| Crynodeb: | <p>Donaldson and Thomas defined Donaldson-Thomas (DT) invariants for moduli spaces of sheaves on proper Calabi-Yau threefolds in [DT,T]. The definition has been extended to more general moduli spaces of objects in 3-Calabi-Yau categories, and in the last few years there has been an increasing interest in studying refinements of DT invariants. This thesis is devoted to the computation of the generating series of refined DT invariants in some geometric examples.</p>
<p>In Chapter 2 we compute the generating series of refined DT invariants for the moduli space 𝑀<sub>n,r</sub> of higher rank framed sheaves on ℙ<sup>3</sup>, generalizing the result of [BBS] for the Hilbert scheme of <em>n</em> points on ℂ<sup>3</sup>. We describe 𝑀<sub>n,r</sub> as a global critical locus by means of the Beilinson spectral sequence, and compute the generating series using motivic wallcrossing. We interpret the result as a count of r-tuples of weighted 3D partitions in analogy with the refined topological vertex [IKV].</p>
<p>In Chapter 3 we study the refined DT invariants for the moduli space <em>P</em><sup>r</sup><sub>n,d</sub> of higher rank (framed) stable pairs on the resolved conifold. Applying twice the Beilinson spectral sequence, we identify <em>P</em><sup>r</sup><sub>n,d</sub> with the moduli space of PT-stable representations of the r-framed conifold quiver with superpotential, generalizing a result of [NN11]. We compute a product expansion for the generating series of refined DT invariants of <em>P</em><sup>r</sup><sub>n,d</sub>, and we compare our results with the partition function counting U(1)-instantons on the minimal resolution of an A<sub>r-1</sub> surface singularity.</p>
<p>In Chapter 4 we study the universal generating series of motivic DT invariants for deformations of graded 3-Calabi-Yau algebras. We consider Jacobi algebras which are derived equivalent to the resolved conifold and the resolution of a line of A<sub>n</sub>-singularities, and investigate their marginal deformations. The universal series of motivic DT invariants are affected by the deformations, and their variation is interpreted in terms of virtual classes of some special representation varieties associated to the quiver.</p>
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