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A new class of compact Kähler manifolds, called special, is defined: they are the ones having no surjective meromorphic map to an orbifold of general type. The special manifolds are in many respect higher-dimensional generalisations of rational and elliptic curves. For example, we show that being rationally connected or having vanishing Kodaira dimension(More)
Introduction Let X n be a complex projective n-dimensional manifold and˜X its universal cover. The Shafarevich conjecture asserts that˜X is holomorphically convex, i.e. admits a proper holomorphic map onto a Stein space. There are two extremal cases, namely that this map is constant, i.e. ˜ X is compact. This means that π 1 (X) is finite and not much can be(More)
Introduction Given a line bundle L on a projective manifold X, the Nakai-Moishezon criterion says that L is ample if and only if L s · Y > 0 for all s and all irreducible subvarieties Y ⊂ X of dimension s. Examples show that it is not sufficient to assume that L · C > 0 for all curves; line bundles with this property are called strictly nef. If however L =(More)
We introduce a birational invariant κ ++ (X|∆) ≥ κ(X|∆) for orbifold pairs (X|∆) by considering the ∆-saturated Kodaira dimensions of rank-one coherent subsheaves of Ω p X. The difference between these two invariants measures the birational unstability of Ω 1 (X|∆). Assuming conjectures of the LMMP, we obtain a simple geometric description of the invariant(More)
This work establishes a structure theorem for compact Kähler manifolds with semipositive anticanonical bundle. Up to finité etale cover, it is proved that such manifolds split holomorphically and isometrically as a product of Ricci flat varieties and of rationally connected manifolds. The proof is based on a characterization of rationally connected(More)