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Seminar
FIELD Math:Geometry
DATE October 28 (Thu), 2021
TIME 16:30-17:30
PLACE Online
SPEAKER Dimitri Zvonkine
HOST Kim, Bumsig
INSTITUTE  
TITLE Gromov-Witten invariants of complete intersections
ABSTRACT Gromov-Witten invariants of a smooth complex variety do not change as the variety is deformed. Moreover, if a variety X degenerates into a union of two smooth varieties X_1, X_2 intersecting along a smooth divisor D, the degeneration formula by Jun Li allows one to recover a part of Gromov-Witten invariants of X from those of X_1, X_2, and D. The degeneration formula applies nicely to complete intersections. Indeed, if f is one of the polynomial equations defining a complete intersection X, we can deform f into a product g_1 g_2 of two polynomials of smaller degrees, thus degenerating the complete intersection into a union of two simpler complete intersections. However, the degeneration formula has a limitation: it only applies to cohomology classes that are well-defined on the total space of the degeneration. Thus, in general, it is inapplicable to the primitive cohomology of complete intersections. On the other hand it does use the primitive cohomology of D to express the answer. In other words, it is not possible to compute Gromov-Witten invariants by induction by just using this formula. In a joint work with Arguz, Bousseau and Pandharipande we solved this difficulty by introducing nodal Gromov-Witten invariants. We constructed an algorithm that computes all Gromov-Witten invariants of all complete intersections.
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