Explicitly computing Donaldson-Thomas invariants (of CY 3-folds) I am interested in the explicit computation of generating functions of rank 1 and higher rank Donaldson-Thomas (DT) invariants. In particular, I am interested in DT invariants of K3 fibered Calabi-Yau threefolds, and their modularity. Given a specific CY 3-fold and a fixed K3 lattice polarization, how can one systematically compute the DT invariants. What data of the CY 3-fold do I need? I am interested in not just smooth CYs, but smooth ones is a natural starting point, and I don't know how to handle those. 
Are there any references, where such computations have been worked out, maybe in very simple cases.
 A: This answer might just be a list of references, but I hope it helps. The most explicit computations of which I am aware exploit a torus action on the Calabi-Yau 3-fold in question, where the calculus can be reduced to a combinatorial problem: for example, the famous paper Gromov–Witten theory and Donaldson–Thomas theory. I (MSN) by Maulik, Nekrasov, Okounkov, and Pandharipande. The same approach yields explicit computations for stable pairs invariants, as done by Pandharipande and Thomas in another paper, The 3-fold vertex via stable pairs. Finally, one more case where it might be possible to compute Donaldson-Thomas invariants is when you can describe the corresponding moduli space as a critical locus, and compute the perverse sheaf of vanishing cycles, as done by Dimca and Szendrői in The Milnor fibre of the Pfaffian and the Hilbert scheme of four points on $\mathbb C^3$. Joyce and Song have greatly developed the theory of invariants for local vanishing loci, but it might not be a good place to start to look for explicit computations. 
A: I think an important reference for your question is Maulik and Pandharipande's paper on Noether-Lefschetz theory: https://arxiv.org/pdf/0705.1653.pdf
Their results are phrased in terms of Gromov-Witten theory, but the same result applies to Donaldson-Thomas theory. In particular, they tell you how to compute the DT/GW partition function for curve classes in the fiber of a $K3$ fibered Calabi-Yau threefold. The answer involves the partition function for local $K3$ surfaces along with the "Noether-Lefschetz" number of the family of $K3$s defined by the fibration, i.e. the number of times the base of the family hits the Noether-Lefschetz divisor (associated to the fiber curve class) in the moduli space of $K3$ surfaces. The generating function for both the NL numbers and the local $K3$ invariants have modular properties which are explained in that paper.
