Maximally-graded matrix factorizations for an invertible polynomial of chain type

Abstract

In 1977, Orlik–Randell construct a nice integral basis of the middle homology group of the Milnor fiber associated to an invertible polynomial of chain type and they conjectured that it is represented by a distinguished basis of vanishing cycles.The purpose of this paper is to prove the algebraic counterpart of the Orlik–Randell conjecture. Under the homological mirror symmetry, we may expect that the triangulated category of maximally-graded matrix factorizations for the Berglund–Hübsch transposed polynomial admits a full exceptional collection with a nice numerical property. Indeed, we show that the category admits a Lefschetz decomposition with respect to a polarization in the sense of Kuznetsov–Smirnov, whose Euler matrix are calculated in terms of the “zeta function” of the inverse of the polarization.As a corollary, it turns out that the homological mirror symmetry holds at the level of the Grothendieck group in the following sense; the Grothendieck group of the category with the Euler form is isomorphic to the middle homology group with the (quasi-)Seifert form (with a suitable sign).