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*Please note the time change*

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The topic of this talk is to investigate when the tangent sheaf of a complex variety splits as a direct sum of subsheaves.Ìý

We will give applications to the structure of (mildly singular) Fano varieties and to algebraically integrable foliations with zero first Chern class.Ìý

This is partly joint work with H. Guenancia and M. Paun.