Research on Periods of Algebraic Varieties and Hypergeometric Functions

During the period of the project，we have investigated the following subjects and obtained the following results。(I)Mirror Symmetry Conjecture for Calabi-Yau manifolds，(Ii)Counting Curves of higher genus in Rational Elliptic Surfaces，(Iii)Lie theoretic aspects on Painleve equations，Algebro-geometric aspects on Painleve equations，GKZ hypergeometric systems and their Grobner deformations。As for(I)，we have been studying Gromov-Witteninvariants for certain Calabi-Yau3-folds and computed a part of A-model prepotentials by means of the theta function of the E I D28锡D2-lattice while their B-model prepotential had already calculated by GKZ hypergometric series.As a result，we have checked MSC mathematically for those cases。Developing further，in(Ii)we have investigated counting problems of curves of higher genus in rational elliptic surfaces.We propose the holomorphic anomaly@equation(HAE)，which the prepotential should satisfy。By using the Jacobi‘s triple product formula and the relative Lefschetz decomposition，we have checked the prepotential satisfies the HAE.As for the studies of Painleve equations((Iii))，our group have been developing Lie theoretic approach and algeblo-geometric approach，which clarify the relations among Painleve equations，the symmetry of Affine Weyl groups and the geometry of rational surfaces.

During the period of the project, we have investigated the following subjects and obtained the following results. (i)Mirror Symmetry Conjecture for Calabi-Yau manifolds, (ii)Counting Curves of higher genus in Rational Elliptic Surfaces, (iii)Lie theoretic aspects on Painleve equations, Algebro-geometric aspects on Painleve equations, GKZ hypergeometric systems and their Grobner deformations. As for (i), we have been studying Gromov-Witteninvariants for certain Calabi-Yau 3-folds and computed a part of A-model prepotentials by means of the theta function of the EィイD28ィエD2-lattice while their B-model prepotential had already calculated by GKZ hypergometric series. As a result, we have checked MSC mathematically for those cases. Developing further, in (ii)we have investigated counting problems of curves of higher genus in rational elliptic surfaces. We propose the holomorphic anomaly equation(HAE), which the prepotential should satisfy. By using the Jacobi's triple product formula and the relative Lefschetz decomposition, we have checked the prepotential satisfies the HAE.As for the studies of Painleve equations ((iii)), our group have been developing Lie theoretic approach and algeblo-geometric approach, which clarify the relations among Painleve equations, the symmetry of Affine Weyl groups and the geometry of rational surfaces.