Gopakumar-Vafa BPS invariants, Hilbert schemes and quasimodular forms. I

We prove a closed formula for leading Gopakumar- Vafa BPS invariants of local Calabi-Yau geometries given by the canonical line bundles of toric Fano surfaces. It shares some sim- ilar features with Gottsche-Yau-Zaslow formula: Connection with Hilbert schemes, connection with quasimodular forms, and qua- dratic property after suitable transformation. In Part I of this paper we will present the case of projective plane, more general cases will be presented in Part II. The problem of counting curves in algebraic varieties dates back to the nineteenth century. Classical examples include the famous 27 lines on a cubic surface and the 2875 lines on a quintic 3-fold. Through the interaction with string theory, spectacular progresses in this classi- cal branch of algebraic geometry have been made over the years since 1990's. First of all, Gromov-Witten theory and its various variants have laid the foundation for the modern treatment of many classical enumerative problems and greatly expanded the range of enumerative problems being considered. See (PT) for an introduction to some of the exciting developments. The study of the problems of counting curves in surfaces and Calabi- Yau 3-folds have served both as motivations and applications of Gromov- Witten theory. We will recall some results in these directions in §2 and §3 to serve as motivations for this work. We emphasize on the following three salient features for the curving counting problems for algebraic surfaces: connection with Hilbert schemes, connection with quasimod- ular forms, and quadratic properties of the node polynomials after suit- able transformation. Our main results will show that these features are also shared in the curve counting problems of some noncompact Calabi- Yau 3-folds, arising as the the total space of the canonical line bundles of toric Fano surfaces. 1