{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/72545"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/72545","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Line Bundles on Projective Homogeneous Spaces","abstract":"The topic of my thesis is the geometry of projective homogeneous spaces G/H for a semisimple algebraic group G in characteristic p $&gt;$ 0, where H is a subgroup scheme containing a Borel subgroup B. In characteristic p $&gt;$ 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 $&gt;$ 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.","abstract_html":"The topic of my thesis is the geometry of projective homogeneous spaces G/H for a semisimple algebraic group G in characteristic p $&amp;gt;$ 0, where H is a subgroup scheme containing a Borel subgroup B. In characteristic p $&amp;gt;$ 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&#x27;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&#x27;s vanishing theorem we discovered, that G/H with H non-reduced can be used to construct new counterexamples to Kodaira&#x27;s vanishing theorem in characteristic p $&amp;gt;$ 0. For G of type $D\\sb5$ and H the extension of B by the first Frobenius kernel of <span class=\"etd-inline-math\">P\\sb&alpha;</span>, where <span class=\"etd-inline-math\">P\\sb&alpha;</span> 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 <span class=\"etd-inline-math\">{\\cal L}\\otimes&omega;\\sb{G/H}</span> has negative Euler characteristic. This also answers an old question of Raynaud.","abstract_has_math":true,"creators":["Lauritzen, Niels Thomas Hjort"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Haboush, W.J.,"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-17T23:17:48Z","date_published":"2014-12-17T23:17:48Z","updated_at":"2026-07-22T22:26:07Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI9411681"],"render_values":[{"text":"(UMI)AAI9411681","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/72545","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Haboush, W.J.,"]},{"key":"dc:creator","label":"Author","values":["Lauritzen, Niels Thomas Hjort"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-17T23:17:48Z","10000-01-01","1993"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/72545","(UMI)AAI9411681"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The topic of my thesis is the geometry of projective homogeneous spaces G/H for a semisimple algebraic group G in characteristic p $&gt;$ 0, where H is a subgroup scheme containing a Borel subgroup B. In characteristic p $&gt;$ 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 $&gt;$ 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.","Made available in DSpace on 2014-12-17T23:17:48Z (GMT). No. of bitstreams: 1 9411681.pdf: 2005765 bytes, checksum: 83c3b1abc990808a8181fd22439380c7 (MD5) Previous issue date: 1993","Embargo set by: Seth Robbins for item 72713 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","53 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1993."]},{"key":"dc:title","label":"Title","values":["Line Bundles on Projective Homogeneous Spaces"]}]}],"canonical_facts":{"dc:contributor":["Haboush, W.J.,"],"dc:creator":["Lauritzen, Niels Thomas Hjort"],"dc:date":["2014-12-17T23:17:48Z","10000-01-01","1993"],"dc:description":["The topic of my thesis is the geometry of projective homogeneous spaces G/H for a semisimple algebraic group G in characteristic p $&gt;$ 0, where H is a subgroup scheme containing a Borel subgroup B. In characteristic p $&gt;$ 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 $&gt;$ 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.","Made available in DSpace on 2014-12-17T23:17:48Z (GMT). No. of bitstreams: 1 9411681.pdf: 2005765 bytes, checksum: 83c3b1abc990808a8181fd22439380c7 (MD5) Previous issue date: 1993","Embargo set by: Seth Robbins for item 72713 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","53 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1993."],"dc:identifier":["http://hdl.handle.net/2142/72545","(UMI)AAI9411681"],"dc:subject":["Mathematics"],"dc:title":["Line Bundles on Projective Homogeneous Spaces"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:07Z"}