In 1973 V.L.Popov classified affine SL(2)-embeddings. He proved that a locally transitive SL(2)-action on a normal affine three-dimensional variety X is uniquely determined by a pair (p q , r), where 0 < p q ≤ 1 is an uncancelled fraction and r is a positive integer. Here r is the order of the stabilizer of a generic point. In this paper we show that the variety X is toric, i.e. admits a locall...

The zeta function of this mirror pair of families over finite fields have been studied in great detail by Dwork [6] for n = 2, 3 and more recently by Candelas, de la Ossa and Rodriguez Villegas [8] for n = 4. The zeta function and its roots are closely related to the solutions of the Picard-Fuch differential equations. In this course, we study the zeta function of the family Xλ for all n and it...

We prove a conjecture of Artur Elezi [4] in a generalized form suggested by Givental [5]. Namely, our main result relates genus-0 Gromov–Witten invariants of a bundle space with such invariants of the base, provided that the fiber is a toric manifold. When the base is the point, a new proof of mirror theorems by A. Givental [6] and H. Iritani [9] for toric manifolds is obtained. 1 Formulations ...

The Crepant Resolution Conjecture of Ruan and Bryan–Graber asserts that certain generating functions for genus-zero Gromov–Witten invariants of an orbifold X can be obtained from their counterparts for a crepant resolution of X by analytic continuation followed by specialization of parameters. In this paper we use mirror symmetry to determine the relationship between the genus-zero Gromov–Witte...

We introduce an integral structure in orbifold quantum cohomology associated to the K-group and the b Γ-class. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the Landau-Ginzburg model under mirror symmetry. By assuming the existence of an integral structure, we give a natural explanation for the specialization to a ro...