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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 UrbanaChampaign 
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 Bthe reduced ones are the ordinary parabolic subgroups P $\supseteq$ B. Examples of nonreduced 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 nonreduced 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 UrbanaChampaign, 1993. 
URI:  http://hdl.handle.net/2142/72545 
Other Identifier(s):  (UMI)AAI9411681 
Date Available in IDEALS:  20141217 
Date Deposited:  1993 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois