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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 construct a compactification $M^{mu ss}$ of the Uhlenbeck-Donaldson type for the moduli space of slope stable framed bundles. This is a kind of a moduli space of slope semistable framed sheaves. We show that there exists a projective morphism $gam
We use the toric degeneration of Bott-Samelson varieties and the description of cohomolgy of line bundles on toric varieties to deduce vanishings results for the cohomology of lines bundles on Bott-Samelson varieties.
{bf Construction.} For a dominating polynomial mapping {$F: K^nto K^l$} with an isolated critical value at 0 ($K$ an algebraically closed field of characteristic zero) we construct a closed {it bundle} $G_F subset T^{*}K^n $. We restrict $ G_F $ over
We investigate degenerations of syzygy bundles on plane curves over $p$-adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the speci
We study vector bundles on flag varieties over an algebraically closed field $k$. In the first part, we suppose $G=G_k(d,n)$ $(2le dleq n-d)$ to be the Grassmannian manifold parameterizing linear subspaces of dimension $d$ in $k^n$, where $k$ is an a