Derived symplectic structures in generalized Donaldson–Thomas theory and categorification
<p>This thesis presents a series of results obtained in [13, 18, 19, 23{25, 87]. In [19], we prove a Darboux theorem for derived schemes with symplectic forms of degree <em>k</em> < 0, in the sense of [142]. We use this to show that the classical scheme <em>X = t<su...
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| Format: | Thesis |
| Language: | English |
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2014
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| _version_ | 1826316715903942656 |
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| author | Bussi, V |
| author2 | Joyce, D |
| author_facet | Joyce, D Bussi, V |
| author_sort | Bussi, V |
| collection | OXFORD |
| description | <p>This thesis presents a series of results obtained in [13, 18, 19, 23{25, 87]. In [19], we prove a Darboux theorem for derived schemes with symplectic forms of degree <em>k</em> < 0, in the sense of [142]. We use this to show that the classical scheme <em>X = t<sub>0</sub>(X)</em> has the structure of an <em>algebraic d-critical locus</em>, in the sense of Joyce [87]. Then, if (<em>X, s</em>) is an <em>oriented d-critical locus</em>, we prove in [18] that there is a natural perverse sheaf P•<sub><em>X,s</em></sub> on <em>X</em>, and in [25], we construct a natural motive <em>MF<sub>X,s</sub></em>, in a certain quotient ring M<sup>μ</sup><sub>X</sub> of the μ-equivariant motivic Grothendieck ring M<sup>μ</sup><sub>X</sub>, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants [102]. In [13], we obtain similar results for <em>k</em>-shifted symplectic derived Artin stacks.</p> <p>We apply this theory to <em>categorifying</em> Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to categorifying Lagrangian intersections in a complex symplectic manifold using perverse sheaves, and to prove the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with 'orientation data', as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory [102], and on intersections <em>L</em>∩<em>M</em> of oriented Lagrangians <em>L,M</em> in an algebraic symplectic manifold (<em>S</em>,ω). In [23] we show that if (<em>S</em>,ω) is a complex symplectic manifold, and <em>L,M</em> are complex Lagrangians in <em>S</em>, then the intersection <em>X</em>= <em>L</em>∩<em>M</em>, as a complex analytic subspace of <em>S</em>, extends naturally to a complex analytic d-critical locus (<em>X, s</em>) in the sense of Joyce [87]. If the canonical bundles <em>K<sub>L</sub>,K<sub>M</sub></em> have square roots K<sup>1/2</sup><sub>L</sub>, K<sup>1/2</sup><sub>M</sub> then (<em>X, s</em>) is oriented, and we provide a direct construction of a perverse sheaf <em>P•<sub>L,M</sub></em> on <em>X</em>, which coincides with the one constructed in [18].</p> <p>In [24] we have a more in depth investigation in <em>generalized Donaldson-Thomas invariants DT<sup>α</sup>(τ)</em> defined by Joyce and Song [85]. We propose a new algebraic method to extend the theory to algebraically closed fields <b>K</b> of characteristic zero, rather than <b>K = C</b>, and we conjecture the extension of generalized Donaldson-Thomas theory to compactly supported coherent sheaves on noncompact quasi-projective Calabi-Yau 3-folds, and to complexes of coherent sheaves on Calabi-Yau 3-folds.</p> |
| first_indexed | 2024-03-07T08:18:33Z |
| format | Thesis |
| id | oxford-uuid:54896cc4-b3fa-4d93-9fa9-2a842ad5e4df |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-12-09T03:50:00Z |
| publishDate | 2014 |
| record_format | dspace |
| spelling | oxford-uuid:54896cc4-b3fa-4d93-9fa9-2a842ad5e4df2024-12-08T13:17:54ZDerived symplectic structures in generalized Donaldson–Thomas theory and categorificationThesishttp://purl.org/coar/resource_type/c_db06uuid:54896cc4-b3fa-4d93-9fa9-2a842ad5e4dfAlgebraic geometryEnglishOxford University Research Archive - Valet2014Bussi, VJoyce, D<p>This thesis presents a series of results obtained in [13, 18, 19, 23{25, 87]. In [19], we prove a Darboux theorem for derived schemes with symplectic forms of degree <em>k</em> < 0, in the sense of [142]. We use this to show that the classical scheme <em>X = t<sub>0</sub>(X)</em> has the structure of an <em>algebraic d-critical locus</em>, in the sense of Joyce [87]. Then, if (<em>X, s</em>) is an <em>oriented d-critical locus</em>, we prove in [18] that there is a natural perverse sheaf P•<sub><em>X,s</em></sub> on <em>X</em>, and in [25], we construct a natural motive <em>MF<sub>X,s</sub></em>, in a certain quotient ring M<sup>μ</sup><sub>X</sub> of the μ-equivariant motivic Grothendieck ring M<sup>μ</sup><sub>X</sub>, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants [102]. In [13], we obtain similar results for <em>k</em>-shifted symplectic derived Artin stacks.</p> <p>We apply this theory to <em>categorifying</em> Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to categorifying Lagrangian intersections in a complex symplectic manifold using perverse sheaves, and to prove the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with 'orientation data', as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory [102], and on intersections <em>L</em>∩<em>M</em> of oriented Lagrangians <em>L,M</em> in an algebraic symplectic manifold (<em>S</em>,ω). In [23] we show that if (<em>S</em>,ω) is a complex symplectic manifold, and <em>L,M</em> are complex Lagrangians in <em>S</em>, then the intersection <em>X</em>= <em>L</em>∩<em>M</em>, as a complex analytic subspace of <em>S</em>, extends naturally to a complex analytic d-critical locus (<em>X, s</em>) in the sense of Joyce [87]. If the canonical bundles <em>K<sub>L</sub>,K<sub>M</sub></em> have square roots K<sup>1/2</sup><sub>L</sub>, K<sup>1/2</sup><sub>M</sub> then (<em>X, s</em>) is oriented, and we provide a direct construction of a perverse sheaf <em>P•<sub>L,M</sub></em> on <em>X</em>, which coincides with the one constructed in [18].</p> <p>In [24] we have a more in depth investigation in <em>generalized Donaldson-Thomas invariants DT<sup>α</sup>(τ)</em> defined by Joyce and Song [85]. We propose a new algebraic method to extend the theory to algebraically closed fields <b>K</b> of characteristic zero, rather than <b>K = C</b>, and we conjecture the extension of generalized Donaldson-Thomas theory to compactly supported coherent sheaves on noncompact quasi-projective Calabi-Yau 3-folds, and to complexes of coherent sheaves on Calabi-Yau 3-folds.</p> |
| spellingShingle | Algebraic geometry Bussi, V Derived symplectic structures in generalized Donaldson–Thomas theory and categorification |
| title | Derived symplectic structures in generalized Donaldson–Thomas theory and categorification |
| title_full | Derived symplectic structures in generalized Donaldson–Thomas theory and categorification |
| title_fullStr | Derived symplectic structures in generalized Donaldson–Thomas theory and categorification |
| title_full_unstemmed | Derived symplectic structures in generalized Donaldson–Thomas theory and categorification |
| title_short | Derived symplectic structures in generalized Donaldson–Thomas theory and categorification |
| title_sort | derived symplectic structures in generalized donaldson thomas theory and categorification |
| topic | Algebraic geometry |
| work_keys_str_mv | AT bussiv derivedsymplecticstructuresingeneralizeddonaldsonthomastheoryandcategorification |