ИСТИНА

For a germ of a quasihomogeneous function with an isolated critical point at the origin invariant with respect to an appropriate action of a finite abelian group (an admissible one), H.Fan, T.Jarvis, and Y.Ruan defined the so-called quantum cohomology group. This group is defined in terms of the vanishing cohomology groups of Milnor fibres of restrictions of the function to fixed point sets of elements of the group. The quantum cohomology group is considered as the main object of the so called quantum singularity theory or FJRW-theory. Fan, Jarvis, and Ruan studied some structures on the quantum cohomology group which generalize similar structures in the usual singularity theory. An important role in singularity theory is played by such concepts as the (integral) Milnor lattice, the monodromy operator, the Seifert form and the intersection form. Analogues of these concepts have not yet been considered in the FJRW-theory. We define an orbifold version of the monodromy operator on the quantum (co)homology group and a lattice which is invariant with respect to the orbifold monodromy operator and is considered as an orbifold version of the Milnor lattice. The action of the orbifold monodromy operator on it can be considered as an analogue of the integral monodromy operator. Moreover, we define orbifold versions of the Seifert form and of the intersection form. To define these concepts we use the language of group rings. An appropriate change of the basis in the group ring allows to give a decomposition of a certain extension of the quantum (co)homology group into parts isomorphic to (co)homology groups of certain suspensions of the restrictions of the function under consideration to fixed point sets. This permits to define analogues of the Seifert and intersection form on this extension. We show that the intersection of this decomposition with the quantum (co)homology group respects the relations between the monodromy, the Seifert and the intersection form.