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We consider a uniform $r$-bundle $E$ on a complex rational homogeneous space $X$ %over complex number field $mathbb{C}$ and show that if $E$ is poly-uniform with respect to all the special families of lines and the rank $r$ is less than or equal to some number that depends only on $X$, then $E$ is either a direct sum of line bundles or $delta_i$-unstable for some $delta_i$. So we partially answer a problem posted by Mu~{n}oz-Occhetta-Sol{a} Conde. In particular, if $X$ is a generalized Grassmannian $mathcal{G}$ and the rank $r$ is less than or equal to some number that depends only on $X$, then $E$ splits as a direct sum of line bundles. We improve the main theorem of Mu~{n}oz-Occhetta-Sol{a} Conde when $X$ is a generalized Grassmannian by considering the Chow rings. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert-M{u}lich-Barth theorem on rational homogeneous spaces.
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Given a rational homogeneous variety G/P where G is complex simple and of type ADE, we prove that all tangent bundles T_{G/P} are simple, meaning that their only endomorphisms are scalar multiples of the identity. This result combined with Hitchin-Kobayashi correspondence implies stability of these tangent bundles with respect to the anticanonical polarization. Our main tool is the equivalence of categories between homogeneous vector bundles on G/P and finite dimensional representations of a given quiver with relations.
We prove that for an irreducible representation $tau:GL(n)to GL(W)$, the associated homogeneous ${bf P}_k^n$-vector bundle $W_{tau}$ is strongly semistable when restricted to any smooth quadric or to any smooth cubic in ${bf P}_k^n$, where $k$ is an algebraically closed field of characteristic $ eq 2,3$ respectively. In particular $W_{tau}$ is semistable when restricted to general hypersurfaces of degree $geq 2$ and is strongly semistable when restricted to the $k$-generic hypersurface of degree $geq 2$.
This paper investigates the cohomological property of vector bundles on biprojective space. We will give a criterion for a vector bundle to be isomorphic to the tensor product of pullbacks of exterior products of differential sheaves.
In char $k = p >0$, A. Langer proved a strong restriction theorem (in the style of H. Flenner) for semistable sheaves to a very general hypersurface of degree $d$, on certain varieties, with the condition that `char $k > d$. He remarked that to remove this condition, it is enough to answer either of the following questions affirmatively: {it For the syzygy bundle $sV_d$ of ${mathcal O}(d)$, is $sV_d$ semistable for arbitrary $n, d$ and $p = {char} k$?, or is there a good estimate on $mu_{max}(sV_d^*)$?} Here we prove that (1) the bundle $sV_d$ is semistable, for a certain infinite set of integers $dgeq 0$, and (2) for arbitrary $d$, there is a good enough estimate on $mu_{max}(sV_d^*)$ in terms of $d$ and $n$. In particular one obtains Langers theorem, in arbitrary characeristic.
In this article, we solve the problem of constructing moduli spaces of semistable principal bundles (and singul
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