We construct full strong exceptional collections of line bundles on smooth toric Fano Deligne-Mumford stacks of Picard number at most two and of any Picard number in dimension two. It is hoped that the approach of this paper will eventually lead to the proof of the existence of such collections on all smooth toric nef-Fano Deligne-Mumford stacks.

In this paper we study the moduli stack of complexes of vector bundles (with chain isomorphisms) over a smooth projective variety X via derived algebraic geometry. We prove that if X is a Calabi-Yau variety of dimension d then this moduli stack has a (1−d)-shifted Poisson structure. In the case d = 1, we construct a natural foliation of the moduli stack by 0-shifted symplectic substacks. We show that our construction recovers various known Poisson structures associated to complex elliptic curves, including the Poisson structure on Hilbert scheme of points on elliptic quantum projective planes studied by Nevins and Stafford, and the Poisson structures on the moduli spaces of stable triples over an elliptic curves considered by one of us. We also relate the latter Poisson structures to the semi-classical limits of the elliptic Sklyanin algebras studied by Feigin and Odesskii. * huazheng@maths.hku.hk † apolish@uoregon.edu Moduli spaces of complexes and perfect complexesThe main purpose of this section is to briefly recall the construction of the moduli stack of objects in a dg category, due to Töen and Vaquié [33], and to set up some notation involving dg-categories and derived stacks that will be used later. Moduli of objects in dg-categoriesLet k be a field of characteristic zero. We denote by C (k) the category of (unbounded) cochain complexes of k-modules. It carries a standard model structure ([12, Definition 2.3.3]). For L, M ∈ C (k) and n ∈ Z, let Hom n (L, M ) denote ⊂ ǫM 1

Abstract. In this short note we construct Calabi-Yau threefolds with nonabelian fundamental groups of order 64 as quotients of the small resolutions of certain complete intersections of quadrics in P 7 that were first considered by M. Gross and S. Popescu.

Hilbert scheme topological invariants of plane curve singularities are identified to framed threefold stable pair invariants. As a result, the conjecture of Oblomkov and Shende on HOMFLY polynomials of links of plane curve singularities is given a Calabi-Yau threefold interpretation. The motivic Donaldson-Thomas theory developed by M. Kontsevich and the third author then yields natural motivic invariants for algebraic knots. This construction is motivated by previous work of V. Shende, C. Vafa and the first author on the large N -duality derivation of the above conjecture.