Problems on varieties

1.从代数几何、算术几何和复解析几何的角度出发，得到了复曲面簇、Fano簇、正特征椭圆曲面和对数格式的交上同调等重要结果.从数论和算术几何的观点出发，对代数簇上的有理点，特别是代数曲面，以及p-adic解析曲面，得到了新的结果.从代数分析、表示论和数学物理的观点出发，在代数簇、p-adic对称空间、非平衡模型、量子群表示和共形场论等方面对等变完整系统的研究取得了进展.从整体分析、微分几何和复几何的观点出发，我们在调和映射、调和叶理、电磁场的几何和量子遍历性、爱因斯坦-厄米特度规、箭图以及自对偶度规等方面取得了新的结果。我们能够获得这些结果，要归功于通过该基金可以方便地访问有关品种的大量结果的数据库。不同观点的专家自由地聚集在一起，提供个人见解和想法，以建立综合结果。我们试图将我们的结果传达给世界各地的相关专家。

1. From the viewpoints of algebraic geometry, arithmetic geometry and complex analytic geometry, various important results were obtained on the intersection cohomology of toric varieties, Fano varieties, elliptic surfaces in positive characteristics and logarithmic schemes.2. From the viewpoints of number theory and arithmetic geometry, new results were obtaied on rational points on algebraic varieties, algebraic surfaces in particular, as well as p-adic analytic surfaces.3. From the viewpoints of algebraic analysis, representation theory and mathematical physics, progress was made on equivariant holonomic systems on alegbraic varieties, p-adic symmetric spaces, nonequilibrium models, representations of quantum groups and conformal field theory.4. From the viewpoints of global analysis, differential geometry and complex geometry, new results were obtained on harmonic maps, harmonic foliations, the geometry and quantum ergodicity of eletro-magnetic fields, Einstein-Hermitian metrics, quivers as well as self-dual metrics.We were able to obtain these results thanks to easy access to databases on voluminous results on varieties made possible through this grant. Specialists on various different viewpoints freely got together, provided individual insights and and ideas to build up comprehensive results. We tried to communicate our results to relevant experts throughout the world.