0\) then the necessary condition \((*)\) is also sufficient for the existence of a (determined by \(D\)) e.f.s. structure on \(X\). This paper continues the study of the case \(D.c_2 = 0\) started in part I of this paper (loc. cit.). If \({\pi}_1(X)\) is infinite then \(X\) is an étale quotient of either an abelian threefold or the product of a K3 surface and an elliptic curve; and in the remaining case \(D.c_2 = 0\) and \({\pi}_1(X)\) finite the question is to find when (\(\ast\)) is sufficient for the existence of an e.f.s. structure on \(X\). The main proposition of the present paper yields the following answer:

Corollary 1. Let \(X\) be a Calabi-Yau threefold with \({\pi}_1(X)\) finite and \(D\) a nef integral divisor on \(X\) as in (\(\ast\)) such that \(D.c_2 = 0\). Then \(D\) determines an e.f.s. structure on \(X\) iff \(e(X) > 2r\) where \(e(X)\) is the Euler characteristic of \(X\) and \(r\) is the number of rational surfaces \(E \subset X\) such that \(D| E \equiv 0\).

In the proof, there are used results from part I of this paper (loc. cit.), the classification results by K. Oguiso [Math. Z. 221, No. 3, 437-448 (1996; Zbl 0852.14012) and Doc. Math., J. DMV 1, 417-447 (1996; Zbl 0864.14020)], and the existence theorems of S. S. Roan for crepant resolutions of 3-dimensional Gorenstein orbifolds [see Topology 35, No. 2, 489-508 (1996; Zbl 0872.14034)].

Corollary 1 and results from the author’s two papers cited above imply that e.f.s. structures on Calabi-Yau threefolds with finite \({\pi}_1\) are stable under small deformations of the complex structure (see corollary 2).

The existence of an elliptic fibre space (e.f.s.) structure on the smooth Calabi-Yau threefold \(X\) implies:

\((*)\) There exists a nef divisor \(D\) on \(X\) such that \(D^3 = 0\), \(D^2 \not\equiv 0\) and \(D.c_2 \geq 0\).

In Invent. Math. 107, No. 3, 561-583 (1992; Zbl 0766.14035); correction ibid. 114, No. 1, 231-233 (1993; Zbl 0794.14016) and in part I of this paper (loc. cit.), P. M. H. Wilson has shown that if \(D.c_2 > 0\) then the necessary condition \((*)\) is also sufficient for the existence of a (determined by \(D\)) e.f.s. structure on \(X\). This paper continues the study of the case \(D.c_2 = 0\) started in part I of this paper (loc. cit.). If \({\pi}_1(X)\) is infinite then \(X\) is an étale quotient of either an abelian threefold or the product of a K3 surface and an elliptic curve; and in the remaining case \(D.c_2 = 0\) and \({\pi}_1(X)\) finite the question is to find when (\(\ast\)) is sufficient for the existence of an e.f.s. structure on \(X\). The main proposition of the present paper yields the following answer:

Corollary 1. Let \(X\) be a Calabi-Yau threefold with \({\pi}_1(X)\) finite and \(D\) a nef integral divisor on \(X\) as in (\(\ast\)) such that \(D.c_2 = 0\). Then \(D\) determines an e.f.s. structure on \(X\) iff \(e(X) > 2r\) where \(e(X)\) is the Euler characteristic of \(X\) and \(r\) is the number of rational surfaces \(E \subset X\) such that \(D| E \equiv 0\).

In the proof, there are used results from part I of this paper (loc. cit.), the classification results by K. Oguiso [Math. Z. 221, No. 3, 437-448 (1996; Zbl 0852.14012) and Doc. Math., J. DMV 1, 417-447 (1996; Zbl 0864.14020)], and the existence theorems of S. S. Roan for crepant resolutions of 3-dimensional Gorenstein orbifolds [see Topology 35, No. 2, 489-508 (1996; Zbl 0872.14034)].

Corollary 1 and results from the author’s two papers cited above imply that e.f.s. structures on Calabi-Yau threefolds with finite \({\pi}_1\) are stable under small deformations of the complex structure (see corollary 2).

Reviewer: A.Iliev (Sofia)