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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
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