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 Title: Line Bundles on Projective Homogeneous Spaces Author(s): Lauritzen, Niels Thomas Hjort Doctoral Committee Chair(s): Haboush, W.J., Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics Abstract: The topic of my thesis is the geometry of projective homogeneous spaces G/H for a semisimple algebraic group G in characteristic p $>$ 0, where H is a subgroup scheme containing a Borel subgroup B. In characteristic p $>$ 0 there are an infinite number of subgroup schemes containing B--the reduced ones are the ordinary parabolic subgroups P $\supseteq$ B. Examples of non-reduced parabolic subgroup schemes are extensions of B by Frobenius kernels of P. Using an algebraic analogue of the fixed point formula of Atiyah and Bott, we give a formula for the Euler character of a homogeneous line bundle on G/H generalizing Weyl's character formula. The canonical line bundle on G/H is rarely negative ample. A consequence of this is, that G/H is Frobenius split only when H is an extension of a parabolic subgroup by a Frobenius kernel of G. In an attempt to generalize Kempf's vanishing theorem we discovered, that G/H with H non-reduced can be used to construct new counterexamples to Kodaira's vanishing theorem in characteristic p $>$ 0. For G of type $D\sb5$ and H the extension of B by the first Frobenius kernel of $P\sb\alpha$, where $P\sb\alpha$ is the minimal parabolic subgroup having (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)as its only positive root, we give an example of an ample line bundle $\cal{L}$ on G/H such that ${\cal L}\otimes\omega\sb{G/H}$ has negative Euler characteristic. This also answers an old question of Raynaud. Issue Date: 1993 Type: Text Description: 53 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1993. URI: http://hdl.handle.net/2142/72545 Other Identifier(s): (UMI)AAI9411681 Date Available in IDEALS: 2014-12-17 Date Deposited: 1993
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