Quantum cohomology of complete intersections

TL;DR

This paper explores the structure of quantum cohomology for complete intersections, revealing simplified algebraic forms in high dimensions and deriving new enumerative formulas for rational curves.

Contribution

It demonstrates a simplified form of quantum cohomology for high-dimensional complete intersections and derives new enumerative formulas for rational curves.

Findings

Quantum cohomology simplifies for large-dimensional complete intersections.

Derived enumerative formulas for lines, conics, and twisted cubics.

Identified conditions under which the algebra takes a simple form.

Abstract

The quantum cohomology algebra of a projective manifold X is the cohomology H(X,Q) endowed with a different algebra structure, which takes into account the geometry of rational curves in X. We show that this algebra takes a remarkably simple form for complete intersections when the dimension is large enough w.r.t. the degree. As a reward we get a number of surprising enumerative formulas relating lines, conics and twisted cubics on X.