Compactification problems in the theory of characteristic currents associated with a singular connection

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dc.contributor.advisorHarvey, F. Reeseen_US
dc.creatorZweck, Johnen_US
dc.date.accessioned2009-06-03T23:53:03Zen_US
dc.date.available2009-06-03T23:53:03Zen_US
dc.date.issued1993en_US
dc.description.abstractA compactification of the Chern-Weil theory for bundle maps developed by Harvey and Lawson is described. For each smooth section $\nu$ of the compactification $\IP(\underline{\doubc}\oplus F)\to X$ of a rank n complex vector bundle $F\to X$ with connection, and for each Ad-invariant polynomial $\phi$ on ${\bf gl}\sb{n},$ there are associated current formulae generalizing those of Harvey and Lawson. These are of the form $$\eqalign{\phi(\Omega\sb{\it F}) &+\rm \nu\sp*(Res\sb\infty(\phi))\ Div\sb\infty(\nu) - \phi(\Omega\sb0)\ -\cr&\qquad\rm Res\sb0(\phi)\ Div\sb0(\nu) = {\it dT}\quad on\ {\it X},\cr}$$where Div$\sb0(\nu)$ and Div$\sb\infty(\nu)$ are integrally flat currents supported on the zero and pole sets of $\nu,$ where Res$\sb0(\phi)$ and Res$\sb\infty(\phi)$ are smooth residue forms which can be calculated in terms of the curvature $\Omega\sb{F}$ of F, where T is a canonical transgression form with coefficients in $L\sbsp{\rm loc}{1},$ and where $\phi(\Omega\sb0)$ is an $L\sbsp{\rm loc}{1}$ form canonically defined in terms of a singular connection naturally associated to $\nu.$ These results hold for $C\sp\infty$-meromorphic sections $\nu$ which are atomic. The notion of an atomic section of a vector bundle was first introduced and studied by Harvey and Semmes. The formulae obtained include a generalization of the Poincare-Lelong formula to $C\sp\infty$-meromorphic sections of a bundle of arbitrary rank. Analogous results hold for real vector bundles and for quaternionic line bundles.en_US
dc.format.extent152 p.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.callnoThesis Math. 1993 Zwecken_US
dc.identifier.citationZweck, John. "Compactification problems in the theory of characteristic currents associated with a singular connection." (1993) Diss., Rice University. <a href="https://hdl.handle.net/1911/16700">https://hdl.handle.net/1911/16700</a>.en_US
dc.identifier.urihttps://hdl.handle.net/1911/16700en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectMathematicsen_US
dc.titleCompactification problems in the theory of characteristic currents associated with a singular connectionen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentMathematicsen_US
thesis.degree.disciplineNatural Sciencesen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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