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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 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
In this paper we study the existence of sections of universal bundles on rational homogeneous varieties -- called nestings -- classifying them completely in the case in which the Lie algebra of the automorphism group of the variety is simple of class
In this paper, we study syzygies of rational homogeneous varieties. We extend Manivels result that a $p$-th power of an ample line bundle on a flag variety satisfies Propery $(N_p)$ to many rational homogeneous varieties of type $B$, $C$, $D$, and $G_2$.
We prove that the kernel bundle of the evaluation morphism of global sections, namely the syzygy bundle, of a sufficiently ample line bundle on a smooth projective variety is slope stable with respect to any polarization. This settles a conjecture of Ein-Lazarsfeld-Mustopa.
Complete intersections inside rational homogeneous varieties provide interesting examples of Fano manifolds. For example, if $X = cap_{i=1}^r D_i subset G/P$ is a general complete intersection of $r$ ample divisors such that $K_{G/P}^* otimes mathcal