Springer Science+Business Media New York
SPLITTING CRITERIA FOR VECTOR BUNDLES ON
MINUSCULE HOMOGENEOUS VARIETIES
Abstract. We prove that a vector bundle on a minuscule homogeneous variety splits into
a direct sum of line bundles if and only if its restriction to the union of two-dimensional
Schubert subvarieties splits. A case-by-case analysis is done. The result largely genera-
lizes Horrocks’ splitting criterion for vector bundles on projective spaces.
The goal of this note is to prove the following (cf. 2.3, 3.4):
Theorem. A vector bundle on a minuscule homogeneous variety X splits if and
only if its restriction to the 2-dimensional Schubert subvarieties of X splits.
For clarity, we point out that we are considering Schubert subvarieties with
respect to a ﬁxed Borel subgroup, so there are no families of subvarieties involved
in the discussion. Our approach to prove the result diﬀers from the previous
methods: we essentially use the Frobenius splitting of the Schubert varieties ,
. To our knowledge, this important feature has not been used before to study
the splitting of vector bundles on homogeneous varieties.
The theorem is proved by repeatedly applying the following general fact.
Theorem. Let T be a scheme deﬁned over an algebraically closed ﬁeld k and S
be a closed subscheme of it; moreover, consider a locally free sheaf V
on T . We
assume that H
)=k and H
splits into a direct sum of invertible sheaves if and only if V
The list of minuscule ﬂag varieties can be found in, e.g., [12, Chap. 5, §2]. For
the groups of type A
one gets respectively the Grassmannians, the even
dimensional quadrics, and the spinor varieties. There are two ‘exceptional’ cases:
the Cayley plane and the Freudenthal variety, corresponding to the groups E
A case-by-case analysis shows that most of the minuscule varieties have only
one irreducible 2-dimensional Schubert subvariety, isomorphic to P
. There are two
exceptions: the Grassmannians Gr(k, n), 1 < k < n − 1, and the 4-dimensional
Received August 7, 2015. Accepted February 10, 2016.
Corresponding Author: M. Halic, e-mail: firstname.lastname@example.org.
-01 -93 -6 71 z
Vol. 22, No.
, 2017, pp.