Homogeneous Irreducible ACM Bundles On The Rational Homogeneous Spaces Of Picard Number 1

Vector bundles are important objects in the research of algebraic geometry and related to the properties of algebraic varieties.In 1956,Grothendieck proved that the vector bundles on the projective line are splitting.A natural question is to consider when a vector bundle splits on general varieties.In 1964,Horrocks answered this question for projective spaces.He showed that a rank r vector bundle on a projective space P~nsplits if and only if it has no intermediate cohomology.Such vector bundles are called arithmetically Cohen-Macaulay(ACM for short)bundles.We would like to know whether this holds true for other varieties.The answer is no.There are many varieties with unsplit ACM bundles.Hence the fundamental problem is to classify the indecomposable ACM bundles on a projective variety.However,it is still a hard problem to give a classification of all indecomposable ACM bundles.Thus we hope to give a classification in some special cases or get some information of the moduli space of indecomposable ACM bundles.In this paper,we study vector bundles on homogeneous spaces of Picard number1.In particular,we consider the irreducible homogeneous vector bundles on them.For these special vector bundles,we use the Borel-Bott-Weil theorem to transform the computation of cohomology into the regularity of the highest weight of the irreducible representation.Then we get a numerical criterion of irreducible homo-geneous vector bundles.Furthermore,we prove that the moduli space of irreducible homogenous projective bundles consists of finite points.Through such characteri-zation,we get an algorithm of the classification of irreducible homogeneous ACM bundles.We give the complete classification for the irreducible homogeneous ACM bundles with special highest weight for the isotropic Grassmannians.Generally,we give an estimate of upper bound of the rank of irreducible homogeneous ACM bundles for isotropic Grassmannians.For the exceptional Grassmannians,we can give the complete classification of irreducible homogeneous ACM bundles.Based on the classification results,we can show that most rational homogeneous spaces are of wild type.Such result implies that there are huge amounts of indecomposable ACM bundles on these rational homogeneous spaces.