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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 $$k^+(X)=k(X)$$.

##### MSC:
 14E05 Rational and birational maps 14J35 $$4$$-folds
##### Keywords:
Kodaira dimension; birational geometry
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##### References:
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