Categories of singularities of invertible polynomials

Abstract

We study the categories of singularities coming from Landau-Ginzburg models given by the invertible polynomials. Such categories appear on the B-side of the Berglund-H\"ubsch mirror symmetry. We provide an efficient method of computing morphism spaces in these categories and explicitly construct full strongly exceptional collections in the cases of small dimensions ($n\le 3$). Finally, we use this construction in order to prove Orlov's conjecture stating that such collections can be chosen to have block decompositions of size one more than the number of variables.