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 k < 0, in the sense of [142]. We use this to show that the classical scheme X = t<sub>0</sub>(X) has the structure of an algebraic d-critical locus, in the sense of Joyce [87]. Then, if (X, s) is an oriented d-critical locus, we prove in [18] that there is a natural perverse sheaf P·<sub>X,s</sub> on X, and in [25], we construct a natural motive MF<sub>X,s</sub>, 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 k-shifted symplectic derived Artin stacks. We apply this theory to categorifying 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 L??M of oriented Lagrangians L,M in an algebraic symplectic manifold (S,ω). In [23] we show that if (S,ω) is a complex symplectic manifold, and L,M are complex Lagrangians in S, then the intersection X= L??M, as a complex analytic subspace of S, extends naturally to a complex analytic d-critical locus (X, s) in the sense of Joyce [87]. If the canonical bundles K<sub>L</sub>,K<sub>M</sub> have square roots K<sup>1/2</sup><sub>L</sub>, K<sup>1/2</sup><sub>M</sub> then (X, s) is oriented, and we provide a direct construction of a perverse sheaf P·<sub>L,M</sub> on X, which coincides with the one constructed in [18]. In [24] we have a more in depth investigation in generalized Donaldson-Thomas invariants DT<sup>α</sup>(τ) 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:669779 |
| Date | January 2014 |
| Creators | Bussi, Vittoria |
| Contributors | Joyce, Dominic |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:54896cc4-b3fa-4d93-9fa9-2a842ad5e4df |
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