Counting generic genus-\(0\) curves on Hirzebruch surfaces (Q2781281)

In quantum cohomology theory for general symplectic manifolds, the so-called virtual fundamental class construction is known to play an essential rôle in defining (and computing) the Gromov-Witten invariants of the underlying symplectic manifold. However, in concrete applications of the general theory, it is often quite hard to figure out to what extent the contributions coming from nodal pseudo-holomorphic curves affect the various Gromov-Witten invariants. On the other hand, as the Gromov-Witten invariants appear as structure constants of the according quantum cohomology ring of the underlying symplectic manifold, it is very important to compute them properly. The paper under review aims at illustrating this crucial problem by means of a suitable class of concrete examples. The author shows that the classical Hirzebruch surfaces \(F_k\), \(k\in\mathbb{N}\), provide striking examples of symplectic manifolds, whose Gromov-Witten invariants might well count (nodal) curves in the boundary components of the involved moduli spaces of stable curves. More precisely, it is explained in great detail how to obtain the Gromov-Witten invariants and the quantum cohomology ring of all Hirzebruch surfaces \(F_k= \mathbb{P}_{\mathbb{C} \mathbb{P}^1}({\mathcal O}(k)\oplus {\mathcal O})\), together with the contributions from nodal curves. NEWLINENEWLINENEWLINEThe result is then compared to V. V. Batyrev's ad hoc construction of a possible quantum cohomology ring for toric manifolds, which was carried out in 1993 [\textit{V. V. Batyrev}, Astérisque, 218, 9-34 (1993; Zbl 0806.14041)], however without taking into account the contributions from nodal curves. This comparison reveals the substantial difference between the two approaches and demonstrates, in particular, why Batyrev's special counting argument for toric varieties does not work in the general framework. In other words, the author shows that the two approaches yield different quantum cohomology rings for the class of Hirzebruch surfaces, for instance.