# zbMATH — the first resource for mathematics

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
##### Keywords:
regular 3-folds of general type; birational map
Full Text:
##### References:
  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  F. A. Bogomolov, Holomorphic tensors and vector bundles on projective varieties , Math. USSR-Izv. 13 (1979), 499-555. · Zbl 0439.14002  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  M. Hanamura, Pluricanonical maps of minimal $$3$$-folds , Proc. Japan Acad. 61 (1985), no. 4, 116-118. · Zbl 0598.14032  J. Kollár, Higher direct images of dualizing sheaves, I , Ann. of Math. (2) 123 (1986), no. 1, 11-42. JSTOR: · Zbl 0598.14015  T. Mabuchi, Invariant $$\beta$$ and uniruled threefolds , J. Math. Kyoto Univ. 22 (1982), no. 3, 503-554. · Zbl 0511.32016  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  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  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  S. Zucker, Remarks on a theorem of Fujita , J. Math. Soc. Japan 34 (1982), no. 1, 47-54. · Zbl 0503.14002
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.