Singular instantons and parabolic bundles over complex surfaces
<p>Let (X,Σ) be a pair consisting of a Riemannian 4-manifold and an embedded surface and G a compact connected Lie group. A singular G-instanton on X\Σ is one not extending to X and the singularity is encoded in a conjugacy class in G, the holonomy along Σ.</p> <p>When X = C<sup...
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| Format: | Thesis |
| Language: | English |
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1993
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| _version_ | 1826311148787466240 |
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| author | Munari, A |
| author_facet | Munari, A |
| author_sort | Munari, A |
| collection | OXFORD |
| description | <p>Let (X,Σ) be a pair consisting of a Riemannian 4-manifold and an embedded surface and G a compact connected Lie group. A singular G-instanton on X\Σ is one not extending to X and the singularity is encoded in a conjugacy class in G, the holonomy along Σ.</p>
<p>When X = C<sup>2</sup> and Σ is a complex line we show that if the holonomy along Σ has a certain property there is a natural correspondence between euclidean G-instantons on C<sup>2</sup>\Σ and suitable holomorphic objects on (P<sup>2</sup>, Σ, D<sup>∞</sup>) called parabolic framings, where D<sup>∞</sup> = P<sup>2</sup>\C<sup>2</sup>. As a corollary we obtain a classification of G-instantons on S<sup>4</sup>\S<sup>2</sup> for the round metric in terms of parabolic framings which in the trivial case of no singularities gives a new proof of Donaldson Theorem on framed bundles [D1], and a classification of hyperbolic G-monopoles with arbitrary mass and charge in terms of based rational maps of P<sup>1</sup> into flag manifolds proving conjectures of Atiyah [A1] and Garland-Murray [GM].</p>
<p>Using the techniques developed and Simpson existence Theorem, [S], when (X, Σ) is a compact complex Kähler pair and G = U(n) or SU(n) we also obtain a natural correspondence between irreducible instantons on X\Σ and stable parabolic vector bundles on (X, Σ) proving the “Parabolic bundles conjecture” of Kronheimer [K].</p>
<p>We then construct moduli spaces of stable parabolic and framed parabolic bundles and of singular instantons and carry out a comparison of the deformation theories where the correspondence is proved showing that there always is an infinitesimal isomorphism which is a diffeomorphism at smooth points. As a consequence we obtain the smoothness of all moduli spaces of irreducible G-instantons on S<sup>4</sup>\ S<sup>2</sup>, sufficient topological conditions for the existence of G-instantons on S<sup>4</sup>\S<sup>2</sup> with prescribed holonomy along S<sup>2</sup> and necessary and sufficient topological conditions when G = SU(n) proving a conjecture of Kronheimer-Mrowka [KM1]. We also deduce that the extended moduli spaces of instantons on S<sup>4</sup>\S<sup>2</sup>, where the holonomy along S<sup>2</sup> is allowed to vary with fixed centralizer, are products so that the diffeomorphism type of the moduli space depends on the holonomy along S<sup>2</sup> only up to its centralizer in G. Corresponding results for hyperbolic monopoles also follow.</p> |
| first_indexed | 2024-03-07T08:04:05Z |
| format | Thesis |
| id | oxford-uuid:becddf07-3e01-4c18-a067-0af1c6539892 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T08:04:05Z |
| publishDate | 1993 |
| record_format | dspace |
| spelling | oxford-uuid:becddf07-3e01-4c18-a067-0af1c65398922023-10-17T12:57:30ZSingular instantons and parabolic bundles over complex surfacesThesishttp://purl.org/coar/resource_type/c_db06uuid:becddf07-3e01-4c18-a067-0af1c6539892Moduli theoryGeometry, AlgebraicEnglishHyrax Deposit1993Munari, A<p>Let (X,Σ) be a pair consisting of a Riemannian 4-manifold and an embedded surface and G a compact connected Lie group. A singular G-instanton on X\Σ is one not extending to X and the singularity is encoded in a conjugacy class in G, the holonomy along Σ.</p> <p>When X = C<sup>2</sup> and Σ is a complex line we show that if the holonomy along Σ has a certain property there is a natural correspondence between euclidean G-instantons on C<sup>2</sup>\Σ and suitable holomorphic objects on (P<sup>2</sup>, Σ, D<sup>∞</sup>) called parabolic framings, where D<sup>∞</sup> = P<sup>2</sup>\C<sup>2</sup>. As a corollary we obtain a classification of G-instantons on S<sup>4</sup>\S<sup>2</sup> for the round metric in terms of parabolic framings which in the trivial case of no singularities gives a new proof of Donaldson Theorem on framed bundles [D1], and a classification of hyperbolic G-monopoles with arbitrary mass and charge in terms of based rational maps of P<sup>1</sup> into flag manifolds proving conjectures of Atiyah [A1] and Garland-Murray [GM].</p> <p>Using the techniques developed and Simpson existence Theorem, [S], when (X, Σ) is a compact complex Kähler pair and G = U(n) or SU(n) we also obtain a natural correspondence between irreducible instantons on X\Σ and stable parabolic vector bundles on (X, Σ) proving the “Parabolic bundles conjecture” of Kronheimer [K].</p> <p>We then construct moduli spaces of stable parabolic and framed parabolic bundles and of singular instantons and carry out a comparison of the deformation theories where the correspondence is proved showing that there always is an infinitesimal isomorphism which is a diffeomorphism at smooth points. As a consequence we obtain the smoothness of all moduli spaces of irreducible G-instantons on S<sup>4</sup>\ S<sup>2</sup>, sufficient topological conditions for the existence of G-instantons on S<sup>4</sup>\S<sup>2</sup> with prescribed holonomy along S<sup>2</sup> and necessary and sufficient topological conditions when G = SU(n) proving a conjecture of Kronheimer-Mrowka [KM1]. We also deduce that the extended moduli spaces of instantons on S<sup>4</sup>\S<sup>2</sup>, where the holonomy along S<sup>2</sup> is allowed to vary with fixed centralizer, are products so that the diffeomorphism type of the moduli space depends on the holonomy along S<sup>2</sup> only up to its centralizer in G. Corresponding results for hyperbolic monopoles also follow.</p> |
| spellingShingle | Moduli theory Geometry, Algebraic Munari, A Singular instantons and parabolic bundles over complex surfaces |
| title | Singular instantons and parabolic bundles over complex surfaces |
| title_full | Singular instantons and parabolic bundles over complex surfaces |
| title_fullStr | Singular instantons and parabolic bundles over complex surfaces |
| title_full_unstemmed | Singular instantons and parabolic bundles over complex surfaces |
| title_short | Singular instantons and parabolic bundles over complex surfaces |
| title_sort | singular instantons and parabolic bundles over complex surfaces |
| topic | Moduli theory Geometry, Algebraic |
| work_keys_str_mv | AT munaria singularinstantonsandparabolicbundlesovercomplexsurfaces |