| mots-clés : feuilletages holomorphes – fibré vectoriel décomposé – revêtement universel – courbes rationnelles – théorie de Mori – variété rationnellement connexe – variété uniréglée – images directes de faisceaux – positivité des faisceaux cohérents – espace fibré |
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| Two applications of positivity to the classification theory of complex projective varieties |
The subject of this thesis is to investigate two very natural questions in complex algebraic geometry.
The first question asks if the universal covering of a compact Kähler manifold with a split tangent bundle is a product of two manifolds. We will establish a structure theory for manifolds with a split tangent bundle and use covering families of rational curves to show the existence of a fibre space structure. A discussion of the fibre space structure allows to give an affirmative answer to the question for several classes of manifolds.
The second question asks if the positivity of a line bundle implies the positivity of the direct image of the adjoint line bundle under a flat projective morphism. We will see that the answer to this question depends on the positivity of the line bundle and its relation to the geometry of the morphism. We will show under a variety of conditions that the answer is to the affirmative. |
| mots-clés en anglais : holomorphic foliations – split vector bundle – universal covering – rational curves – Mori theory – rationally connected variety – uniruled variety – direct images of sheaves – positivity of coherent sheaves – fibre space |
