zbMATH — the first resource for mathematics

Subsheaves of the cotangent bundle. (English) Zbl 1108.14009
Let \(X\) be a smooth complex projective variety and let \(L\) be a line bundle on \(X\); the Kodaira-Iitaka dimension kod\((X,L)\) of \((X,L)\) is the maximum dimension of the image of the rational map \(\varphi_n\), defined by the linear system \(| nL| \); we set kod\((X,L)=-\infty\) if \(| nL| = \emptyset\) for every \(n\).
The Kodaira dimension \(k(X)\) of \(X\) is the Kodaira-Iitaka dimension of the pair \((X,K_X)\), where \(K_X\) is the canonical bundle of \(X\). A related invariant \(k^+(X)\) was defined by F. Campana [J. Algebr. Geom. 4, No. 3, 487–502 (1995; Zbl 0845.32027)] as the maximum of the Kodaira-Iitaka dimensions of the pairs \((X, \det {\mathcal F})\), with \({\mathcal F}\) a coherent subsheaf \({\mathcal F} \subset \Omega^p_X\) for some \(p>0\).
It is clear from the definitions that \(k^+(X) \geq k(X)\) and it is not difficult to show that equality does not hold in general; however it is natural to conjecture that the difference is due to the presence of rational curves.
More precisely the conjecture states that \(k^+(X)\) equals the Kodaira dimension of the rational quotient of \(X\), that is the target of the maximal rationally connected fibration of \(X\). Campana showed that this conjecture follows from the minimal model program and the conjecture which claims that a smooth algebraic variety is uniruled if and only if its Kodaira dimension is negative.
In the paper under review another invariant, \(k_1^+(X)\), is considered, taking in the definition of \(k^+\) only line bundles \(L \subset \Omega^p_X\); the author proves that, if \(X\) is a variety of dimension not greater than four with non negative Kodaira dimension, then \(k_1^+(X)=k(X)\) (Theorem 1.1).
This theorem is a consequence of results on the Kodaira-Iitaka dimension of pairs \((X,K_X+L)\) (Theorems 1.2 and 1.3), and of pairs \((X,L)\) such that \(k(X)=0\) (Theorem 1.4) and of a positivity result (Theorem 2.1), which states that, on a non-uniruled variety a line bundle which is a quotient of \(\Omega^1_X\) is pseudo-effective.
F. Campana andTh. Peternell have recently proved [Geometric stability of the cotangent bundle and the universal cover of a projective manifold, math.AG/0405093] the stronger result that if \(X\) is a variety of dimension not greater than four with non negative Kodaira dimension, then \(k^+(X)=k(X)\).

14E05 Rational and birational maps
14J35 \(4\)-folds
Full Text: DOI
[1] F. Bogomolov: “Holomorphic Tensors and Vector Bundles on Projective Varieties”, Math. USSR Izv., Vol. 13, (1979), pp. 499-555. http://dx.doi.org/10.1070/IM1979v013n03ABEH002076 · Zbl 0439.14002
[2] S. Boucksom, J.P. Demailly, M. Paun and T. Peternell: “The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension”, math.AG/0405285. · Zbl 1267.32017
[3] F. Campana: “Réducation d’Albanèse d’un morphisme propre et faiblement kählérien. II. Groupes d’automorphismes relatifs”, Compositio Math., Vol. 54(3), (1985), pp. 399-416. · Zbl 0609.32008
[4] F. Campana: “Connexité rationelle des variétés de Fano”, Ann. Sci. E.N.S., Vol. 25, (1992), pp. 539-545. · Zbl 0783.14022
[5] F. Campana: “Fundamental Group and Positivity of Cotangent Bundles of Compact Kähler Manifolds”, J. Algebraic Geom., Vol. 4, (1995), pp. 487-502. · Zbl 0845.32027
[6] F. Campana: “Orbifolds, Special Varieties and Classification Theory”, Ann. Inst. Fourier, Grenoble, Vol. 54(3), (2004), pp. 499-630. · Zbl 1062.14014
[7] F. Campana and T. Peternell: “Geometric Stability of the Cotangent Bundle and the Universal Cover of a Projective Manifold”, math.AG/0405093. · Zbl 1218.14030
[8] J.P. Demailly, T. Peternell and M. Schneider: “Pseudo-effective Line Bundles on compact Kähler Manifolds”, Intern. J. Math., Vol. 12(6), (2001), pp. 689-741. · Zbl 1111.32302
[9] T. Ekedahl: T. Ekedahl: “Foliations and inseparable morphisms” (english), In: Algebraic geometry, Proc. Summer Res. Inst., (Brunswick/Maine 1985), Proc. Symp. Pure Math., Vol. 46(2), Amer. Math. Soc., Providence, RI, 1987, pp. 139-149.
[10] T. Graber, J. Harris and J. Starr: “Families of rationally connected varieties”, J. Amer. Math. Soc., Vol. 16, (2003), pp. 57-67. http://dx.doi.org/10.1090/S0894-0347-02-00402-2 · Zbl 1092.14063
[11] P. Griffiths: “Periods of Integrals on Algebraic Manifolds III”, Publ. Math. I.H.E.S., Vol. 38, (1970), pp. 125-180. · Zbl 0212.53503
[12] S. Iitaka: Algebraic Geometry, Graduate Texts in Math., Vol. 76, Springer, 1982. · Zbl 0491.14006
[13] Y. Kawamata: “Characterization of Abelian Varieties”, Comp. Math., Vol. 43, (1981), pp. 253-276. · Zbl 0471.14022
[14] G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat: Toroidal Embeddings I, Lectures Notes in Math., Vol. 339, Springer Verlag, 1973. · Zbl 0271.14017
[15] J. Kollár: “Higher Direct Images of Dualizing Sheaves II”, Ann. Math., Vol. 124, (1986), pp. 171-202. · Zbl 0605.14014
[16] J. Kollár: “Nonrational Hypersurfaces”, J. Am. Math. Soc., Vol. 8(1), (1995), pp. 241-249. · Zbl 0839.14031
[17] J. Kollár: Shafarevich maps and automorphic forms, Princeton University Press, 1995. · Zbl 0871.14015
[18] K. Matsuki, Introduction to the Mori program, Springer-Verlag, New York, 2002. · Zbl 0988.14007
[19] Y. Miyaoka: “The Chern classes and Kodaira dimension of a minimal variety”, In: Proc. Sympos. Alg. Geom., Sendai 1985, Adv. Stud. Pure Math, Vol. 10, Kynokuniya, Tokyo, 1985, pp. 449-476.
[20] S. Mori: “Classification of higher-dimensional varieties”, In: Algebraic geometry, Bowdoin 1985 (Brunswick/Maine 1985), Proc. Symp. Pure Math., Vol. 46(1), Amer. Math. Soc., Providence, RI, 1987, pp. 269-331.
[21] Y. Namikawa: “On deformations of Calabi-Yau 3-folds with terminal singularities”, Topology, Vol. 33(3), (1994), pp. 429-446. http://dx.doi.org/10.1016/0040-9383(94)90021-3 · Zbl 0813.14004
[22] J.H.M. Steenbrink: Mixed Hodge Structure on the Vanishing Cohomology, Real and Complex Singularities, Nordic Summer School, Oslo, 1976, pp. 525-563.
[23] K. Ueno: Classification Theory of Algebraic Varieties and Compact Complex Spaces, Lectures Notes in Math., Vol. 439, Springer Verlag, 1975. · Zbl 0299.14007
[24] E. Viehweg: “Die Additivität der Kodaira Dimension für projektive Faserräume über Varietäten des allgemeinen Typs”, J. Reine Angew. Math., Vol. 330, (1982), pp. 132-142.
[25] E. Viehweg and K. Zuo: “On the isotriviality of families of projective manifolds over curves Complex Spaces”, J. Alg. Geom., Vol. 10, (2001), pp. 781-799. · Zbl 1079.14503
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.