Gromov-Witten invariants of a class of toric varieties.

*(English)*Zbl 1085.14519Toric varieties often provide an excellent class of examples in the study of complicated questions in algebraic geometry. On the one hand, due to their correspondence to fans, many questions can be reduced to completely combinatorial properties, on the other hand, they represent an ample cross-section of all varieties. These nice properties can be also effectively exploited for the computation of Gromov-Witten invariants.

Let \(X\) be a nonsingular complex toric variety. When, in addition, \(X\) is Fano (i.e. the set of generators of the fan is convex), one can identify \(QH^*(X)\simeq\mathbb{Q} [C]\otimes_{\mathbb{Q}} H^*(X,\mathbb{Q})\) as \(\mathbb{Q} [C]\)-modules, where \(C\) denotes the semigroup of effective curve classes on \(X\). V.V. Batyrev [Asterisque 218, 9–34 (1993; Zbl 0806.14041)] described the structure of \(QH^*(X)\) for toric manifolds in terms of generators and relations; to fully understand the quantum product, one has to find an expression for \(QH^*(X)\) for any cocycle from \(H^*(X,\mathbb{Q})\) (a so-called quantum Giambelli formula). Both ingredients would yield all structure constants of the quantum multiplication in \(X\), i.e., its three-point Gromov-Witten invariants.

In the paper under review, the author proves a quantum Giambelli formula for smooth projective toric varieties \(X\) with the extra condition that the variety itself, all its toric subvarieties, and all nonsingular toric varieties \(X'\) such that \(X\) is the blow-up of \(X'\) along an irreducible toric subvariety, are Fano. In the first part, these class is described in combinatorial terms, and it it shown that such a variety is always an iterated blow-up of a product of projective spaces. The second part gives the proof of the quantum Giambelli formula in this situation. It is based on Batyrev’s representation of \(QH^*(X)\) and and doesn’t need the construction of a virtual fundamental class.

Let \(X\) be a nonsingular complex toric variety. When, in addition, \(X\) is Fano (i.e. the set of generators of the fan is convex), one can identify \(QH^*(X)\simeq\mathbb{Q} [C]\otimes_{\mathbb{Q}} H^*(X,\mathbb{Q})\) as \(\mathbb{Q} [C]\)-modules, where \(C\) denotes the semigroup of effective curve classes on \(X\). V.V. Batyrev [Asterisque 218, 9–34 (1993; Zbl 0806.14041)] described the structure of \(QH^*(X)\) for toric manifolds in terms of generators and relations; to fully understand the quantum product, one has to find an expression for \(QH^*(X)\) for any cocycle from \(H^*(X,\mathbb{Q})\) (a so-called quantum Giambelli formula). Both ingredients would yield all structure constants of the quantum multiplication in \(X\), i.e., its three-point Gromov-Witten invariants.

In the paper under review, the author proves a quantum Giambelli formula for smooth projective toric varieties \(X\) with the extra condition that the variety itself, all its toric subvarieties, and all nonsingular toric varieties \(X'\) such that \(X\) is the blow-up of \(X'\) along an irreducible toric subvariety, are Fano. In the first part, these class is described in combinatorial terms, and it it shown that such a variety is always an iterated blow-up of a product of projective spaces. The second part gives the proof of the quantum Giambelli formula in this situation. It is based on Batyrev’s representation of \(QH^*(X)\) and and doesn’t need the construction of a virtual fundamental class.

Reviewer: Olaf Teschke (Berlin)

##### MSC:

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

14J45 | Fano varieties |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |