Title:

Derived symplectic structures in generalized DonaldsonThomas theory and categorification

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_{0}(X) has the structure of an algebraic dcritical locus, in the sense of Joyce [87]. Then, if (X, s) is an oriented dcritical locus, we prove in [18] that there is a natural perverse sheaf P·_{X,s} on X, and in [25], we construct a natural motive MF_{X,s}, in a certain quotient ring M^{μ}_{X} of the μequivariant motivic Grothendieck ring M^{μ}_{X}, and used in Kontsevich and Soibelman's theory of motivic DonaldsonThomas invariants [102]. In [13], we obtain similar results for kshifted symplectic derived Artin stacks. We apply this theory to categorifying DonaldsonThomas invariants of CalabiYau 3folds, 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 CalabiYau 3fold equipped with 'orientation data', as required in Kontsevich and Soibelman's motivic DonaldsonThomas 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 dcritical locus (X, s) in the sense of Joyce [87]. If the canonical bundles K_{L},K_{M} have square roots K^{1/2}_{L}, K^{1/2}_{M} then (X, s) is oriented, and we provide a direct construction of a perverse sheaf P·_{L,M} on X, which coincides with the one constructed in [18]. In [24] we have a more in depth investigation in generalized DonaldsonThomas invariants DT^{α}(τ) defined by Joyce and Song [85]. We propose a new algebraic method to extend the theory to algebraically closed fields K of characteristic zero, rather than K = C, and we conjecture the extension of generalized DonaldsonThomas theory to compactly supported coherent sheaves on noncompact quasiprojective CalabiYau 3folds, and to complexes of coherent sheaves on CalabiYau 3folds.
