GROMOV–WITTEN INVARIANTS OF COMPLETE INTERSECTIONS

In our joint work with Julia Arguz, Pierrick Bosso and Rahul Pandaripande, we are constructing an algorithm for calculating all Gromov-Witten invariants of arbitrary genus of any smooth complete intersection on X. The main - long-known - calculation method is Jun Lee's degeneracy formula. If there is a family that degenerates X into the union of two smooth varieties that intersect transversally along a smooth divisor, then the Gromov-Witten invariants of X can be expressed in terms of the Gromov-Witten invariants of these two varieties and the divisor. The main problem: during degeneration, some cohomology classes disappear, and then Jun Lee's formula is not applicable to them. The purpose of the article is to circumvent this difficulty by using additional properties of Gromov-Witten invariants, namely, their invariance under monodromy. We prove that this additional information is exactly enough to recover all invariants.