Moishezon twistor spaces on \(4\mathbb{CP}^{2}\). (English) Zbl 1317.32038
Let \(M\) be a four-dimensional compact real oriented manifold which admits a self-dual conformal structure. It is a classical construction to associate to \(M\) a three-dimensional compact complex manifold \(Z\), the so-called twistor space. It was shown by Campana that \(Z\) can only be Moishezon when \(M\) is either \(S^4\) or \(n\mathbb{CP}^2\), the connected sum of \(n\) copies of the projective plane.
Classification results for Moishezon twistor spaces \(Z\) are known in the case \(M=S^4\) by N. H. Kuiper [Ann. Math. (2) 50, 916–924 (1949; Zbl 0041.09303)], in the cases \(M=\mathbb{CP}^2\) and \(M=2\mathbb{CP}^2\) by Y. S. Poon [J. Differ. Geom. 24, 97–132 (1986; Zbl 0583.53054)], and in the case \(M=3\mathbb{CP}^2\) by Y.S. Poon. [J. Differ. Geom. 36, No. 2, 451–491 (1992; Zbl 0742.53024)], B. Kreußler and H. Kurke [Compos. Math. 82, No. 1, 25–55 (1992; Zbl 0766.53049)] and {C. LeBrun} [J. Differ. Geom. 34, No. 1, 223–253 (1991; Zbl 0725.53067)].
In the present article, the author continues the work on the classification of Moishezon twistor spaces by treating the case \(M=4\mathbb{CP}^2\). The classification in this case is achieved by considering the anticanonical map \(\Phi\) of \(Z\). It is shown that only three possibilities can occur: \(\Phi\) is either birational, or a rational double cover (\(Z\) is of double solid type), or the general fiber of \(\Phi\) is a smooth rational curve.
The author studies these possibilities in great detail: In each possible case, he gives an explicit description of the image of \(\Phi\). If \(Z\) is of double solid type, he furthermore gives explicit equations for the branch divisor of \(\Phi\).
The article also computes the dimension of the moduli spaces of Moishezon twistor spaces on \(4\mathbb{CP}^{2}\).
Classification results for Moishezon twistor spaces \(Z\) are known in the case \(M=S^4\) by N. H. Kuiper [Ann. Math. (2) 50, 916–924 (1949; Zbl 0041.09303)], in the cases \(M=\mathbb{CP}^2\) and \(M=2\mathbb{CP}^2\) by Y. S. Poon [J. Differ. Geom. 24, 97–132 (1986; Zbl 0583.53054)], and in the case \(M=3\mathbb{CP}^2\) by Y.S. Poon. [J. Differ. Geom. 36, No. 2, 451–491 (1992; Zbl 0742.53024)], B. Kreußler and H. Kurke [Compos. Math. 82, No. 1, 25–55 (1992; Zbl 0766.53049)] and {C. LeBrun} [J. Differ. Geom. 34, No. 1, 223–253 (1991; Zbl 0725.53067)].
In the present article, the author continues the work on the classification of Moishezon twistor spaces by treating the case \(M=4\mathbb{CP}^2\). The classification in this case is achieved by considering the anticanonical map \(\Phi\) of \(Z\). It is shown that only three possibilities can occur: \(\Phi\) is either birational, or a rational double cover (\(Z\) is of double solid type), or the general fiber of \(\Phi\) is a smooth rational curve.
The author studies these possibilities in great detail: In each possible case, he gives an explicit description of the image of \(\Phi\). If \(Z\) is of double solid type, he furthermore gives explicit equations for the branch divisor of \(\Phi\).
The article also computes the dimension of the moduli spaces of Moishezon twistor spaces on \(4\mathbb{CP}^{2}\).
Reviewer: Florian Schrack (Bayreuth)
MSC:
| 32L25 | Twistor theory, double fibrations (complex-analytic aspects) |
| 32J17 | Compact complex \(3\)-folds |
| 32Q57 | Classification theorems for complex manifolds |
| 32G05 | Deformations of complex structures |
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