Grassmannians, Flag Varieties, Schur Functors and Cohomology

In this chapter we introduce the basic theory of flag varieties, and describe general results regarding cohomology groups of line bundles, with an emphasis on vanishing statements. We develop the theory in the relative setting, which offers significant flexibility for the inductive arguments that we employ. We also introduce Schur functors and explain their relationship to direct images of line bundles associated to dominant weights. We end with a brief discussion of Grothendieck duality and some applications, that will be instrumental in our calculation of Ext modules in subsequent chapters.