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Register a new userNormal bundles on the exceptional sets of simple small resolutions
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We study the normal bundles of the exceptional sets of isolated simple small singularities in the higher dimension when the Picard group of the exceptional set is $mathbb{Z}$ and the normal bundle of it has some good filtration. In particular, for the exceptional set is a projective space with the split normal bundle, we generalized Nakayama and Andos results to higher dimension. Moreover, we also generalize Laufers results of rationality and embedding dimension to higher dimension.
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We investigate maximal exceptional sequences of line bundles on (P^1)^3, i.e. those consisting of 2^r elements. For r=3 we show that they are always full, meaning that they generate the derived category. Everything is done in the discrete setup: Exceptional sequences of line bundles appear as special finite subsets s of the Picard group Z^r of (P^1)^r, and the question of generation is understood like a process of contamination of the whole Z^r out of an infectious seed s.
In this paper, we prove that the normal bundle of a general Brill-Noether space curve of degree $d$ and genus $g geq 2$ is stable if and only if $(d,g) otin { (5,2), (6,4) }$. When $gleq1$ and the characteristic of the ground field is zero, it is classical that the normal bundle is strictly semistable. We show that this fails in characteristic $2$ for all rational curves of even degree.
We compute the motivic Donaldson-Thomas theory of small crepant resolutions of toric Calabi-Yau 3-folds.
In this paper, we prove a singular version of the Donaldson-Uhlenbeck-Yau theorem over normal projective varieties and normal complex subvarieties of compact Kahler manifolds that are smooth outside a codimension three analytic subset. As a consequence, we deduce the polystability of (dual) tensor products of stable reflexive sheaves, and we give a new proof of the Bogomolov-Gieseker inequality over such spaces, along with a precise characterization of the case of equality. In addition, we improve several previously known algebro-geometric results on normalized tautological classes. We also study the limiting behavior of semistable bundles over a degenerating family of normal projective varieties. In the case of a family of stable bundles, we explain how the singular Hermitian-Yang-Mills connections obtained here fit into the degeneration picture. These can also be characterized from the algebro-geometric perspective. As an application, we apply the results to the degeneration of stable bundles through the deformation to projective cones, and we explain how our results are related to the Mehta-Ramanathan restriction theorem.
We compute the symplectic reductions for the action of Sp_2n on several copies of C^2n and for all coregular representations of Sl_2. If it exists we give at least one symplectic resolution for each example. In the case Sl_2 acting on sl_2+C^2 we obtain an explicit description of Fus and Namikawas example of two non-equivalent symplectic resolutions connected by a Mukai flop.
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