A theory of generalized Donaldson-Thomas invariants
Donaldson–Thomas invariants <i><b>DT<sup>α</sup>(τ)</b></i> are integers which `count' <b>τ</b>-stable coherent sheaves with Chern character <i><b>α</b></i> on a Calabi–Yau 3-fold <i><b>X</b></i>, w...
| المؤلفون الرئيسيون: | , |
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| التنسيق: | Journal article |
| منشور في: |
American Mathematical Society
2011
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| _version_ | 1826262575348711424 |
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| author | Joyce, D Song, Y |
| author_facet | Joyce, D Song, Y |
| author_sort | Joyce, D |
| collection | OXFORD |
| description | Donaldson–Thomas invariants <i><b>DT<sup>α</sup>(τ)</b></i> are integers which `count' <b>τ</b>-stable coherent sheaves with Chern character <i><b>α</b></i> on a Calabi–Yau 3-fold <i><b>X</b></i>, where <i><b>τ</b></i> denotes Gieseker stability for some ample line bundle on <i><b>X</b></i>. They are unchanged under deformations of <i><b>X</b></i>. The conventional definition works only for classes <i><b>α</b></i> containing no strictly <i><b>τ</b></i>-semistable sheaves. Behrend showed that <i><b>DT<sup>α</sup>(τ)</b></i> can be written as a weighted Euler characteristic <i><b>χ(M<sup>α</sup><sub>st</sub>(τ),ν<sub>M<sup>α</sup><sub>st</sub>(τ)</sub>)</b></i> of the stable moduli scheme <i><b>M<sup>α</sup><sub>st</sub>(τ)</b></i> by a constructible function <i><b>ν<sub>M<sup>α</sup><sub>st</sub>(τ)</sub></b></i> we call the `Behrend function'. This book studies generalized Donaldson–Thomas invariants <i><b>DT<sup>α</sup>(τ)</b></i>. They are rational numbers which `count' both <i><b>τ</b></i>-stable and <i><b>τ</b></i>-semistable coherent sheaves with Chern character <i><b>α</b></i> on <i><b>X</b></i>; strictly <i><b>τ</b></i>-semistable sheaves must be counted with complicated rational weights. The <i><b>DT<sup>α</sup>(τ)</b></i> are defined for all classes <i><b>α</b></i>, and are equal to <i><b>DT<sup>α</sup>(τ)</b></i> when it is defined. They are unchanged under deformations of <i><b>X</b></i>, and transform by a wall-crossing formula under change of stability condition <i><b>τ</b></i>. To prove all this we study the local structure of the moduli stack <i><b>M</b></i> of coherent sheaves on <i><b>X</b></i>. We show that an atlas for <i><b>M</b></i> may be written locally as <i><b>Crit(ƒ)</b></i> for <i><b>ƒ:U→C</b></i> holomorphic and <i><b>U</b></i> smooth, and use this to deduce identities on the Behrend function <i><b>ν<sub>M</sub></b></i>. We compute our invariants <i><b>DT<sup>α</sup>(τ)</b></i> in examples, and make a conjecture about their integrality properties. We also extend the theory to abelian categories mod-<i><b>KQ/I</b></i> of representations of a quiver <i><b>Q</b></i> with relations <i><b>I</b></i> coming from a superpotential <i><b>W</b></i> on <i><b>Q</b></i>, and connect our ideas with Szendrői's noncommutative Donaldson–Thomas invariants, and work by Reineke and others on invariants counting quiver representations. Our book is closely related to Kontsevich and Soibelman's independent paper Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. |
| first_indexed | 2024-03-06T19:38:23Z |
| format | Journal article |
| id | oxford-uuid:1fcd89e0-3aec-4241-b461-a15da8e44da5 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T19:38:23Z |
| publishDate | 2011 |
| publisher | American Mathematical Society |
| record_format | dspace |
| spelling | oxford-uuid:1fcd89e0-3aec-4241-b461-a15da8e44da52022-03-26T11:24:06ZA theory of generalized Donaldson-Thomas invariantsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:1fcd89e0-3aec-4241-b461-a15da8e44da5Symplectic Elements at OxfordAmerican Mathematical Society2011Joyce, DSong, YDonaldson–Thomas invariants <i><b>DT<sup>α</sup>(τ)</b></i> are integers which `count' <b>τ</b>-stable coherent sheaves with Chern character <i><b>α</b></i> on a Calabi–Yau 3-fold <i><b>X</b></i>, where <i><b>τ</b></i> denotes Gieseker stability for some ample line bundle on <i><b>X</b></i>. They are unchanged under deformations of <i><b>X</b></i>. The conventional definition works only for classes <i><b>α</b></i> containing no strictly <i><b>τ</b></i>-semistable sheaves. Behrend showed that <i><b>DT<sup>α</sup>(τ)</b></i> can be written as a weighted Euler characteristic <i><b>χ(M<sup>α</sup><sub>st</sub>(τ),ν<sub>M<sup>α</sup><sub>st</sub>(τ)</sub>)</b></i> of the stable moduli scheme <i><b>M<sup>α</sup><sub>st</sub>(τ)</b></i> by a constructible function <i><b>ν<sub>M<sup>α</sup><sub>st</sub>(τ)</sub></b></i> we call the `Behrend function'. This book studies generalized Donaldson–Thomas invariants <i><b>DT<sup>α</sup>(τ)</b></i>. They are rational numbers which `count' both <i><b>τ</b></i>-stable and <i><b>τ</b></i>-semistable coherent sheaves with Chern character <i><b>α</b></i> on <i><b>X</b></i>; strictly <i><b>τ</b></i>-semistable sheaves must be counted with complicated rational weights. The <i><b>DT<sup>α</sup>(τ)</b></i> are defined for all classes <i><b>α</b></i>, and are equal to <i><b>DT<sup>α</sup>(τ)</b></i> when it is defined. They are unchanged under deformations of <i><b>X</b></i>, and transform by a wall-crossing formula under change of stability condition <i><b>τ</b></i>. To prove all this we study the local structure of the moduli stack <i><b>M</b></i> of coherent sheaves on <i><b>X</b></i>. We show that an atlas for <i><b>M</b></i> may be written locally as <i><b>Crit(ƒ)</b></i> for <i><b>ƒ:U→C</b></i> holomorphic and <i><b>U</b></i> smooth, and use this to deduce identities on the Behrend function <i><b>ν<sub>M</sub></b></i>. We compute our invariants <i><b>DT<sup>α</sup>(τ)</b></i> in examples, and make a conjecture about their integrality properties. We also extend the theory to abelian categories mod-<i><b>KQ/I</b></i> of representations of a quiver <i><b>Q</b></i> with relations <i><b>I</b></i> coming from a superpotential <i><b>W</b></i> on <i><b>Q</b></i>, and connect our ideas with Szendrői's noncommutative Donaldson–Thomas invariants, and work by Reineke and others on invariants counting quiver representations. Our book is closely related to Kontsevich and Soibelman's independent paper Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. |
| spellingShingle | Joyce, D Song, Y A theory of generalized Donaldson-Thomas invariants |
| title | A theory of generalized Donaldson-Thomas invariants |
| title_full | A theory of generalized Donaldson-Thomas invariants |
| title_fullStr | A theory of generalized Donaldson-Thomas invariants |
| title_full_unstemmed | A theory of generalized Donaldson-Thomas invariants |
| title_short | A theory of generalized Donaldson-Thomas invariants |
| title_sort | theory of generalized donaldson thomas invariants |
| work_keys_str_mv | AT joyced atheoryofgeneralizeddonaldsonthomasinvariants AT songy atheoryofgeneralizeddonaldsonthomasinvariants AT joyced theoryofgeneralizeddonaldsonthomasinvariants AT songy theoryofgeneralizeddonaldsonthomasinvariants |