Sections of homogeneous vector bundles

Abstract

In this work we give a method for computing sections of homogeneous vector bundles on any rational homogeneous variety G/P of type ADE. 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. Our result generalizes the work of Ottaviani and Rubei (2006) [OR06].

Double spinor Calabi-Yau varieties

“…Moreover, with the previous notations, we have det(U ) = det(L) ⊗ U /L, det(U ) = det(L) ⊗ U /L, and the quadratic form induces a natural duality between U /L and U /L. From this one easily deduces that p * (1). And then…”

“…The half-spin representations are self dual when n is even, and dual one to the other when n is odd. It follows from the usual Bruhat decomposition that the Chow ring of S ± is free, and the dimension of its k-dimensional component is equal to the number of strict partitions of k with parts smaller than n. In particular the Picard group has rank one, and L is a generator; we therefore denote L = O S ± (1). We also let U be the rank n vector bundle obtained by restricting the tautological bundle of G(n, 2n).…”

“…1 ↑ D-equivalent in the sequel. 1. They are the only two Hartshorne varieties among the rational homogeneous spaces, if we define a Hartshorne variety to be a smooth variety Z ⊂ P N of dimension n = 2 3 N which is not a complete intersection (recall that Hartshorne's conjecture predicts that n > 2 3 N is impossible) [28,Corollary 2.16].…”

Principal parts bundles on projective spaces and quiver representations

“…Hence one has the following result. We will apply Theorem A of [3], which in this context says that if for some i = h + 1, . .…”

On the prolongations of homogeneous vector bundles

“…In 1988, Purohit 48 HÜLYA KADIO ¼ GLU [13] showed that there is a one-to-one correspondence between homogeneous vector bundles and linear representations. Various other studies about homogeneous vector bundles can be found in the literature ( [2], [8], [11]). Moreover, in 1972 R. W. Brockett and H. J. Sussmann described how the tangent bundle of a homogeneous space can be viewed as a homogeneous space [5].…”