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Threefolds of non-negative Kodaira dimension with sectional genus less than or equal to 15. (English) Zbl 0636.14014
From the text: “Let X be a smooth connected n-dimensional subvariety of complex projective space, $${\mathbb{P}}^ N$$. Assume that X has non-negative Kodaira dimension, i.e. that some positive power of the canonical bundle $$K_ X$$ has a non-trivial holomorphic section. In this paper we use the results of A. J. Sommese [J. Reine Angew. Math. 329, 16-41 (1981; Zbl 0509.14044)] to investigate what the numerical invariants of such X are under the assumption that the sectional genus g of X (i.e. the genus of $$X\cap {\mathbb{P}}^{N-n+1}$$ for a generic linear $${\mathbb{P}}^{N- n+1}\subseteq {\mathbb{P}}^ N)$$ is small. This problem is studied most thoroughly under the assumptions that $$n=3$$ and $$g\leq 15$$, but a number of partial results for arbitrary g and n are shown.
In § 0 we recall background material and especially the quoted results of Sommese. The latter results relate the surface $$S=X\cap {\mathbb{P}}^{N- n+2}$$ for a general linear $${\mathbb{P}}^{N-n+2}$$ to its minimal model S’. This lets us use the arguments from the theory of minimal surface to prove a number of results about the invariants of X. We also generalize a result of P. Griffiths and J. Harris [Ann. Math., II. Ser. 108, 461-505 (1978; Zbl 0423.14001)] by relaxing a hypothesis about a projective n-fold X from $$h^{n,O}(X)\neq O''$$ to “X is of non- negative Kodaira dimension”.
In § 1 we prove a number of general results. One example is the following theorem. Let X be an n-dimensional connected submanifold of $${\mathbb{P}}^ N$$ not contained in any hyperplane. Let d denote the degree of X in $${\mathbb{P}}^ N$$ and assume that $$K^ t_ X\approx {\mathcal O}_ X$$ for some $$t\neq 0$$. If $$d<n(N+1)$$ then the order of the fundamental group of X is finite and bounded by $$(n^ 2+n-2)/(n(N+1)-d)$$. In particular X is simply connected if $$d\leq n(N-(n-1)/2).''$$
In section 2 and 3 the threefolds in $${\mathbb{P}}^ 5$$ and $${\mathbb{P}}^ 6$$ are dealt with in several tables.
Reviewer: E.Stagnaro

##### MSC:
 14J10 Families, moduli, classification: algebraic theory 14J30 $$3$$-folds 14J40 $$n$$-folds ($$n>4$$)
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##### References:
 [1] H.F. Baker , Principles of Geometry , vol. V , Cambridge , 1933 . Zbl 0008.21906 | JFM 59.0620.04 · Zbl 0008.21906 · www.emis.de [2] W. Barth - C. Peters - A. Van De Ven , Compact Complex Surfaces , Springer-Verlag , Berlin - Heidelberg - New York ( 1984 ). MR 749574 | Zbl 0718.14023 · Zbl 0718.14023 [3] W. Barth , Larsen’s theorem on homotopy groups of projective manifolds Of small embedding codimension , Proc. Sympos. Pure Math. , 29 ( 1975 ), pp. 307 - 313 . MR 377123 | Zbl 0309.14017 · Zbl 0309.14017 [4] W. Fulton , Intersection Theory , Springer-Verlag , Berlin - Heidelberg - New York ( 1984 ). MR 732620 | Zbl 0541.14005 · Zbl 0541.14005 [5] P.A. Griffiths - J. Harris , Residues and zero cycles on algebraic varieties , Ann. of Math. , 108 ( 1978 ), pp. 461 - 505 . MR 512429 | Zbl 0423.14001 · Zbl 0423.14001 · doi:10.2307/1971184 [6] A. Beauville , Variétés kähleriennes dont la première classe de Chern est nulle , J. Differential Geom. , 18 ( 1983 ), pp. 755 - 782 . MR 730926 | Zbl 0537.53056 · Zbl 0537.53056 [7] R. Hartshorne , Ample subvarieties of algebraic varieties , Lecture Notes in Math. 156 , Berlin - Heidelberg - New York ( 1970 ). MR 282977 | Zbl 0208.48901 · Zbl 0208.48901 · doi:10.1007/BFb0067839 · eudml:203415 [8] R. Hartshorne , Algebraic Geometry , Springer-Verlag , Berlin - Heidelberg - New York ( 1977 ). MR 463157 | Zbl 0367.14001 · Zbl 0367.14001 [9] A. Holme - M. Schneider , A computer aided approach to codimension 2 subvarieties of Pn, n \succcurleq 6 , preprint. · Zbl 0581.14035 [10] Y. Kawamata , A generalization of Kodaira-Ramanujam’s vanishing theorem , Math. Ann. , 261 ( 1982 ), pp. 43 - 46 . MR 675204 | Zbl 0476.14007 · Zbl 0476.14007 · doi:10.1007/BF01456407 · eudml:182862 [11] M. Reid , Canonical 3-folds , Algebraic Geometry , ed. by A. Beauville, Siythoff & Noordholf , Netherlands ( 1980 ). MR 605348 · Zbl 0451.14014 [12] B. Shiffman - A.J. Sommese , Vanishing theorems on complex manifolds , to appear in Progr. Math. , Birkhauser . MR 782484 | Zbl 0578.32055 · Zbl 0578.32055 [13] A.J. Sommese , Complex subspaces of homogeneous complex manifolds.- II: Homotopy results , Nagoya Math. J. , 86 ( 1982 ), pp. 101 - 129 . Article | MR 661221 | Zbl 0497.32026 · Zbl 0497.32026 · minidml.mathdoc.fr [14] A.J. Sommese , Hyperplane sections of projective surfaces. - I : The adjunction mapping , Duke Math. J. , 46 ( 1979 ), pp. 377 - 401 . Article | MR 534057 | Zbl 0415.14019 · Zbl 0415.14019 · doi:10.1215/S0012-7094-79-04616-7 · minidml.mathdoc.fr [15] A.J. Sommese , On the minimality of hyperplane sections of projective threefolds , J. Reine Angew. Math. , 329 ( 1981 ), pp. 16 - 41 . MR 636441 | Zbl 0509.14044 · Zbl 0509.14044 · doi:10.1515/crll.1981.329.16 · eudml:183533 [16] E. Viehweg , Vanishing theorems , J. Reine Angew. Math. , 355 ( 1982 ),. pp. 1 - 8 . MR 667459 | Zbl 0485.32019 · Zbl 0485.32019 · doi:10.1515/crll.1982.335.1 · crelle:GDZPPN002199688 · eudml:152458
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