Moduli Spaces and Special Functions

研究者松本启二利用代数曲线的Prym簇构造了周期映射和由某些代数簇族的周期映射的逆导出的自守形式。事实上，证明了光滑三次曲面族的周期映射可以用亏格为10的曲线的Prym簇的周期来表示。利用Prym簇的theta常数表示了四维复球上的自守形式，并利用这些曲线的Prym簇构造了复射影线的4重覆盖族到5维复球的周期映射. 5维复球上的自守形式给出了这个周期映射的逆，用与Prym簇相关的θ常数表示.斋藤睦美证明了仿射调簇上的微分算子环与A-超几何微分方程系统的对称代数是反同构的，并在这些对称下对A-超几何微分方程系统进行了组合分类.他研究了分次环gr（D（R_A））是可分次生成的条件，给出了作为D（R_A）-模的任意音调簇上的函数环R_A的合成因子.岛田一郎证明了：如果代数纤维空间中每个奇异纤维的奇异性都不坏，可以构造出基空间的第二同伦群到任意一般纤维的基本群的边界同态。他表明，基本组的补充结果超曲面是交换的。他还表明，任何supersingular K3表面可以表示为一个分支的双重覆盖的射影平面。

The head of investigator MATSUMOTO Keiji constructed period maps and automorphic forms derived from the inverses of period maps for certain families of algebraic varieties by using Prym varieties of algebraic curves.In fact, it was shown that the period map for the family of smooth cubic surfaces could be expressed in terms of periods of the Prym varieties for curves of genus 10. Automorphic forms on the 4-dimensional complex ball giving the inverse of this period map were expressed by theta constants associated to the Prym varieties.For the family of the 4-fold coverings of the complex projective line branching at eight points, the period map from this family to the 5-dimensional complex ball was constructed by using the Prym varieties of these curves. Automorphic forms on the 5-dimensional complex ball giving the inverse of this period map were expressed by theta constants associated to the Prym varieties.SAITO Mutsumi showed that the ring of differential operators on affine tone varieties and the algebra of symmetries of the system of A-hypergeometric differential equations were anti-isomorphic, and classified systems of A-hypergeometric differential equations combinatonally under these symmetries. He studied the condition that the graded ring gr(D(R_A)) was finitely generated, and gave the composition factors of the ring R_A of functions on any tone variety as a D(R_A)-module.SHIMADA Ichiro showed that if the singularity of each singular fiber was not bad for an algebraic fiber space, the boundary homomorphism from the second homotopy group of the base space to the fundamental group of any general fiber could be constructed. He showed that the fundamental group of the complement of a resultant hypersurface was commutative. He also showed that any supersingular K3 surface could be expressed as a branched double cover of the projective plane.