Discrete derived categories were studied initially by Vossieck [42] and later by Bobiński, Geiß, Skowroński [9]. In this article, we describe the homomorphism hammocks and autoequivalences on these categories. We classify silting objects and bounded t-structures.

The familiar Fourier-Mukai technique can be extended to an equivariant setting where a finite group G acts on a smooth projective variety X. In this paper we compare the group of invariant autoequivalences Aut(D b (X)) G with the group of autoequivalences of D G (X). We apply this method in three cases: Hilbert schemes on K3 surfaces, Kummer surfaces and canonical quotients.

Abstract. In this paper it is shown that the existence of two independent holomorphic first integrals for foliations by curves on (C 3 , 0) is not a topological invariant. More precisely, we provide an example of two topologically equivalent foliations such that only one of them admits two independent holomorphic first integrals. The existence of invariant surfaces over which the induced foliation possesses infinitely many separatrices possibly constitutes the sole obstruction for the topological invariance of complete integrability and a characterization of foliations admitting this type of invariant surfaces is also given.

We study objects in triangulated categories which have a two-dimensional graded endomorphism algebra. Given such an object, we show that there is a unique maximal triangulated subcategory, in which the object is spherical. This general result is then applied to algebraic geometry.Comment: 21 pages. Identical to published version. There is a separate article with examples from representation theory, see arXiv:1502.0683

We consider the Berglund-Hübsch transpose of a bimodal invertible polynomial and construct a triangulated category associated to the compactification of a suitable deformation of the singularity. This is done in such a way that the corresponding Grothendieck group with the (negative) Euler form can be described by a graph which corresponds to the Coxeter-Dynkin diagram with respect to a distinguished basis of vanishing cycles of the bimodal singularity. * Supported by the DFG priority program SPP 1388 "Representation Theory" (Eb 102/6-1).