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Dualité homologique projective et résolutions catégoriques des singularités

Abstract : Let X be an algebraic variety with Gorenstein rational singularities. A crepant resolution of X is often considered to be a minimal resolution of singularities for X. Unfortunately, crepant resolution of singularities are very rare. For instance, determinantal varieties of skew-symmetric matrices never admit crepant resolution of singularities. In this thesis, we discuss various notions of categorical crepant resolution of singularities as defined by Alexander Kuznetsov. Conjecturally, these resolutions are minimal from the categorical point of view. We introduce the notion of wonderful resolution of singularities and we prove that a variety endowed with such a resolution admits a weakly crepant resolution of singularities. As a corollary, we prove that all determinantal varieties (square, as well as symmetric and skew-symmetric) admit weakly crepant resolution of singularities. Finally, we study some quartics hypersurfaces which come from the Tits-Freudenthal magic square. Though they do no admit any wonderful resolution of singularities, we are still able to prove that they have a weakly crepant resolution of singularities. This last result should be of interest in order to construct homological projective duals for some symplectic Grassmannians over the composition algebras.
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Roland Abuaf. Dualité homologique projective et résolutions catégoriques des singularités. Mathématiques générales [math.GM]. Université de Grenoble, 2013. Français. ⟨NNT : 2013GRENM057⟩. ⟨tel-01167557⟩

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