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Global 2-forms on regular 3-folds of general type. (English) Zbl 0838.14032
Let \(V\) be a smooth projective variety of \(\mathbb{C}\)-dimension = 3. The author is motivated by the following problem: Assume that \(V\) is of general type. Then find some universal integer \(N\) such that \(|NK |\) defines a birational map where \(K\) is the canonical bundle of \(V\). For irregular 3-folds \(V\) of general type, the problem was settled by Kollár. Therefore, in this paper, the author concentrates on regular 3-folds \(V\) of general type, i.e. \(h^1 (V,{\mathcal O}_V) = 0\). Let \({\mathcal H}^2 : = H^0 (V, \Omega^2_V) \otimes_\mathbb{C} {\mathcal O}_V \subset \Omega^2_V\) be the subsheaf of \(\Omega^2_V\) spanned by \(H^0 (V, \Omega^2_V)\). It is clear that \[ \text{rank } {\mathcal H}^2 \leq 3. \] The main result of this paper is the following theorem:
The above problem admits a positive answer for regular 3-folds \(V\) of general type, provided rank \({\mathcal H}^2 = 1\) or 3. Some discussion when rank \({\mathcal H}^2 = 2\) is carried out.

MSC:
14J30 \(3\)-folds
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[1] W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. · Zbl 0718.14023
[2] F. A. Bogomolov, Holomorphic tensors and vector bundles on projective varieties , Math. USSR-Izv. 13 (1979), 499-555. · Zbl 0439.14002
[3] A. R. Fletcher, Contributions to Riemann-Roch on projective \(3\)-folds with only canonical singularities and applications , Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 221-231. · Zbl 0662.14026
[4] M. Hanamura, Pluricanonical maps of minimal \(3\)-folds , Proc. Japan Acad. 61 (1985), no. 4, 116-118. · Zbl 0598.14032
[5] J. Kollár, Higher direct images of dualizing sheaves, I , Ann. of Math. (2) 123 (1986), no. 1, 11-42. JSTOR: · Zbl 0598.14015
[6] T. Mabuchi, Invariant \(\beta\) and uniruled threefolds , J. Math. Kyoto Univ. 22 (1982), no. 3, 503-554. · Zbl 0511.32016
[7] S. Mori, Classification of higher-dimensional varieties , Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., Part 1, vol. 46, Amer. Math. Soc., Providence, 1987, pp. 269-331. · Zbl 0656.14022
[8] M. Reid, Young person’s guide to canonical singularities , Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., Part 1, vol. 46, Amer. Math. Soc., Providence, 1987, pp. 345-414. · Zbl 0634.14003
[9] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces , Algebraic Varieties and Analytic Varieties (Tokyo, 1981), Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 329-353. · Zbl 0513.14019
[10] S. Zucker, Remarks on a theorem of Fujita , J. Math. Soc. Japan 34 (1982), no. 1, 47-54. · Zbl 0503.14002
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