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Classifying affine line bundles on a compact complex space
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| Zeitschriftentitel: | Complex Manifolds |
|---|---|
| Personen und Körperschaften: | |
| In: | Complex Manifolds, 6, 2019, 1, S. 103-117 |
| Format: | E-Article |
| Sprache: | Unbestimmt |
| veröffentlicht: |
Walter de Gruyter GmbH
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| Schlagwörter: |
| author_facet |
Plechinger, Valentin Plechinger, Valentin |
|---|---|
| author |
Plechinger, Valentin |
| spellingShingle |
Plechinger, Valentin Complex Manifolds Classifying affine line bundles on a compact complex space Geometry and Topology |
| author_sort |
plechinger, valentin |
| spelling |
Plechinger, Valentin 2300-7443 Walter de Gruyter GmbH Geometry and Topology http://dx.doi.org/10.1515/coma-2019-0005 <jats:title>Abstract</jats:title><jats:p>The classification of affine line bundles on a compact complex space is a difficult problem. We study the affine analogue of the Picard functor and the representability problem for this functor. Let <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_001.png" /> be a compact complex space with <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_002.png" />. We introduce the affine Picard functor <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_003.png" /> which assigns to a complex space <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_004.png" /> the set of families of linearly <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_005.png" />-framed affine line bundles on <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_006.png" /> parameterized by <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_007.png" />. Our main result states that the functor <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_008.png" /> is representable if and only if the map <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_009.png" /> is constant. If this is the case, the space which represents this functor is a linear space over <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_010.png" /> whose underlying set is <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_011.png" />, where <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_012.png" /> is a Poincaré line bundle normalized at <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_013.png" />. The main idea idea of the proof is to compare the representability of <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_014.png" /> to the representability of a functor considered by Bingener related to the deformation theory of <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_015.png" />-cohomology classes. Our arguments show in particular that, for <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_016.png" /> = 1, the converse of Bingener’s representability criterion holds</jats:p> Classifying affine line bundles on a compact complex space Complex Manifolds |
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Walter de Gruyter GmbH, 2019 |
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Walter de Gruyter GmbH, 2019 |
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2019 |
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Walter de Gruyter GmbH |
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ai |
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ai |
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Complex Manifolds |
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49 |
| title |
Classifying affine line bundles on a compact complex space |
| title_unstemmed |
Classifying affine line bundles on a compact complex space |
| title_full |
Classifying affine line bundles on a compact complex space |
| title_fullStr |
Classifying affine line bundles on a compact complex space |
| title_full_unstemmed |
Classifying affine line bundles on a compact complex space |
| title_short |
Classifying affine line bundles on a compact complex space |
| title_sort |
classifying affine line bundles on a compact complex space |
| topic |
Geometry and Topology |
| url |
http://dx.doi.org/10.1515/coma-2019-0005 |
| publishDate |
2019 |
| physical |
103-117 |
| description |
<jats:title>Abstract</jats:title><jats:p>The classification of affine line bundles on a compact complex space is a difficult problem. We study the affine analogue of the Picard functor and the representability problem for this functor. Let <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_001.png" /> be a compact complex space with <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_002.png" />. We introduce the affine Picard functor <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_003.png" /> which assigns to a complex space <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_004.png" /> the set of families of linearly <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_005.png" />-framed affine line bundles on <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_006.png" /> parameterized by <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_007.png" />. Our main result states that the functor <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_008.png" /> is representable if and only if the map <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_009.png" /> is constant. If this is the case, the space which represents this functor is a linear space over <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_010.png" /> whose underlying set is <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_011.png" />, where <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_012.png" /> is a Poincaré line bundle normalized at <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_013.png" />. The main idea idea of the proof is to compare the representability of <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_014.png" /> to the representability of a functor considered by Bingener related to the deformation theory of <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_015.png" />-cohomology classes. Our arguments show in particular that, for <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_016.png" /> = 1, the converse of Bingener’s representability criterion holds</jats:p> |
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| description | <jats:title>Abstract</jats:title><jats:p>The classification of affine line bundles on a compact complex space is a difficult problem. We study the affine analogue of the Picard functor and the representability problem for this functor. Let <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_001.png" /> be a compact complex space with <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_002.png" />. We introduce the affine Picard functor <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_003.png" /> which assigns to a complex space <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_004.png" /> the set of families of linearly <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_005.png" />-framed affine line bundles on <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_006.png" /> parameterized by <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_007.png" />. Our main result states that the functor <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_008.png" /> is representable if and only if the map <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_009.png" /> is constant. If this is the case, the space which represents this functor is a linear space over <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_010.png" /> whose underlying set is <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_011.png" />, where <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_012.png" /> is a Poincaré line bundle normalized at <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_013.png" />. The main idea idea of the proof is to compare the representability of <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_014.png" /> to the representability of a functor considered by Bingener related to the deformation theory of <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_015.png" />-cohomology classes. Our arguments show in particular that, for <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_016.png" /> = 1, the converse of Bingener’s representability criterion holds</jats:p> |
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| spelling | Plechinger, Valentin 2300-7443 Walter de Gruyter GmbH Geometry and Topology http://dx.doi.org/10.1515/coma-2019-0005 <jats:title>Abstract</jats:title><jats:p>The classification of affine line bundles on a compact complex space is a difficult problem. We study the affine analogue of the Picard functor and the representability problem for this functor. Let <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_001.png" /> be a compact complex space with <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_002.png" />. We introduce the affine Picard functor <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_003.png" /> which assigns to a complex space <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_004.png" /> the set of families of linearly <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_005.png" />-framed affine line bundles on <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_006.png" /> parameterized by <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_007.png" />. Our main result states that the functor <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_008.png" /> is representable if and only if the map <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_009.png" /> is constant. If this is the case, the space which represents this functor is a linear space over <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_010.png" /> whose underlying set is <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_011.png" />, where <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_012.png" /> is a Poincaré line bundle normalized at <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_013.png" />. The main idea idea of the proof is to compare the representability of <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_014.png" /> to the representability of a functor considered by Bingener related to the deformation theory of <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_015.png" />-cohomology classes. Our arguments show in particular that, for <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_coma-2019-0005_eq_016.png" /> = 1, the converse of Bingener’s representability criterion holds</jats:p> Classifying affine line bundles on a compact complex space Complex Manifolds |
| spellingShingle | Plechinger, Valentin, Complex Manifolds, Classifying affine line bundles on a compact complex space, Geometry and Topology |
| title | Classifying affine line bundles on a compact complex space |
| title_full | Classifying affine line bundles on a compact complex space |
| title_fullStr | Classifying affine line bundles on a compact complex space |
| title_full_unstemmed | Classifying affine line bundles on a compact complex space |
| title_short | Classifying affine line bundles on a compact complex space |
| title_sort | classifying affine line bundles on a compact complex space |
| title_unstemmed | Classifying affine line bundles on a compact complex space |
| topic | Geometry and Topology |
| url | http://dx.doi.org/10.1515/coma-2019-0005 |